Sticky-reflected stochastic heat equation driven by colored noise
UDC 519.21 We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval $[0,1]$ driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usu...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6282 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512320814317568 |
|---|---|
| author | Konarovskyi, V. Konarovskyi, V. Konarovskyi, V. |
| author_facet | Konarovskyi, V. Konarovskyi, V. Konarovskyi, V. |
| author_sort | Konarovskyi, V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:02:08Z |
| description | UDC 519.21
We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval $[0,1]$ driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients. |
| doi_str_mv | 10.37863/umzh.v72i9.6282 |
| first_indexed | 2026-03-24T03:26:55Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i9.6282
UDC 519.21
V. Konarovskyi (Univ. Leipzig, Germany and Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
STICKY-REFLECTED STOCHASTIC HEAT EQUATION
DRIVEN BY COLORED NOISE
СТОХАСТИЧНЕ РIВНЯННЯ ТЕПЛОПРОВIДНОСТI
З ЛИПКИМ ВIДБИТТЯМ, КЕРОВАНЕ КОЛЬОРОВИМ ШУМОМ
We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval [0, 1] driven by colored
noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real
line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution
has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to
other types of SPDEs with discontinuous coefficients.
Доведено iснування розв’язку стохастичного рiвняння теплопровiдностi на вiдрiзку [0, 1] з липким вiдбиттям, керо-
ваного кольоровим шумом. Даний процес може бути iнтерпретований як нескiнченновимiрний аналог броунiвського
руху на дiйснiй прямiй iз липким вiдбиттям, але тепер розв’язок пiдпорядковується звичайному стохастичному рiв-
нянню теплопровiдностi за винятком точок, в яких вiн досягає нуля. В нулi шум не впливає на розв’язок, а перенос
штовхає його так, щоб розв’язок залишався додатним. Доведення ґрунтується на новому пiдходi, який може бу-
ти застосований до iнших типiв стохастичних диференцiальних рiвнянь з частинними похiдними iз розривними
коефiцiєнтами.
1. Introduction. In this paper, we study the existence of a continuous function X : [0, 1]\times [0,\infty ) \rightarrow
\rightarrow [0,\infty ) that is a weak solution to the SPDE
\partial Xt
\partial t
=
1
2
\partial 2Xt
\partial u2
+ \lambda \BbbI \{ Xt=0\} + f(Xt) + \BbbI \{ Xt>0\} Q \.Wt (1.1)
with Neumann
X \prime
t(0) = X \prime
t(1) = 0, t \geq 0, (1.2)
or Dirichlet
Xt(0) = Xt(1) = 0, t \geq 0, (1.3)
boundary conditions and the initial condition
X0(u) = g(u), u \in [0, 1], (1.4)
where \.W is a space-time white noise, the functions g \in \mathrm{C}[0, 1] and \lambda \in \mathrm{L}2 := \mathrm{L}2[0, 1] are non-
negative, f is a continuous function from [0,\infty ) to [0,\infty ) which has a linear growth and f(0) = 0,
and Q is a nonnegative definite self-adjoint Hilbert – Schmidt operator on \mathrm{L}2. We will also assume
that in the case of the Dirichlet boundary conditions g(0) = g(1) = 0.
The equation appears as a sticky-reflected counterpart of the reflected SPDE introduced in [14,
22]. We assume that a solution obeys the stochastic heat equation being strictly positive, but reaching
zero, its diffusion vanishes and an additional drift at zero pushes the process to be positive. The form
of equation (1.1) is similar to the form of the SDE for a sticky-reflected Brownian motion on the real
c\bigcirc V. KONAROVSKYI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9 1195
1196 V. KONAROVSKYI
line
dx(t) = \lambda \BbbI \{ x(t)=0\} dt+ \BbbI \{ x(t)>0\} \sigma dw(t), (1.5)
and we expect that the local behavior of X at zero is very similar to the behavior of the sticky-
reflected Brownian motion x. Remark that SDE (1.5) admits only a weak unique solution because
of its discontinuous coefficients, see, e.g., [4, 7, 19]. The approaches which are applicable to sticky
processes in finite-dimensional spaces can not be used for solving of SPDE (1.1). For instance,
Engelbert and Peskir in [7] showed that equation (1.5) admits a weak (unique) solution, where their
approach was based on the time change for a reflected Brownian motion. This method is very
restrictive and can be applied only for the sticky dynamics in a one-dimensional state space. An
equation for sticky-reflected dynamics for higher (finite) dimensions was considered by Grothaus and
co-authors in [8, 12, 13], where they used the Dirichlet form approach [10, 21]. This approach was
based on a priori knowledge of the invariant measure. Since the space is infinite-dimensional in our
case, finding of the invariant measure seems a very complicated problem (see, e.g., [9, 24] for the
form of invariant measure for the reflected stochastic heat equation driven by the white noise).
In this paper, we propose a new method for the proof of existence of weak solutions to equations
describing sticky-reflected behavior. This approach is a modification of the method proposed by the
author in [18], and is based on a property of quadratic variation of semimartingales.
The paper leaves a couple of important open problems. The first problem is the uniqueness of a
solution to SPDE (1.1) – (1.4). Similarly to the one-dimensional SDE for sticky-reflected Brownian
motion (1.5), where the strong uniqueness is failed [4, 7], we do not expect the strong uniqueness for
the SPDE considered here. However, we believe that the weak uniqueness holds.
Another interesting question is the existence of solutions to a similar sticky-reflected heat equation
driven by the white-noise. It seems that the method proposed here can be adapted to the case of such
an SPDE. For this we need a similar statement to Theorem 3.1, that remains an open problem.
1.1. Definition of solution and main result. For convenience of notation we introduce a
parameter \alpha 0 which equals to 1 in the case of the Neumann boundary conditions (1.2) and 0 in the
case of Dirichlet boundary conditions (1.3). Let us also introduce for k \geq 1 the space \mathrm{C}k[0, 1] of
k-times continuously differentiable functions on (0, 1) which together with their derivatives up to the
order k can be extended to continuous functions on [0, 1]. We will write \varphi \in \mathrm{C}k
\alpha 0
[0, 1] if additionally
\varphi (\alpha 0)(0) = \varphi (\alpha 0)(1) = 0, where \varphi (0) = \varphi and \varphi (1) = \varphi \prime .
Denote the inner product in the space \mathrm{L}2 by \langle \cdot , \cdot \rangle and the corresponding norm by \| \cdot \| . Let us
give a definition of a weak solution to SPDE (1.1).
Definition 1.1. We say that a continuous function X : [0, 1] \times [0,\infty ) \rightarrow [0,\infty ) is a (weak)
solution to SPDE (1.1) – (1.4), if X0 = g and for every \varphi \in \mathrm{C}2
\alpha 0
[0, 1] the process
\scrM \varphi
t = \langle Xt, \varphi \rangle - \langle X0, \varphi \rangle -
1
2
t\int
0
\langle Xs, \varphi
\prime \prime \rangle ds -
-
t\int
0
\langle \lambda \BbbI \{ Xs=0\} , \varphi \rangle ds -
t\int
0
\langle f(Xs), \varphi \rangle ds, t \geq 0,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1197
is an (\scrF X
t )-martingale with quadratic variation
[M\varphi ]t =
t\int
0
\bigm\| \bigm\| Q(\BbbI \{ Xs>0\} \varphi )
\bigm\| \bigm\| 2 ds, t \geq 0.
Hereinafter \{ ek, k \geq 1\} will denote the basis in \mathrm{L}2 consisting of eigenvectors of the nonnegative
definite self-adjoint operator Q. Let \{ \mu k, k \geq 1\} be the corresponding family of eigenvalues of Q.
We note that
\sum \infty
k=1
\mu 2k <\infty , since Q is a Hilbert – Schmidt operator. Introduce the function
\chi 2 :=
\infty \sum
k=1
\mu 2ke
2
k, (1.6)
where the series trivially converges in \mathrm{L}1[0, 1] and a.e. The main result of this paper is the following
theorem.
Theorem 1.1 (Existence of solutions). If
\lambda \BbbI \{ \chi >0\} = \lambda a.e., (1.7)
then SPDE (1.1) – (1.4) admits a weak solution.
Remark 1.1. Condition (1.7) means that the drift \lambda has to be equal to zero for those u for which
the noise vanishes.
Remark 1.2. The equation can admit a solution even if condition (1.7) does not hold. The reason
is that the existence can be failed if Xt(u) = 0 for u \in [0, 1] such that \lambda (u) > 0 and \chi (u) = 0
because of the term \lambda \BbbI \{ Xt=0\} and the absence of the noise for such u. However, if the initial
condition is strictly positive for such u, then the solution could stay always strictly positive for such
u, by the comparison principle for the classical heat equation. Therefore, the solution will exist.
Take, for example, Q = 0 and f = 0. Then a weak solution to the heat equation
\partial Xt
\partial t
=
1
2
\partial 2Xt
\partial u2
considered with corresponding boundary and initial conditions is a solution to SPDE (1.1) – (1.4) if
Xt(u) > 0, t > 0, u \in (0, 1). But the strong positivity of X is valid, e.g., under the assumption
the strong positivity of the initial condition. Hence, SPDE (1.1) – (1.4) has a weak solution even if
\lambda > 0 for, e.g., Q = 0, f = 0 and g > 0.
We will construct a solution to equation (1.1) as a limit of polygonal approximation similarly to
the approach done in [11]. The main difficulty here is that coefficients are discontinuous. So, we
cannot pass to the limit directly. In the next section, we will explain the key idea which allows to
overcome this difficulty.
1.2. Key idea of passing to the limit. We demonstrate our idea of passing to the limit in the case
of discontinuous coefficients using the equation for a sticky-reflected Brownian motion in \BbbR
dx(t) = \lambda \BbbI \{ x(t)=0\} dt+ \BbbI \{ x(t)>0\} \sigma dw(t), t \geq 0,
x(0) = x0,
(1.8)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1198 V. KONAROVSKYI
where w is a standard Brownian motion in \BbbR and \lambda , \sigma , x0 are positive constants. It is known that
this equation has only a unique weak solution (see, e.g., [7]).
Let us show that a solution to SDE (1.8) can be constructed as a weak limit of solutions to
equations with “good” coefficients. The first three steps proposed here are rather standard and the
last step shows how one can overcome the problem of the discontinuity of the coefficients.
Step I. Approximating sequence. Consider a nondecreasing continuously differentiable function
\kappa : \BbbR \rightarrow \BbbR such that \kappa (s) = 0, s \leq 0, and \kappa (s) = 1, s \geq 1. Denote \kappa \varepsilon (s) := \kappa
\Bigl( s
\varepsilon
\Bigr)
, s \in \BbbR , and
consider the SDE
dx\varepsilon (t) = \lambda
\bigl(
1 - \kappa 2\varepsilon (x\varepsilon (t))
\bigr)
dt+ \kappa \varepsilon (x\varepsilon (t))\sigma dw(t), t \geq 0,
x\varepsilon (0) = x0.
(1.9)
This SDE has a unique strong solution for every \varepsilon > 0.
Step II. Tightness in an appropriate space. Consider the processes
a\varepsilon (t) : = \lambda
t\int
0
\bigl(
1 - \kappa 2\varepsilon (x\varepsilon (s))
\bigr)
ds, t \geq 0,
\eta \varepsilon (t) : =
t\int
0
\sigma \kappa \varepsilon (x\varepsilon (s))dw(s), t \geq 0,
and
[\eta \varepsilon ]t =
t\int
0
\sigma 2\kappa 2\varepsilon (x\varepsilon (s))ds, t \geq 0,
where [\eta \varepsilon ] is the quadratic variation of the martingale \eta \varepsilon .
By the uniform boundedness of the coefficients of SDE (1.9), one can show that the family
\{ (x\varepsilon , a\varepsilon , \eta \varepsilon , [\eta \varepsilon ]), \varepsilon > 0\} is tight in (\mathrm{C}[0,\infty ))4 . By Prokhorov’s theorem, one can choose a sub-
sequence (xm, am, \eta m, [\eta m]) := (x\varepsilon m , a\varepsilon m , \eta \varepsilon m , [\eta \varepsilon m ]), m \geq 1, which converges to (x, a, \eta , \rho ) in
(\mathrm{C}[0,\infty ))4 in distribution as m \rightarrow \infty . By the Skorokhod representation theorem, we may assume
that (xm, am, \eta m, [\eta m]) \rightarrow (x, a, \eta , \rho ) a.s. as m\rightarrow \infty .
Step III. Properties of the limit process. One can see that for every T > 0 there exist a random
element \.\rho in \mathrm{L}2[0, T ] and a subsequence N such that
\sigma 2\kappa 2m(xm) \rightarrow \.\rho in the weak topology of \mathrm{L}2[0, T ] (1.10)
along N, for every t \in [0, T ],
x(t) = x0 + a(t) + \eta (t), \rho (t) =
t\int
0
\.\rho (s)ds, a(t) = \lambda
\biggl(
t - 1
\sigma 2
\rho (t)
\biggr)
, (1.11)
and \eta is a continuous square-integrable martingale with quadratic variation \rho . We may assume that
N = \BbbN .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1199
Step IV. Identification of quadratic variation and drift. Because of the discontinuity of the
coefficients of equation (1.8), we cannot conclude directly that \rho (t) =
\int t
0
\sigma 2\BbbI \{ x(s)>0\} ds and a(t) =
= \lambda
\int t
0
\BbbI \{ x(s)=0\} ds, t \in [0, T ], which would imply that x is a weak solution to SDE (1.8). We
propose to overcome this problem as follows. Let us use the following facts:
a) if x(t), t \geq 0, is a continuous nonnegative semimartingale with quadratic variation
[x]t =
t\int
0
\sigma 2(s)ds, t \geq 0,
then a.s.1
[x]t =
t\int
0
\sigma 2(s)\BbbI \{ x(s)>0\} ds, t \geq 0;
b) if sm \rightarrow s in \BbbR , then \kappa 2m(sm)\BbbI (0,+\infty )(s) \rightarrow \BbbI (0,+\infty )(s) in \BbbR as m\rightarrow \infty .
So, using (1.10), a), b) and the dominated convergence theorem, we get a.s.
\rho (t) =
t\int
0
\.\rho (s)ds =
t\int
0
\.\rho (s)\BbbI \{ x(s)>0\} ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
t\int
0
\sigma 2\kappa 2m(xm(s))\BbbI \{ x(s)>0\} ds =
t\int
0
\sigma 2\BbbI \{ x(s)>0\} ds, t \in [0, T ].
Hence, (1.11) implies
a(t) = \lambda
\biggl(
t - 1
\sigma 2
\rho (t)
\biggr)
= \lambda
t\int
0
\BbbI \{ x(s)=0\} ds, t \in [0, T ].
Consequently,
x(t) = x0 + \lambda
t\int
0
\BbbI \{ x(s)=0\} ds+ \eta (t), t \geq 0,
where \eta is a continuous square-integrable martingale with quadratic variation
[\eta ]t =
t\int
0
\sigma 2\BbbI \{ x(s)>0\} ds, t \geq 0,
that means that x is a weak solution to (1.8).
Content of the paper. To show the existence of a weak solution to SPDE (1.1) – (1.4), we will
follow the argument above. Step I will be done in Subsection 2.1. More precisely, we will construct
1See also Lemma A.1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1200 V. KONAROVSKYI
a family of processes which will approximate a solution to SPDE (1.1) – (1.4). The approximating
sequence is similar to one considered in [11]. Subsection 2.2 is devoted to the tightness that is
Step II of our argument. Step III is done in Subsection 3.1, where we show that the limit process
satisfies equalities similar to (1.11) (see Proposition 3.1 there). An analog of property a) above is
stated for some infinite-dimensional semimartingales in Theorem 3.1 in Subsection 3.2. The proof of
the existence theorem is given in Subsection 3.3, where we use the approach described in Step IV.
Auxiliary statements are proved in the appendix.
1.3. Preliminaries. We will denote the inner product and the corresponding norm in a Hilbert
space H by \langle \cdot , \cdot \rangle H and \| \cdot \| H , respectively.
For an essentially bounded function \psi \in L\infty we define the multiplication operator [\psi \cdot ] on \mathrm{L}2 as
follows:
([\psi \cdot ]h) (u) = \psi (u)h(u), u \in [0, 1], h \in \mathrm{L}2.
Let A be an operator on \mathrm{L}2 and \varphi 1, . . . , \varphi n \in \mathrm{L}2 such that the product \varphi 1 . . . \varphi n belongs to \mathrm{L}2.
To ease notation, we will always write A\varphi 1 . . . \varphi n for A (\varphi 1 . . . \varphi n) .
Denote the space of Hilbert – Schmidt operators on \mathrm{L}2 by \scrL 2. Remark that \scrL 2 furnished with the
inner product
\langle A,B\rangle \scrL 2
=
\infty \sum
k=1
\langle Aek, Bek\rangle , A,B \in \scrL 2,
is a Hilbert space, where the norm does not depend on the choice of basis in \mathrm{L}2. The family of
operators \{ ek \odot el, k, l \geq 1\} form a basis in \scrL 2. Here, for every \varphi ,\psi \in \mathrm{L}2, \varphi \odot \psi denotes the
operator on \mathrm{L}2 defined as (\varphi \odot \psi )g = \langle g, \psi \rangle \varphi , g \in \mathrm{L}2.
We will consider the set \BbbR n\times n of all (n \times n)-matrices with real enters as a Hilbert space with
the Hilbert – Schmidt inner product \langle A,B\rangle \BbbR n\times n =
\sum n
k,l=1
Ak,lBk,l.
The indicator function will be defined as usually
\BbbI S(x) =
\Biggl\{
1, if x \in S,
0, if x \not \in S.
If \phi : E1 \rightarrow E2 is a function and S is a subset of E2, then \BbbI \{ \phi \in S\} will denote the function
x \mapsto \rightarrow \BbbI S(\phi (x)) from E1 to E2.
Given a Hilbert space H, we write HT := \mathrm{L}2 ([0, T ], H) for the class of all Bochner integrable
functions \Phi : [0, T ] \rightarrow H with
\| \Phi \| H,T =
\left( T\int
0
\| \Phi s\| 2Hds
\right)
1
2
<\infty .
One can show that the space HT equipped with the inner product
\langle \Phi ,\Psi \rangle H,T =
T\int
0
\langle \Phi s,\Psi s\rangle Hds, \Phi ,\Psi \in HT ,
is a Hilbert space.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1201
Considering a sequence \{ Zn\} n\geq 1 in HT , we will say that Zn \rightarrow Z a.e. as n \rightarrow \infty , if
\mathrm{L}\mathrm{e}\mathrm{b}T \{ t \in [0, T ] : Zn
t \not \rightarrow Zt in H, n \rightarrow \infty \} = 0, where \mathrm{L}\mathrm{e}\mathrm{b}T denotes the Lebesgue measure on
[0, T ].
Let L \in \scrL T
2 , Z \in \mathrm{L}T
2 , and S be a Borel measurable subset of \BbbR . It is easily seen that
Lt
\bigl[
\BbbI \{ Zt\in S\} \cdot
\bigr]
, t \in [0, T ], where Lt
\bigl[
\BbbI \{ Zt\in S\} \cdot
\bigr]
is the composition of two operators, is well-defined
and belongs to L \in \scrL T
2 . We will denote such a function shortly by L\cdot
\bigl[
\BbbI \{ Z\cdot \in S\} \cdot
\bigr]
.
Let I be equal to [0, T ], [0,\infty ) or [0, T ]\times [0, 1]. The space of all continuous functions from I to
a Polish space E with the topology of uniform convergence on compact sets is denoted by \mathrm{C} (I, E) .
If I = [0, T ] or [0,\infty ), and E = \BbbR , then we simply write \mathrm{C}[0, T ] or \mathrm{C}[0,\infty ) instead of \mathrm{C} (I,\BbbR ) .
We will denote the right continuous complete filtration generated by continuous processes \xi 1(t),
t \in I, . . . , \xi n(t), t \in I, by (\scrF \xi 1,...,\xi n
t )t\in I . Remark that such a filtration exists by Lemma 7.8 [17].
2. Finite sticky reflected particle system. In this section, we construct a sequence of random
processes which will be used for the approximation of a solution to SPDE (1.1) – (1.4).
Let n \geq 1 be fixed. We set \pi nk = \BbbI [ k - 1
n
, k
n)
, k \in [n] := \{ 1, . . . , n\} . Let Wt, t \geq 0, be a
cylindrical Wiener process in \mathrm{L}2. Define the Wiener processes on \BbbR as follows:
wn
k (t) :=
\surd
n
t\int
0
\langle \pi nk , QdWs\rangle , t \geq 0, k \in [n],
and note that their joint quadratic variation is
[wn
k , w
n
l ]t = n\langle Q\pi nk , Q\pi nl \rangle t =: qnk,lt, t \geq 0.
Let also \lambda nk := n\langle \lambda , \pi nk \rangle \BbbI \{ qnk,k>0\}
2 and gnk := n\langle g, \pi nk \rangle , k \in [n].
Consider the SDE
dxnk(t) =
1
2
\Delta nxnk(t)dt+ \lambda nk\BbbI \{ xn
k (t)=0\} dt+
+f(xnk(t))dt+
\surd
n\BbbI \{ xn
k (t)>0\} dw
n
k (t), k \in [n], (2.1)
satisfying the initial condition
xnk(0) = gnk , k \in [n], (2.2)
where \Delta nxnk = (\Delta nxn)k = n2
\bigl(
xnk+1 + xnk - 1 - 2xnk
\bigr)
and
xn0 (t) = \alpha 0x
n
1 (t), xnn+1(t) = \alpha 0x
n
n(t), t \geq 0. (2.3)
We will construct a solution to SPDE (1.1) – (1.4) as a weak limit in \mathrm{C} ([0,\infty ),\mathrm{C}[0, 1]) of pro-
cesses
\~Xn
t (u) = (un - k + 1)xnk(t) + (k - nu)xnk - 1(t), t \in [0, T ], u \in \pi nk , k \in [n]. (2.4)
2We add the indicator \BbbI \Bigl\{
qn
k,k
>0
\Bigr\} into the definition of \lambda n
k because we need the additional condition that \lambda n
k = 0 if
qnk,k = 0 for the existence of solution to SDE (2.1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1202 V. KONAROVSKYI
Remark that equation (2.1) has discontinuous coefficients. So the classical theory of SDE cannot
be applied in our case. The existence of the solution will follow from Theorem 2.1 which we state
below.
2.1. SDE for sticky-reflected particle system. The aim of this section is to prove the existence of
solutions to (2.1), (2.2). We will formulate the problem in slightly general form. So, let n \in \BbbN and
gk, \lambda k, k \in [n], be nonnegative numbers. We also consider a family of Brownian motions wk(t),
t \geq 0, k \in [n] (with respect to the same filtration), with joint quadratic variation
[wk, wl]t = qk,lt, t \geq 0.
Let as before f : [0,\infty ) \rightarrow [0,\infty ) be a continuous function with linear growth and f(0) = 0.
Consider the SDE
dyk(t) =
1
2
\Delta nyk(t)dt+ \lambda k\BbbI \{ yk(t)=0\} dt+
+f(yk(t))dt+ \BbbI \{ yk(t)>0\} dwk(t), k \in [n], (2.5)
with initial condition
yk(0) = gk, k \in [n], (2.6)
and the boundary conditions
y0(t) = \alpha 0y1(t), yn+1(t) = \alpha 0yn(t), t \geq 0. (2.7)
Theorem 2.1. Let qk,k = 0 imply \lambda k = 0 for every k \in [n]. Then there exists a family of non-
negative (real-valued) continuous processes yk(t), t \geq 0, k \in [n], in \BbbR which is a weak (martingale)
solution to (2.1), (2.2), that is, yk(0) = gk, for each k \in [n],
\scrN k(t) := yk(t) - gk -
1
2
t\int
0
\Delta nyk(s)ds -
- \lambda k
t\int
0
\BbbI \{ yk(s)=0\} ds -
t\int
0
f(yk(s))ds, t \geq 0,
is an (\scrF y
t )-martingale, and the joint quadratic variation of \scrN k and \scrN l, k, l \in [n], equals
[\scrN k,\scrN l]t = qk,l
t\int
0
\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds, t \geq 0.
We are going to construct a solution to the SDE approximating the coefficients by Lipschitz
continuous functions and using the method described in Subsection 1.2.
Let us take a nondecreasing function \kappa \in \mathrm{C}1(\BbbR ) such that \kappa (x) = 0 for x \leq 0, and \kappa (x) = 1 for
x \geq 1. Let also \theta \in \mathrm{C}1(\BbbR ) be a nonnegative function with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \theta \in [ - 1, 1] and
\int +\infty
- \infty
\theta (x)dx = 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1203
For every \varepsilon > 0 we introduce the functions \kappa \varepsilon (x) = \kappa
\Bigl( x
\varepsilon
\Bigr)
, x \in \BbbR , and \theta \varepsilon (x) =
1
\varepsilon
\theta
\Bigl( x
\varepsilon
\Bigr)
, x \in \BbbR .
Setting f\varepsilon (x) =
\int +\infty
0
\theta \varepsilon (x - y)f(y)dy, x \in \BbbR , we consider the SDE
dy\varepsilon k(t) =
1
2
\Delta ny\varepsilon k(t)dt+ \lambda k
\bigl(
1 - \kappa 2\varepsilon (y
\varepsilon
k(t))
\bigr)
dt+
+f\varepsilon (y
\varepsilon
k(t))dt+ \kappa \varepsilon (y
\varepsilon
k(t))dwk(t), (2.8)
y\varepsilon k(0) = gk, k \in [n].
Since equation (2.1) has locally Lipschitz continuous coefficients with linear growth, it has a unique
strong solution.
Our goal is to to show that the sequence \{ y\varepsilon = (y\varepsilon k)
n
k=1\} \varepsilon >0 has a subsequence which converges
in distribution to a week solution to (2.1). We denote for every k \in [n]
a\varepsilon k(t) = \lambda k
t\int
0
\bigl(
1 - \kappa 2\varepsilon (y
\varepsilon
k(s))
\bigr)
ds, t \geq 0,
and
\eta \varepsilon k(t) =
t\int
0
\kappa \varepsilon (y
\varepsilon
k(s))dwk(s), t \geq 0.
Set a\varepsilon = (a\varepsilon k)
n
k=1 and \eta \varepsilon = (\eta \varepsilon k)
n
k=1.
The quadratic variation [\eta \varepsilon ]t , t \geq 0, of the \BbbR n-valued martingale \eta \varepsilon takes values in the space
of nonnegative definite (n\times n)-matrices with entries
[\eta \varepsilon k, \eta
\varepsilon
l ]t =
t\int
0
\sigma \varepsilon k,l(s)ds,
where \sigma \varepsilon k,l(s) = \kappa \varepsilon (y
\varepsilon
k(s))\kappa \varepsilon (y
\varepsilon
l (s))qk,l.
Remark 2.1. According to the choice of the approximating sequence for a, the equality
a\varepsilon k(t) = \lambda k
\biggl(
t - 1
qk,k
[\eta \varepsilon k]t
\biggr)
, t \geq 0,
holds for every k \in [n] satisfying qk,k > 0.
Consider the following metric space \scrW \BbbR n := (\mathrm{C} ([0,\infty ),\BbbR n))3 \times \mathrm{C} ([0,\infty ),\BbbR n\times n) .
Lemma 2.1. The family \{ (y\varepsilon m , a\varepsilon m , \eta \varepsilon m , [\eta \varepsilon m ]) , m \geq 1\} is tight in \scrW \BbbR n , where \varepsilon m, m \geq 1,
is any sequence convergent to zero.
Proof. In order to prove the statement, it is sufficient to show that each family of coordinate
processes of (y\varepsilon m , a\varepsilon m , \eta \varepsilon m , [\eta \varepsilon m ]) , m \geq 1, is tight in the corresponding space. We will only show
the tightness for \{ y\varepsilon m , m \geq 1\} . The tightness for other families can be obtained similarly.
According to the Aldous tightness criterion [2] (Theorem 1), it is enough to show that for every
T > 0, any family of stopping times \tau m, m \geq 1, bounded by T and any sequence \delta m decreasing to
zero
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1204 V. KONAROVSKYI
y\varepsilon m(\tau m + \delta m) - y\varepsilon m(\tau m) \rightarrow 0 in probability as m\rightarrow \infty ,
and \{ y\varepsilon m(t), m \geq 1\} is tight in \BbbR n for each t \in [0, T ].
The conditions above trivially follow from the convergence
\BbbE
\Bigl[
\| y\varepsilon m(\tau m + \delta m) - y\varepsilon m(\tau m)\| 2\BbbR n
\Bigr]
\rightarrow 0 as m\rightarrow \infty ,
and the uniform boundedness of \BbbE
\Bigl[
\| y\varepsilon m(t)\| 2\BbbR n
\Bigr]
in m \geq 1 for every t \in [0, T ].
Using the fact that there exists a constant C > 0 such that | f\varepsilon m(x)| \leq C(1+ | x| ), x \in \BbbR , m \geq 1,
the inequality
\langle y\varepsilon m(t),\Delta ny\varepsilon m(t)\rangle = -
n - 1\sum
k=1
(y\varepsilon mk+1(t) - y\varepsilon mk (t))2 -
- \alpha 0(y
\varepsilon m
1 (t) + y\varepsilon mn (t)) \leq 2\varepsilon m\alpha 0
for all t \in [0, T ], the Itô formula and Gronwall’s lemma, one can check that for every p \geq 1 there
exists a constant Cp,T,n, depending on p, T and n, such that
\BbbE
\Bigl[
\| y\varepsilon m(t)\| 2p\BbbR n
\Bigr]
\leq Cp,T,n, t \in [0, T ]. (2.9)
Next, by the Itô formula and the optional sampling Theorem 7.12 [17], we have
\BbbE
\bigl[
\| y\varepsilon m(\tau m + \delta m) - y\varepsilon m(\tau m)\| 2\BbbR n
\bigr]
\leq \BbbE
\left[ \tau m+\delta m\int
\tau m
\langle y\varepsilon m(r),\Delta ny\varepsilon m(r)\rangle \BbbR ndr
\right] +
+2\BbbE
\left[ \tau m+\delta m\int
\tau m
\langle y(r), \lambda
\bigl(
1 - \kappa 2\varepsilon m(y\cdot (r))
\bigr)
\rangle \BbbR ndr
\right] +
+2\BbbE
\left[ \tau m+\delta m\int
\tau m
\langle y\varepsilon m(r), f\varepsilon m(y\varepsilon m\cdot (r))\rangle \BbbR ndr
\right] +
+\BbbE
\left[ \tau m+\delta m\int
\tau m
n\sum
k=1
\kappa 2\varepsilon m(y
\varepsilon m
k (r))qk,kdr
\right] (2.10)
for all m \geq 1. Using Hölder’s inequality, and estimate (2.9) one can conclude that
\BbbE
\Bigl[
\| y\varepsilon m(\tau m + \delta m) - y\varepsilon m(\tau m)\| 2\BbbR n
\Bigr]
\rightarrow 0 as m\rightarrow \infty .
Lemma 2.1 is proved.
By Lemma 2.1 and Prokhorov’s theorem, there exists a sequence \{ \varepsilon m\} m\geq 1 converging to zero
such that the sequence \mathrm{y}\varepsilon m := (y\varepsilon m , a\varepsilon m , \eta \varepsilon m , [\eta \varepsilon m ]), m \geq 1, converges to a random element
\mathrm{y} := (y, a, \eta , \rho ) in \scrW \BbbR n in distriburion. By the Skorokhod representation Theorem 3.1.8 [6], one
can choose a probability space (\~\Omega , \~\scrF , \~\BbbP ) and determine there a family of random elements \~\mathrm{y}, \~\mathrm{y}\varepsilon m ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1205
m \geq 1, taking values in \scrW \BbbR n such that \mathrm{L}\mathrm{a}\mathrm{w} \~\mathrm{y} = \mathrm{L}\mathrm{a}\mathrm{w} \mathrm{y}, \mathrm{L}\mathrm{a}\mathrm{w} \~\mathrm{y}\varepsilon m = \mathrm{L}\mathrm{a}\mathrm{w} \mathrm{y}\varepsilon m , m \geq 1, and
\~\mathrm{y}\varepsilon m \rightarrow \mathrm{y} in \scrW \BbbR n a.s. So, without loss of generality, we will assume that \mathrm{y}\varepsilon m \rightarrow \mathrm{y} in \scrW \BbbR n a.s.
as m \rightarrow \infty . Since the sequence \{ \varepsilon m\} m\geq 1 will be fixed to the end of this section, we will write m
instead of \varepsilon m in order to simplify the notation.
Let y = (yk)
n
k=1, a = (ak)
n
k=1, \eta = (\eta k)
n
k=1, \rho = (\rho k,l)
n
k,l=1.
Lemma 2.2. (i) The coordinate processes yk(t), t \geq 0, k \in [n], of y are nonnegative and
yk(t) = gk +
1
2
t\int
0
\Delta nyk(s)ds+ ak(t) +
t\int
0
f(yk(s))ds+ \eta k(t), t \geq 0, k \in [n].
(ii) For every k \in [n] such that qk,k > 0 one has
ak = \lambda k
\biggl(
t - 1
qk,k
\rho k,k
\biggr)
.
(iii) For every k \in [n] and T > 0 there exists a random element \.ak in \mathrm{L}2([0, T ],\BbbR ) such that
a.s.
ak(t) =
t\int
0
\.ak(s)ds, t \in [0, T ].
(iv) For every k, l \in [n] and T > 0 there exists a random element \.\rho k,l in \mathrm{L}2 ([0, T ],\BbbR ) such
that a.s.
\rho k,l(t) =
t\int
0
\.\rho k,l(s)ds, t \in [0, T ].
(v) For every k \in [n] the process \eta k(t), t \geq 0, is a continuous square-integrable (\scrF \eta
t )-
martingale, and the joint quadratic variation of \eta k and \eta l, k, l \in [n], equals
[\eta k, \eta l]t = \rho k,l(t), t \geq 0.
Proof. We remark that for every k \in [n]
\BbbP [\forall t \geq 0 fm(ymk (t)) \rightarrow f(yk(t)) as m\rightarrow \infty ] = 1,
and for every m \geq 1 and k \in [n] a.s.
ymk (t) = gk +
1
2
t\int
0
\Delta nymk (s)ds+ amk (t) +
t\int
0
fm(ymk (s))ds+ \eta mk (t), t \geq 0.
Passing to the limit and using the dominated convergence theorem, we obtain the equality (i).
The equality in (ii) follows from Remark 2.1 and the convergence in distribution of (amk , [\eta
m
k ])
to (ak, \rho k,k) in (\mathrm{C} ([0,+\infty ),\BbbR ))2 .
We next prove (iii). Let T > 0 be fixed. Denote the ball in \mathrm{L}2 ([0, T ],\BbbR ) with center 0 and
radius r > 0 by BT
r and furnish it with the weak topology of the space \mathrm{L}2 ([0, T ],\BbbR ) , i.e., a
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1206 V. KONAROVSKYI
sequence \{ hm\} m\geq 1 converges to h in BT
r if \langle hm, b\rangle \BbbR ,T \rightarrow \langle h, b\rangle \BbbR ,T for all b \in BT
r . By Alaoglu’s
Theorem V.4.2 [5] and Theorem V.5.1 ibid, BT
r is a compact metric space.
We fix k \in [n] and take r := \lambda k
\surd
T ,
\.amk (t) := \lambda k
\bigl(
1 - \kappa 2m(ymk (t))
\bigr)
, t \in [0, T ].
Then \.amk is a random element in BT
r for every m \geq 1. By the compactness of BT
r , the family
\{ \.amk , m \geq 1\} is tight in BT
r . Consequently, Prokhorov’s theorem implies the existance of a subse-
quence N \subset \BbbN such that \.amk \rightarrow \~ak in BT
r in distribution along N. In particular, for every family
t1, . . . , tl \in [0, T ] and numbers c1, . . . , cl \in \BbbR ,
l\sum
i=1
ci
ti\int
0
\.amk (s)ds =
T\int
0
\Biggl(
l\sum
i=1
ci\BbbI [0,ti](s)
\Biggr)
\.amk (s)ds\rightarrow
\rightarrow
T\int
0
\Biggl(
l\sum
i=1
ci\BbbI [0,ti](s)
\Biggr)
\~ak(s)ds =
l\sum
i=1
ci
ti\int
0
\~ak(s)ds
in \BbbR in distribution along N. Since the family of functions\Biggl\{
x \mapsto \rightarrow h
\Biggl(
l\sum
l=1
cixi
\Biggr)
, x = (xi)
l
i=1 \in \BbbR l : h is continuous and bounded on \BbbR
\Biggr\}
strongly separates points3, Theorem 3.4.5 [6] yields that\left( t1\int
0
\.amk (s)ds, . . . ,
tl\int
0
\.amk (s)ds
\right) \rightarrow
\left( t1\int
0
\~ak(s)ds, . . . ,
tl\int
0
\~ak(s)ds
\right)
in \BbbR l in distribution along N. From the other hand side,\left( t1\int
0
\.amk (s)ds, . . . ,
tl\int
0
\.amk (s)ds
\right) \rightarrow (ak(t1), . . . , ak(tl))
in \BbbR l a.s. along N. This implies that
\mathrm{L}\mathrm{a}\mathrm{w} ak = \mathrm{L}\mathrm{a}\mathrm{w}
\cdot \int
0
\~ak(s)ds. (2.11)
Let us show that there exists a random element \.ak in \mathrm{L}2 ([0, T ],\BbbR ) such that ak =
\int \cdot
0
\.ak(s)ds
a.s. We define the map \Phi : \mathrm{L}2 ([0, T ],\BbbR ) \rightarrow \mathrm{C}[0, T ]
\Phi (h)(t) =
t\int
0
h(s)ds, t \in [0, T ].
3See the definition in [6, p. 113].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1207
Remark that \Phi is a bijective map from \mathrm{L}2 ([0, T ],\BbbR ) to its image
\mathrm{I}\mathrm{m}\Phi = \{ \Phi (h) : h \in \mathrm{L}2 ([0, T ],\BbbR )\} .
By the Kuratowski Theorem A.10.5 [6], the set \mathrm{I}\mathrm{m}\Phi is Borel measurable in \mathrm{C}[0, T ] and the map
\Phi - 1 is Borel measurable. By (2.11), ak \in \mathrm{I}\mathrm{m}\Phi a.s. Thus, we can define \.ak = \Phi - 1(ak). This
completes the proof of (iii).
Similarly, one can prove (iv).
Statement (v) follows from the fact that the limit of local martingales is a local martingale and
the uniform boundedness of \BbbE
\Bigl[
(\eta mk (t))2
\Bigr]
in m. Indeed, for every k, l \in [n] the processes \eta mk
and \eta mk \eta
m
l - [\eta mk , \eta
m
l ] are (\scrF \eta m
t )-martingales for all m \geq 1, and (\eta m, [\eta m], \eta mk \eta
m
l - [\eta mk , \eta
m
l ]) \rightarrow
\rightarrow (\eta , \rho , \eta k\eta l - \rho k,l) a.s. as m \rightarrow \infty . Proposition IX.1.17 [16] implies that \eta k and \eta k\eta l - \rho k,l are
(\scrF (\eta ,\rho )
t )-local martingales. Note that, by the Fisk approximation Theorem 17.17 [17], \scrF (\eta ,\rho )
t = \scrF \eta
t ,
t \geq 0. Using the uniform boundedness of \BbbE
\Bigl[
(\eta mk (t))2
\Bigr]
in m and Fatou’s lemma, one can see that
\eta k is a square-integrable (\scrF \eta
t )-martingale.
Lemma 2.2 is proved.
Proposition 2.1. Let \mathrm{y}(t) = (y(t), a(t), \eta (t), \rho (t)), t \geq 0, be as in Lemma 2.2. Let additionally
\lambda k = 0 if qk,k = 0, k \in [n]. Then
1) for every k, l \in [n] a.s.
\rho k,l(t) = qk,l
t\int
0
\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds, t \geq 0;
2) for every k \in [n] a.s.
ak(t) = \lambda k
t\int
0
\BbbI \{ yk(s)=0\} ds, t \geq 0.
Proof. We take the sequence \{ ymn \} n\geq 1 as in the proof of Lemma 2.2. Again without loss of
generality we may assume that it converges to \mathrm{y} a.s. We first show that a.s.
\rho k,l(t) = qk,l
t\int
0
\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds, t \geq 0.
Recall that a.s.
[\eta mk , \eta
m
l ]t =
t\int
0
\sigma mk,l(s)ds, t \geq 0,
where \sigma mk,l(s) = qk,l\kappa m(ymk (s))\kappa m(yml (s)), and for each T > 0, k, l \in [n] there exist random
elements \.\rho k,l in \mathrm{L}2 ([0, T ],\BbbR ) such that a.s.
\rho k,l(t) =
t\int
0
\.\rho k,l(s)ds, t \in [0, T ],
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1208 V. KONAROVSKYI
by Lemma 2.2.
Let T > 0, k, l \in [n] be fixed. By the convergence of the sequence [\eta mk , \eta
m
l ], m \geq 1, to
\rho k,l in \mathrm{C}[0, T ] a.s., the uniform boundedness of \sigma mk,l, and the density of \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\bigl\{
\BbbI [0,t], t \in [0, T ]
\bigr\}
in
\mathrm{L}2 ([0, T ],\BbbR ), one has that
\BbbP
\bigl[
\sigma mk,l \rightarrow \.\rho k,l in the weak topology of \mathrm{L}2 ([0, T ],\BbbR ) as m\rightarrow \infty
\bigr]
= 1. (2.12)
By Lemma 2.2, yk and yl are nonnegative continuous semimartingales with quadratic variation
[yk, yl]t = \rho k,l(t) =
t\int
0
\.\rho k,l(s)ds, t \in [0, T ]. Thus, Lemma A.1 implies that a.s.
t\int
0
\.\rho k,l(s)ds =
t\int
0
\.\rho k,l(s)\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds, t \in [0, T ].
The latter equality and (2.12) yield that for every t \in [0, T ] a.s.
\rho k,l(t) =
t\int
0
\.\rho k,l(s)ds =
t\int
0
\.\rho k,l(s)\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
t\int
0
\sigma mk,l\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
t\int
0
qk,l\kappa m(ymk (s))\kappa m(yml (s))\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds =
=
t\int
0
qk,l\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds,
where we have used the convergence \kappa m(xm)\BbbI (0,+\infty )(x) \rightarrow \BbbI (0,+\infty )(x) as xm \rightarrow x in \BbbR and the
dominated convergence theorem. Hence, a.s.
\rho k,l(t) =
t\int
0
qk,l\BbbI \{ yk(s)>0\} \BbbI \{ yl(s)>0\} ds, t \geq 0,
and, consequently, according to Lemma 2.2 (ii), a.s.
ak(t) = \lambda k
\biggl(
1 - 1
qk,k
\rho k,k(t)
\biggr)
= \lambda k
t\int
0
\BbbI \{ yk(s)=0\} ds, t \geq 0,
for all k \in [n] such that qk,k \not = 0. If qk,k = 0, then \lambda k = 0 by the assumption of Proposition 2.1.
Therefore, amk = 0 implies that ak = 0.
Proposition 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1209
Proof of Theorem 2.1. The statement of the theorem directly follows from Lemma 2.2 and
Proposition 2.1.
2.2. Tightness. Let a family of nonnegative continuous processes \{ xnk(t), t \geq 0, k \in [n]\} be
a weak solution to SDE (2.1) – (2.3), which exists according to Theorem 2.1. Let the continuous
process \~Xn
t , t \geq 0, taking values in \mathrm{C}[0, 1] be defined by (2.4). We note that \~Xn
t (u) \geq 0 for all
u \in [0, 1], t \geq 0 and n \geq 1.
The aim of this section is to prove the tightness of the family
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
in \mathrm{C} ([0,\infty ),\mathrm{C}[0, 1]) .
The similar problem was considered in [11] (Section 2), where the author study the existence of
solutions to an SPDE with Lipschitz continuous coefficients. The tightness argument there is based
on properties of fundamental solution to the discrete analog of the heat equation and the fact that
coefficients of the equation have at most linear growth. The Lipschitz continuity was not needed for
the proof of the tightness. Since the proof in our case repeats the proof from [11], we will point out
only its main steps. The main statement of this sections reads as follows.
Proposition 2.2. The family of processes
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
is tight in \mathrm{C} ([0,\infty ),\mathrm{C}[0, 1]) .
For the proof of the proposition it is enough to show that the family
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
is tight in
\mathrm{C} ([0, T ],\mathrm{C}[0, 1]) = \mathrm{C} ([0, T ]\times [0, 1],\BbbR ) for every T > 0. So, we fix T > 0, and use Corol-
lary 16.9 [17] which yields the tightness if
\Bigl\{
\~Xn, n \geq 0
\Bigr\}
satisfies the following conditions:
1)
\Bigl\{
\~Xn
0 (0), n \geq 1
\Bigr\}
is tight in \BbbR ;
2) there exist constants \alpha , \beta , C > 0 such that
\BbbE
\Bigl[
| \~Xn
t (u) - \~Xn
s (v)| \alpha
\Bigr]
\leq C
\Bigl(
| t - s| 2+\beta + | u - v| 2+\beta
\Bigr)
for all t, s \in [0, T ], u, v \in [0, 1], and n \geq 1.
The family
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
trivially satisfies the first condition because \~Xn
0 (0) = gn1 is uniformly
bounded in n \geq 1. In order to check the second condition, we first write equation (2.1) in the
integral form. Let \{ pnk,l(t), t \geq 0, k, l \in [n]\} be the fundamental solution of the system of ordinary
differential equations4
d
dt
pnk,l(t) =
1
2
\Delta n
(k)p
n
k,l(t), t > 0, k, l \in [n],
with the initial condition
pnk,l(0) = n\BbbI \{ k=l\} , k, l \in [n],
and the boundary conditions
pn0,l(t) = \alpha 0p
n
1,l(t), pnn+1,l(t) = \alpha 0p
n
n,l(t), t \geq 0, l \in [n],
where the operator \Delta n
(k) = \Delta n is applied to the vector (pk,l(t))
n
k=1 for every l \in [n]. Noting that
\{ \langle Wt,
\surd
n\pi nk \rangle , t \geq 0, k \in [n]\} is a family of standard Brownian motions, it is easily seen that \~Xn
has the same distribution as the solution to the integral equation
4For more details about properties of the fundamental solution to the discrete analog of the heat equation see, e.g., [11]
(Appendix II).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1210 V. KONAROVSKYI
\~Xn
t (u) =
1\int
0
pn(t, u, v)g(v)dv +
t\int
0
1\int
0
pn(t - s, u, v)\~\lambda n(v)\BbbI \{ \~Xn
s (\lceil v\rceil )=0\} dsdv+
+
t\int
0
1\int
0
pn(t - s, u, v)f
\Bigl(
\~Xn
s (\lceil v\rceil )
\Bigr)
dsdv+
+
t\int
0
1\int
0
pn(t - s, u, v)\BbbI \{ \~Xn
s (\lceil v\rceil )>0\} QdWsdu, t \geq 0, u \in [0, 1], (2.13)
where
pn (t, u, v) = (1 - n(\lceil u\rceil - u))pnk,n\lceil v\rceil (t) + (\lceil u\rceil - u)pnk,n\lceil v\rceil - 1(t), t \geq 0,
\~\lambda n(v) = \lambda (v)\BbbI \Bigl\{
qn
n\lceil v\rceil ,n\lceil v\rceil >0
\Bigr\} , v \in [0, 1],
and \lceil v\rceil = \lceil v\rceil n :=
l
n
for v \in \pi nl , l \in [n]. We will denote by \~Xn,i
t (u) the ith term of the right-hand
side of equation (2.13).
Lemma 2.3. For every \gamma > 0 and T > 0 there exists a constant C > 0 such that
\BbbE
\Bigl[ \Bigl(
\~Xn
t (u)
\Bigr) \gamma \Bigr]
\leq C
for all t \in [0, T ], u \in [0, 1] and n \geq 1.
Lemma 2.4. For each \gamma \in \BbbN and T > 0 there exists a constant C > 0 such that
\BbbE
\biggl[ \bigm| \bigm| \bigm| \~Xn,i
t2
(u2) - \~Xn,i
t1
(u1)
\bigm| \bigm| \bigm| 2\gamma \biggr] \leq C
\Bigl(
| t2 - t1|
\gamma
2 + | u2 - u1|
\gamma
2
\Bigr)
for every t1, t2 \in [0, T ], u1, u2 \in [0, 1], n \geq 1 and i \in [4].
To prove Lemmas 2.3 and 2.4, one needs to repeat the proofs of Lemmas 2.1 and 2.2 from [11]
which are based on properties of the fundamental solution pn(t, u, v), t \in [0, T ], u, v \in [0, T ], and
the fact that the coefficients of the equation has at most linear growth. We omit the proof of those
lemmas here.
Proposition 2.2 follows from Lemma 2.4.
Remark 2.2. Let \~Xt, t \geq 0, be a limit point of the sequence
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
in \mathrm{C} ([0,\infty ),\mathrm{C}[0, 1]),
i.e., \~X is a limit in distribution of a subsequence of
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
. Then the map (t, u) \mapsto \rightarrow \~Xt(u)
is a.s. locally Hölder continuous with exponent \alpha \in (0, 1/4), according to Lemma 2.4 and Corol-
lary 16.9 [17]. Moreover, Lemma 2.3 and Lemma 4.11 [17] imply that for every \gamma > 0 and T > 0
there exists a constant C = C(T, \gamma ) such that
\BbbE
\Bigl[ \Bigl(
\~Xt(u)
\Bigr) \gamma \Bigr]
\leq C, t \in [0, T ] u \in [0, 1].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1211
3. Passing to the limit. The goal of the present section is to show that there exists a solution
to SPDE (1.1) – (1.4). The solution will be constructed as a limit point of the family of processes\Bigl\{
\~Xn, n \geq 1
\Bigr\}
from Proposition 2.2, which exists by Prokhorov’s theorem. Since the coefficients of
the equation are discontinuous, we cannot pass to the limit directly. In the next section, we will show
that there exists a subsequence of
\Bigl\{
\~Xn, n \geq 1
\Bigr\}
whose weak limit in \mathrm{C} ([0,\infty ),\mathrm{C}[0, 1]) is a heat
semimartingale5. After that we will prove an analog of the Itô formula and state a property similar to
one for usual \BbbR -valued semimartingales, stated in Lemma A.1, for such heat semimartingales. Then,
using the argument described in Subsection 1.2, we show that \~X solves equation (1.1) – (1.4). In this
section, T > 0 will be fixed.
3.1. Martingale problem for limit points of the discrete approximation. We start from the
introduction of a new metric space where we will consider the convergence. Denote
r0 :=
\bigl(
1 + \| \lambda \| + \| Q\| \scrL 2
\bigr) \surd
T ,
and consider the following balls:
\mathrm{B} (\mathrm{L}2) : =
\Bigl\{
f \in \mathrm{L}T
2 : \| f\| L2,T
\leq r0
\Bigr\}
,
\mathrm{B} (\scrL 2) : =
\Bigl\{
L \in \scrL T
2 : \| L\| \scrL 2,T
\leq r0
\Bigr\}
in the Hilbert spaces \mathrm{L}T
2 and \scrL T
2 , respectively. We furniture those sets with the induced weak
topologies. By Theorem V.5.1 [5], those topological spaces are metrizable. Moreover, by Alaoglu’s
Theorem V.4.2 [5], they are compact metric spaces.
For every n \geq 1 we take the family of processes \{ xnk(t), t \in [0, T ], k \in [n]\} that is a solution
to SDE (2.1) – (2.3). Let \~Xn
t , t \in [0, T ], be the continuous process in \mathrm{C}[0, 1] defined by (2.4), that
is,
\~Xn
t (u) = (un - k + 1)xnk(t) + (k - nu)xnk - 1(t), u \in [0, 1], t \in [0, T ],
and k \in [n] such that
k - 1
n
\leq u <
k
n
. Let us also introduce the process
Xn
t :=
n\sum
k=1
xnk(t)\pi
n
k , t \in [0, T ],
where \pi nk = \BbbI \{ [ k - 1
n
, k
n)\} . Set
\lambda n :=
n\sum
k=1
n\langle \lambda , \pi nk \rangle \BbbI \{ qnk,k>0\} \pi
n
k \in \mathrm{L}2,
Ln
t := Q
\Bigl[
\BbbI \{ Xn
t >0\} \cdot
\Bigr]
\mathrm{p}\mathrm{r}n, t \in [0, T ],
and
\Gamma n
t := (Ln
t )
\ast Ln
t = \mathrm{p}\mathrm{r}n
\Bigl[
\BbbI \{ Xn
t >0\} \cdot
\Bigr]
Q2
\Bigl[
\BbbI \{ Xn
t >0\} \cdot
\Bigr]
\mathrm{p}\mathrm{r}n, t \in [0, T ].
We can trivially estimate \| \lambda n\| \leq \| \lambda \| and
5We call continuous processes in \mathrm{L}2 satisfying (3.9) below a heat semimartingales.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1212 V. KONAROVSKYI
\| \Gamma n
t \| \scrL 2 \leq
\bigm\| \bigm\| \bigm\| Q \Bigl[ \BbbI \{ Xn
t >0\} \cdot
\Bigr]
\mathrm{p}\mathrm{r}n
\bigm\| \bigm\| \bigm\| 2
\scrL 2
\leq \| Q\| 2\scrL 2
, t \in [0, T ].
The latter inequality follows from Lemma A.3. Hence \lambda n\BbbI \{ Xn=0\} and \Gamma n are random elements in
\mathrm{L}T
2 and \scrL T
2 , respectively. Let us consider the random element
\mathrm{X}n :=
\Bigl(
\~Xn, Xn, \lambda n\BbbI \{ Xn=0\} , \BbbI \{ Xn>0\} ,\Gamma
n
\Bigr)
, n \geq 1, (3.1)
in the complete separable metric space
\scrW L2 = \mathrm{C}([0, T ],\mathrm{C}[0, 1])\times \mathrm{C} ([0, T ],\mathrm{L}2)\times \mathrm{B} (\mathrm{L}2)
2 \times \mathrm{B} (\scrL 2) .
The following statement is the main result of this section.
Proposition 3.1. There exists a subsequence of \{ \mathrm{X}n, n \geq 1\} which converges to
\mathrm{X} =
\bigl(
\~X,X, a, \sigma ,\Gamma
\bigr)
in \scrW L2 in distribution. Moreover, the limit \mathrm{X} satisfies the following properties:
(i) \~Xt = Xt in \mathrm{L}2 for all t \in [0, T ] a.s. and a = \lambda (1 - \sigma ) in \mathrm{L}T
2 a.s.;
(ii) there exists a random element L in \scrL T
2 such that
\BbbP
\bigl[
L2 = \Gamma and L is self-adjoint a.e.
\bigr]
= 1
and
\BbbE
\left[ T\int
0
\| Lt\| 2\scrL 2
dt
\right] < +\infty ; (3.2)
(iii) there exists a continuous square-integrable (\scrF X,M
t )-martingale Mt, t \in [0, T ], in \mathrm{L}2 such
that for every \varphi \in \mathrm{C}2
\alpha 0
[0, 1]
\langle Xt, \varphi \rangle = \langle g, \varphi \rangle + 1
2
t\int
0
\bigl\langle
Xs, \varphi
\prime \prime \bigr\rangle ds+ t\int
0
\langle as, \varphi \rangle ds+
+
t\int
0
\langle f(Xs), \varphi \rangle ds+ \langle Mt, \varphi \rangle , t \in [0, T ], (3.3)
and
[\langle M\cdot , \varphi \rangle ]t =
t\int
0
\| Ls\varphi \| 2ds, t \in [0, T ].
Remark 3.1. Due to equality (3.3) and Theorem 1.2 [23], the process
\int t
0
asds, t \in [0, T ], is
(\scrF X,M
t )-adapted.
Proof. We first remark that the families
\bigl\{
\lambda n\BbbI \{ Xn=0\} , n \geq 1
\bigr\}
,
\bigl\{
\BbbI \{ Xn>0\} , n \geq 1
\bigr\}
and \{ \Gamma n, n \geq
\geq 1\} are tight due to the compactness of the spaces where they are defined. Consequently, by
Proposition 2.2 and Proposition 3.2.4 [6], the family
\Bigl\{ \Bigl(
\~Xn, \lambda n\BbbI \{ Xn=0\} , \BbbI \{ Xn>0\} ,\Gamma
n
\Bigr)
, n \geq 1
\Bigr\}
is
also tight. By the Prokhorov theorem, there exists a subsequence N \subset \BbbN such that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1213\Bigl(
\~Xn, \lambda n\BbbI \{ Xn=0\} , \BbbI \{ Xn>0\} ,\Gamma
n
\Bigr)
\rightarrow ( \~X, a, \sigma ,\Gamma )
in distribution along N. Without loss of generality, let us assume that N = \BbbN .
Since
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\bigm\| \bigm\| \~Xn
t - Xn
t
\bigm\| \bigm\| 2 = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\mathrm{m}\mathrm{a}\mathrm{x}
k\in [n]
\bigm| \bigm| xnk(t) - xnk - 1(t)
\bigm| \bigm| 2 \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq \delta \leq 1
n
\mathrm{m}\mathrm{a}\mathrm{x}
| u - u\prime | \leq \delta
\bigm| \bigm| \~Xn
t (u) - \~Xn
t (u
\prime )
\bigm| \bigm| 2,
it is easily to see that \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,T ]
\bigm\| \bigm\| \~Xn
t - Xn
t
\bigm\| \bigm\| 2 d\rightarrow 0, by Skorokhod representation Theorem 3.1.8 [6]
and the uniform convergence of \~Xn to \~X. Hence, \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,T ]
\bigm\| \bigm\| \~Xn
t - Xn
t
\bigm\| \bigm\| 2 \rightarrow 0 in probability as
n\rightarrow \infty . Using Corollary 3.3.3 [6] and the fact that \~Xn, n \geq 1, also converges to \~X in \mathrm{C} ([0, T ],\mathrm{L}2)
in distribution, we have that Xn d\rightarrow \~X =: X in \mathrm{C} ([0, T ],\mathrm{L}2) .
We next note that
\lambda n\BbbI \{ Xn
t =0\} = (\lambda n - \lambda ) \BbbI \{ Xn
t =0\} + \lambda
\Bigl(
1 - \BbbI \{ Xn
t >0\}
\Bigr)
, t \in [0, T ]. (3.4)
By Lemma A.2, (\lambda n - \lambda ) \BbbI \{ Xn=0\} \rightarrow 0 in \mathrm{B} (\mathrm{L}2) a.s. as n \rightarrow \infty . Thus, (3.4) yields \lambda n\BbbI \{ Xn=0\}
d\rightarrow
d\rightarrow \lambda (1 - \sigma ) in \mathrm{B} (\mathrm{L}2) , n\rightarrow \infty . This implies the equality a = \lambda (1 - \sigma ) a.s.
The existence of a convergent subsequence of \{ \mathrm{X}n\} n\geq 1 , and statement (i) are proved.
The statement (ii) directly follows from Lemma A.5.
In order to prove statement (iii) of the proposition, we first define the following \mathrm{L}2-valued
martingale:
Mn
t :=
n\sum
k=1
t\int
0
\surd
n\BbbI \{ xn
k (s)>0\} dw
n
k (s)\pi
n
k =
=
t\int
0
\mathrm{p}\mathrm{r}n
\bigl[
\BbbI \{ Xn
s >0\} \cdot
\bigr]
QdWs =
t\int
0
(Ln
s )
\ast dWs, t \in [0, T ].
Set for \varphi \in \mathrm{L}2
\~\Delta n\varphi := n3
n\sum
k=1
\Delta n\varphi n
k\pi
n
k , (3.5)
where \varphi n
k = \langle \varphi , \pi nk \rangle , \varphi n
0 = \alpha 0\varphi
n
1 and \varphi n
n+1 = \alpha 0\varphi
n
n. Since Xn =
\sum n
k=1
xnk\pi
n
k and the family
\{ xnk , k \in [n]\} solves SDE (2.1) – (2.3), we get for every \varphi \in \mathrm{L}2
\langle Mn
t , \varphi \rangle = \langle Mn
t ,\mathrm{p}\mathrm{r}
n \varphi \rangle =
= \langle Xn
t , \mathrm{p}\mathrm{r}
n \varphi \rangle - \langle gn, \mathrm{p}\mathrm{r}n \varphi \rangle - 1
2
t\int
0
\Bigl\langle
\~\Delta nXn
s ,\mathrm{p}\mathrm{r}
n \varphi
\Bigr\rangle
ds -
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1214 V. KONAROVSKYI
-
t\int
0
\bigl\langle
\lambda n\BbbI \{ Xn
s =0\} , \mathrm{p}\mathrm{r}
n \varphi
\bigr\rangle
ds -
t\int
0
\langle f(Xn
s ),\mathrm{p}\mathrm{r}
n \varphi \rangle ds =
= \langle Xn
t , \varphi \rangle - \langle gn, \varphi \rangle - 1
2
t\int
0
\Bigl\langle
Xn
s , \~\Delta
n\varphi
\Bigr\rangle
ds -
-
t\int
0
\bigl\langle
\lambda n\BbbI \{ Xn
s =0\} , \varphi
\bigr\rangle
ds -
t\int
0
\langle f(Xn
s ), \varphi \rangle ds, t \in [0, T ], (3.6)
and the quadratic variation of the (\scrF Xn
t )-martingale \langle Mn
\cdot , \varphi \rangle equals
[\langle Mn
\cdot , \varphi \rangle ]t =
t\int
0
\bigm\| \bigm\| Q\BbbI \{ Xn
s >0\} \mathrm{p}\mathrm{r}
n \varphi
\bigm\| \bigm\| 2 ds, t \in [0, T ].
Let \~e1(u) = 1, u \in [0, 1], and \~ek(u) =
\surd
2 \mathrm{c}\mathrm{o}\mathrm{s}\pi (k - 1)u, u \in [0, 1], k \geq 2, if \alpha 0 = 1,
and \~ek(u) =
\surd
2 \mathrm{s}\mathrm{i}\mathrm{n}\pi ku, u \in [0, 1], k \geq 1, if \alpha 0 = 0. Then \~ek \in \mathrm{C}2
\alpha 0
[0, 1] for all k \geq 1, and
\{ \~ek, k \geq 1\} form an orthonormal basis in \mathrm{L}2. Since
\bigm\| \bigm\| \bigm\| Q\BbbI \{ Xn
t >0\} \mathrm{p}\mathrm{r}
n \~ek
\bigm\| \bigm\| \bigm\| 2 \leq \| Q\| 2, t \in [0, T ],
k \geq 1, the families \{ \langle Mn
\cdot , \~ek\rangle , n \geq 1\} and \{ [\langle Mn
\cdot , \~ek\rangle ] , n \geq 1\} are tight in \mathrm{C}[0, T ] for every
k \geq 1, by the Aldous tightness criterion. According to the tightness of \{ \mathrm{X}n, n \geq 1\} , we also have
that \{ \langle Xn
\cdot , \~ek\rangle , n \geq 1\} is tight in \mathrm{C}[0, T ] for each k \geq 1. Using Proposition 2.4 [6] and Prokhorov’s
theorem, we can choose a subsequence N \subset \BbbN such that
(\langle Xn
\cdot , \~ek\rangle , \langle Mn
\cdot , \~ek\rangle , [\langle Mn
\cdot , \~ek\rangle ])k\geq 1 \rightarrow
\bigl(
\=Xk, \=Mk, \=Vk
\bigr)
k\geq 1
(3.7)
in
\bigl(
(\mathrm{C}[0, T ])3
\bigr) \BbbN
in distribution along N. In particular, we have that \langle Mn
\cdot , \~ek\rangle
2 - [\langle Mn
\cdot , \~ek\rangle ] , n \geq 1,
is a sequence of martingales which converges to \=M2
k - \=Vk in \mathrm{C}[0, T ] in distribution along N for all
k \geq 1.
We fix m \geq 1 and let ( \=\scrF \=X, \=M, \=V ,m
t )t\in [0,T ] be the complete right continuous filtration generated
by ( \=Xk, \=Mk, \=Vk), k \in [m]. By Proposition IX.1.17 [16], we can conclude that \=Mk and \=M2
k - \=Vk are
continuous local ( \=\scrF \=X, \=M, \=V ,m
t )-martingales for all k \in [m]. Since
\BbbE
\Bigl[
\langle Mn
T , ek\rangle
2
\Bigr]
=
T\int
0
\BbbE
\Bigl[ \bigm\| \bigm\| Q\BbbI \{ Xn
s >0\} \~ek
\bigm\| \bigm\| 2\Bigr] ds \leq \| Q\| 2T,
we have that \BbbE
\bigl[
\=M2
k (T )
\bigr]
< +\infty by Lemma 4.11 [17]. Hence \=M2
k is a continuous square-integrable
( \=\scrF \=X, \=M, \=V ,m
t )-martingale with quadratic variation
\bigl[
\=Mk
\bigr]
= \=V , k \in [m]. From Theorem 17.17 [17],
we can conclude that \scrF \=X, \=M, \=V ,m
t = \=\scrF \=X, \=M,m
t , t \in [0, T ], where ( \=\scrF \=X, \=M,m
t )t\in [0,T ] is the complete
right continuous filtration generated by
\bigl(
\=Xk, \=Mk
\bigr)
, k \in [m]. Since for every t \in [0, T ] the \sigma -
algebra \=\scrF \=X, \=M,m
t increases to \=\scrF \=X, \=M
t as m \rightarrow \infty , Theorem 1.6 [20] yields that \=Mk is a continuous
square-integrable ( \=\scrF \=X, \=M
t )-martingale with quadratic variation [ \=Mk] = \=Vk for each k \geq 1.
Next, we recall that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1215\Bigl(
\~Xn, Xn, \lambda n\BbbI \{ Xn=0\} ,\Gamma
n
\Bigr)
\rightarrow (X,X, a,\Gamma )
in \mathrm{C} ([0, T ],\mathrm{C}[0, 1]) \times \mathrm{C} ([0, T ],\mathrm{L}2) \times \mathrm{B} (\mathrm{L}2) \times \mathrm{B} (\scrL 2) in distribution as n \rightarrow \infty . By Skorokhod
representation Theorem 3.1.8 [6], we may assume that this sequence converges a.s. Therefore, for
every t \in [0, T ] and k \geq 1
\langle Xn
t , \~ek\rangle \rightarrow \langle Xt, \~ek\rangle =: Xk(t), \langle gn, \~ek\rangle \rightarrow \langle g, \~ek\rangle ,
t\int
0
\bigl\langle
\lambda n\BbbI \{ Xn
s =0\} , \~ek
\bigr\rangle
ds\rightarrow
t\int
0
\langle as, \~ek\rangle ds,
t\int
0
\langle f(Xn
s ), \~ek\rangle ds\rightarrow
t\int
0
\langle f(Xs), \~ek\rangle ds,
[\langle Mn
\cdot , \~ek\rangle ]t =
t\int
0
\| Ln
s \~ek\|
2 ds\rightarrow
t\int
0
\| Ls\~ek\| 2 ds =: Vk(t)
a.s. as n\rightarrow \infty . Using Taylor’s formula and the fact that \~ek \in \mathrm{C}3
\alpha 0
[0, 1], k \geq 1, it is easy to see that
for every t \in [0, T ] and k \geq 1
t\int
0
\Bigl\langle
Xn
s , \~\Delta
n\~ek
\Bigr\rangle
ds\rightarrow
t\int
0
\bigl\langle
Xs, \~e
\prime \prime
k
\bigr\rangle
ds a.s. as n\rightarrow \infty .
Consequently, for every t \in [0, T ] the sequence \langle Mn
t , \~ek\rangle , n \geq 1, converges to
Mk(t) := \langle Xt, \~ek\rangle - \langle g, \~ek\rangle -
1
2
t\int
0
\bigl\langle
Xs, \~e
\prime \prime
k
\bigr\rangle
ds -
-
t\int
0
\langle as, \~ek\rangle ds -
t\int
0
\langle f(Xs), \~ek\rangle ds
a.s. as n\rightarrow \infty . Thus, for every m \in \BbbN and ti \in [0, T ], i \in [m],\Bigl( \bigl( \bigl\langle
Xn
ti , \~ek
\bigr\rangle
,
\bigl\langle
Mn
ti , \~ek
\bigr\rangle
, [\langle Mn
\cdot , \~ek\rangle ]ti
\bigr)
i\in [m]
\Bigr)
k\geq 1
\rightarrow
\Bigl(
(Xk(ti),Mk(ti), Vk(ti))i\in [m]
\Bigr)
k\geq 1
in
\bigl(
\BbbR 3m
\bigr) \BbbN
a.s. as n\rightarrow \infty . This and convergence (3.7) imply that
\mathrm{L}\mathrm{a}\mathrm{w}
\Bigl\{
(Xk,Mk, Vk)k\geq 1
\Bigr\}
= \mathrm{L}\mathrm{a}\mathrm{w}
\Bigl\{ \bigl(
\=Xk, \=Mk, \=Vk
\bigr)
k\geq 1
\Bigr\}
in
\Bigl(
(\mathrm{C}[0, 1])3
\Bigr) \BbbN
. Consequently, for every k \geq 1 the process Mk is a continuous square-integrable
( \=\scrF X,M
t )-martingale with quadratic variation
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1216 V. KONAROVSKYI
[Mk]t = Vk(t) =
t\int
0
\| Ls\~ek\| 2 ds, t \in [0, T ],
where ( \=\scrF X,M
t )t\in [0,T ] is the complete right continuous filtration generated by Xk,Mk, k \geq 1.
Now we introduce the following process in \mathrm{L}2 :
Mt :=
\infty \sum
k=1
Mk(t)\~ek, t \in [0, T ]. (3.8)
Remark that Mt, t \in [0, T ], is a well-defined continuous process in \mathrm{L}2. Indeed, by the Burkholder –
Davis – Gundy inequality, Lemma A.4, (3.2) and the dominated convergence theorem, for every
n,m \geq 1,
\BbbE
\Biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\bigm\| \bigm\| \bigm\| \bigm\| n\sum
k=1
Mk(t)\~ek -
n+m\sum
k=1
Mk(t)\~ek
\bigm\| \bigm\| \bigm\| \bigm\| 2
\Biggr]
= \BbbE
\Biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
n+m\sum
k=n+1
M2
k (t)
\Biggr]
\leq
\leq
T\int
0
\BbbE
\left[ n+m\sum
k,l=n+1
\langle Lt\~ek, Lt\~el\rangle
\right] dt =
=
T\int
0
\BbbE
\left[ \infty \sum
k,l=1
\bigl\langle
Lt \widetilde \mathrm{p}\mathrm{r}n,n+m\~ek, Lt \widetilde \mathrm{p}\mathrm{r}n,n+m\~el
\bigr\rangle \right] dt =
=
T\int
0
\BbbE
\Bigl[ \bigm\| \bigm\| Lt \widetilde \mathrm{p}\mathrm{r}n,n+m
\bigm\| \bigm\| 2
\scrL 2
\Bigr]
dt\rightarrow 0
as n,m\rightarrow \infty , where \widetilde \mathrm{p}\mathrm{r}n,n+m is the orthogonal projection in \mathrm{L}2 onto \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ \~ek, k = n+1, . . . , n+
+m\} . This implies the convergence of series (3.8) and the continuity of Mt, t \in [0, T ], in \mathrm{L}2.
Since \=\scrF X,M
t = \scrF X,M
t , t \in [0, T ], and \langle Mt, \~ek\rangle = Mk(t), t \in [0, T ], for all k \geq 1, it is easily
seen that M is a continuous square-integrable (\scrF X,M
t )-martingale in \mathrm{L}2 with quadratic variation
[M ]t =
t\int
0
L2
sds =
t\int
0
\Gamma sds, t \in [0, T ].
This implies statement (iii).
Proposition 3.1 is proved.
3.2. A property of quadratic variation of heat semimartingales. In this section, we will
assume that (\Omega ,\scrF , (\scrF t)t\geq 0,\BbbP ) is a filtered complete probability space, where the filtration (\scrF t)t\geq 0
is complete and right continuous. Let T > 0 be fixed. Consider a continuous (\scrF t)-adapted \mathrm{L}2-valued
process Zt, t \in [0, T ], such that there exist random elements a and L in \mathrm{L}T
2 and \scrL T
2 , respectively,
such that for every \varphi \in \mathrm{C}2
\alpha 0
[0, 1] the processes
\int t
0
\langle as, \varphi \rangle ds, t \in [0, T ], and
\int t
0
\| Ls\varphi \| 2ds, t \in
\in [0, T ], are (\scrF t)-adapted, and
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1217
\scrM \varphi
Z(t) := \langle Zt, \varphi \rangle - \langle Z0, \varphi \rangle -
1
2
t\int
0
\langle Zs, \varphi
\prime \prime \rangle ds -
t\int
0
\langle as, \varphi \rangle ds, t \in [0, T ], (3.9)
is a local (\scrF t)-martingale with quadratic variation
[\scrM \varphi
Z ]t =
t\int
0
\| Ls\varphi \| 2ds, t \in [0, T ].
Note that the assumptions on L implies that the continuous process
\int t
0
\| Ls\| 2\scrL 2
ds, t \in [0, T ], is
well-defined and (\scrF t)-adapted.
We will further consider the case of the Neumann boundary conditions, where \alpha 0 = 1. All
conclusions of this section will be the same for the Dirichlet boundary conditions, where \alpha 0 = 0. Let
\{ \~ek, k \geq 1\} be the family of the eigenfunctions of \Delta on [0, 1] with Neumann boundary conditions.
We recall that \~e1(u) = 1, u \in [0, 1], and \~ek(u) =
\surd
2 \mathrm{c}\mathrm{o}\mathrm{s}\pi (k - 1)u, u \in [0, 1], k \geq 2. Denote the
orthogonal projection in \mathrm{L}2 onto \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ \~ek, k \in [n]\} by \widetilde \mathrm{p}\mathrm{r}n.
Let Zn
t = \widetilde \mathrm{p}\mathrm{r}nZt, t \geq 0, and ant = \widetilde \mathrm{p}\mathrm{r}nat, t \in [0, T ]. We also introduce
\.Zn
t =
n\sum
k=1
\langle Zt, \~ek\rangle \~e\prime k, t \in [0, T ], n \geq 1,
and note that \.Zn, n \geq 1, is a sequence of random elements in \mathrm{L}T
2 .
Lemma 3.1. (i) The equality
\BbbP
\Bigl[
\.Zn, n \geq 1, converges in \mathrm{L}T
2 and a.e. as n\rightarrow \infty
\Bigr]
= 1
holds.
(ii) Set
\.Z := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\.Zn,
where the limit is taken a.e. Then \.Z is a random element in \mathrm{L}T
2 and for every t \in [0, T ]
t\int
0
\| \.Zn
s \| 2ds\rightarrow
t\int
0
\| \.Zs\| 2ds a.s. as n\rightarrow \infty .
Proof. Set zk(t) := \langle Zt, \~ek\rangle , t \in [0, T ], k \geq 1. Then, by the definition of Z, for every k \geq 1
the process
\xi k(t) := zk(t) - zk(0) +
\pi 2(k - 1)2
2
t\int
0
zk(s)ds -
t\int
0
ak(s)ds, t \in [0, T ],
is a continuous local (\scrF t)-martingale with quadratic variation
[\xi k]t =
t\int
0
\| Ls\~ek\| 2ds, t \in [0, T ],
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1218 V. KONAROVSKYI
where ak(s) := \langle as, \~ek\rangle . Denote \sigma 2k,l(t) := \langle Lt\~ek, Lt\~el\rangle , t \in [0, T ], and remark that
Zn
t =
n\sum
k=1
zk(t)\~ek and ant =
n\sum
k=1
ak(t)\~ek, t \in [0, T ], n \geq 1.
By the Itô formula and the polarisation equality, we get
\| Zn
t \| 2 = \| Zn
0 \| 2 -
t\int
0
\| \.Zn
s \| 2ds+ 2
t\int
0
\langle ans , Zn
s \rangle ds+
+
t\int
0
\| Ls \widetilde \mathrm{p}\mathrm{r}n\| 2\scrL 2
ds+\scrM n(t), t \in [0, T ], (3.10)
where \scrM n(t), t \in [0, T ], is a continuous local (\scrF t)-martingale defined as
\scrM n(t) = 2
n\sum
k=1
t\int
0
zk(s)d\xi k(s), t \in [0, T ].
A simple computation gives that
[\scrM n]t = 4
t\int
0
\| LsZ
n
s \| 2ds, t \in [0, T ].
Trivially, \| Zn
t \| 2 \rightarrow \| Zt\| 2 a.s. as n \rightarrow \infty for all t \in [0, T ]. Using the dominated convergence
theorem, we can conclude that
\int t
0
\langle ans , Zn
s \rangle ds\rightarrow
t\int
0
\langle as, Zs\rangle ds a.s. as n\rightarrow \infty . Next, by Lemma A.4
and the dominated convergence theorem,
\int t
0
\| Ls \widetilde \mathrm{p}\mathrm{r}n\| 2\scrL 2
ds \rightarrow
\int t
0
\| Ls\| 2\scrL 2
ds a.s. as n \rightarrow \infty . Next,
we will show that \scrM n(t) converges in probability. Since \scrM n is a local martingale, we need to
choose a localization sequence of (\scrF t)-stopping times defined as follows:
\tau k := \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ t \in [0, T ] :
t\int
0
\| Ls\| 2\scrL 2
ds \geq k
\right\} \wedge T.
Then the processes \scrM n(t \wedge \tau k), t \in [0, T ], n \geq 1, are square-integrable (\scrF t)-martingales for
every k \geq 1, and \tau k \uparrow T as k \rightarrow \infty . By the Burkholder – Davis – Gundy inequality (see, e.g., [15],
Theorem III.3.1), for every k, n,m \geq 1, n < m,
\BbbE
\Biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\bigm| \bigm| \scrM n(t \wedge \tau k) - \scrM m(t \wedge \tau k)
\bigm| \bigm| 2\Biggr] \leq 16\BbbE
\left[ \tau k\int
0
\| Ls \widetilde \mathrm{p}\mathrm{r}n,mZm
s \| 2 ds
\right] ,
where \widetilde \mathrm{p}\mathrm{r}n,m is the orthogonal projection in \mathrm{L}2 onto \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n} \{ \~ek, k = n+ 1, . . . ,m\} . Hence, by the
dominated convergence theorem,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1219
\BbbE
\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
| \scrM n(t \wedge \tau k) - \scrM m(t \wedge \tau k)| 2
\biggr]
\rightarrow 0 as n\rightarrow \infty .
This implies that there exists a continuous square-integrable (\scrF t)-martingale \scrM k(t), t \in [0, T ], such
that
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\bigm| \bigm| \scrM n(t \wedge \tau k) - \scrM k(t)
\bigm| \bigm| \rightarrow 0 in probability as n\rightarrow \infty .
By Lemma B.11 [3],
[\scrM k]t = 4
t\wedge \tau k\int
0
\| LsZs\| 2 ds, t \in [0, T ].
Furthermore, for every k \geq 1 \scrM k = \scrM k+1(\cdot \wedge \tau k) a.s. We define \scrM (t) := \scrM k(t) for t \leq \tau k,
k \geq 1. Trivially, \scrM is a continuous local (\scrF t)-martingale with quadratic variation
[\scrM ]t = 4
t\int
0
\| LsZs\| 2 ds, t \in [0, T ].
Using Lemma 4.2 [17], \scrM n(t) \rightarrow \scrM (t) in probability as n\rightarrow \infty for every t \in [0, T ].
We have obtained that every term, except
\int t
0
\| \.Zn
s \| 2ds, of equality (3.10) converges in proba-
bility. Hence,
\int t
0
\| \.Zn
s \| 2ds also converges in probability. Moreover, this sequence is monotone. By
Lemma 4.2 [17], it converges almost surely. By Fatou’s lemma,
T\int
0
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| \.Zn
s \| 2ds <\infty a.s. (3.11)
This implies the convergence of
\Bigl\{
\.Zn
s (\omega )
\Bigr\}
n\geq 1
in \mathrm{L}2 for almost all s and \omega . Hence \.Zn, n \geq 1,
converges to \.Z a.e. a.s. as n \rightarrow \infty . The equality in the second part of the lemma follows from the
monotone convergence theorem and (3.11). In particular, \| \.Zn\| L2,T \rightarrow \| \.Z\| L2,T . Thus, \.Zn \rightarrow \.Z in
\mathrm{L}T
2 a.s., according to Proposition 2.12 [17].
Proposition 3.2. Let F \in \mathrm{C}2 (\BbbR ) has a bounded second derivative and h \in \mathrm{C}1[0, 1]. Then
\langle F (Zt), h\rangle = \langle F (Z0), h\rangle -
1
2
t\int
0
\Bigl\langle \bigl(
F \prime (Zs)h
\bigr) \prime
, \.Zs
\Bigr\rangle
ds+
t\int
0
\bigl\langle
F \prime (Zs)h, as
\bigr\rangle
ds+
+
1
2
t\int
0
\bigl\langle
Ls
\bigl[
F \prime \prime (Zs)h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds+\scrM F,h(t), t \in [0, T ], (3.12)
where \scrM F,h(t), t \in [0, T ], is a continuous local (\scrF t)-martingale with quadratic variation
[\scrM F,h]t =
t\int
0
\bigm\| \bigm\| LsF
\prime (Zs)h
\bigm\| \bigm\| 2 ds, t \in [0, T ],
and (F \prime (Zs)h)
\prime := F \prime \prime (Zs) \.Zsh+ F \prime (Zs)h
\prime \in \mathrm{L}2.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1220 V. KONAROVSKYI
Proof. As in the proof of Lemma 3.1, we can compute for every n \geq 1
\langle F (Zn
t ) , h\rangle = \langle F (Zn
0 ), h\rangle -
n\sum
k=1
\pi 2(k - 1)2
2
t\int
0
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle
zk(s)ds+
+
n\sum
k=1
t\int
0
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle
ak(s)ds+
1
2
n\sum
k,l=1
t\int
0
\bigl\langle
F \prime \prime (Zn
s )h\~ek, \~el
\bigr\rangle
\sigma 2k,l(s)ds+
+
n\sum
k=1
t\int
0
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle
d\xi k(s), t \in [0, T ].
Consequently,
\langle F (Zn
t ), h\rangle = \langle F (Zn
0 ), h\rangle -
1
2
t\int
0
\Bigl\langle \bigl(
F \prime (Zn
s )h
\bigr) \prime
n
, \.Zn
s
\Bigr\rangle
ds+
t\int
0
\bigl\langle
F \prime (Zn
s )h, a
n
s
\bigr\rangle
ds+
+
t\int
0
\bigl\langle
Ls \widetilde \mathrm{p}\mathrm{r}n \bigl[ F \prime \prime (Zn
s )h\cdot
\bigr]
, Ls \widetilde \mathrm{p}\mathrm{r}n\bigr\rangle \scrL 2
ds+\scrM n
F,h(t), t \in [0, T ],
where
\scrM n
F,h(t) =
n\sum
k=1
t\int
0
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle
d\xi k(s),
and (F \prime (Zn
s )h)
\prime
n =
\sum n
k=1
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle
\~e\prime k. The process \scrM n
F,h(t), t \in [0, T ], is a continuous local
(\scrF t)-martingale with quadratic variation
\bigl[
\scrM n
F,h
\bigr]
t
=
n\sum
k,l=1
t\int
0
\bigl\langle
F \prime (Zn
s )h, \~ek
\bigr\rangle \bigl\langle
F \prime (Zn
s )h, \~el
\bigr\rangle
\sigma 2k,lds =
=
t\int
0
\bigm\| \bigm\| Ls \widetilde \mathrm{p}\mathrm{r}nF \prime (Zn
s )h
\bigm\| \bigm\| 2 ds, t \in [0, T ].
By the boundedness of the second derivative of F we have that there exists a constant C > 0 such
that | F \prime (x)| \leq C(1 + | x| ) and | F (x)| \leq C(1 + | x| 2). Therefore, \langle F (Zn
t ), h\rangle \rightarrow \langle F (Zt), h\rangle a.s.,
n\rightarrow \infty , and F \prime (Zn
t )h\rightarrow F \prime (Zt)h and F \prime \prime (Zn
t )h\rightarrow F \prime \prime (Zt)h in \mathrm{L}2 a.s. as n\rightarrow \infty for all t \in [0, T ].
By the dominated convergence theorem and Lemma A.4,
t\int
0
\bigl\langle
F \prime (Zn
s )h, a
n
s
\bigr\rangle
ds\rightarrow
t\int
0
\bigl\langle
F \prime (Zs)h, as
\bigr\rangle
ds a.s.
and
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1221
t\int
0
\bigl\langle
Ls \widetilde \mathrm{p}\mathrm{r}n \bigl[ F \prime \prime (Zn
s )h\cdot
\bigr]
, Ls \widetilde \mathrm{p}\mathrm{r}n\bigr\rangle \scrL 2
ds\rightarrow
t\int
0
\bigl\langle
Ls
\bigl[
F \prime \prime (Zs)h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds a.s.
as n \rightarrow \infty . Using the localization sequence, one can show that, for every t \in [0, T ], \scrM n
F,h(t) \rightarrow
\rightarrow \scrM F,h(t) in probability as in the proof of the previous lemma.
In order to finish the proof of the proposition, we only need to show the convergence\int t
0
\Bigl\langle \bigl(
F \prime (Zn
s )h
\bigr) \prime
n
, \.Zn
s
\Bigr\rangle
ds to the corresponding term. By Lemma 3.1, it is enough to show that
(F \prime (Zn
\cdot )h)
\prime
n \rightarrow (F \prime (Z\cdot )h)
\prime = F \prime \prime (Z\cdot ) \.Z\cdot h + F \prime (Z\cdot )h
\prime a.e. a.s. as n \rightarrow \infty . But this easily follows
from the integration by parts formula.
Theorem 3.1. Let the process Zt, t \in [0, T ], and the random element L \in \scrL T
2 be as above.
Assume that Zt \geq 0 a.e., t \in [0, T ]. Then the equality
L\cdot
\bigl[
\BbbI \{ Z\cdot \not =0\} \cdot
\bigr]
= L in \scrL T
2 a.s.
holds.
Proof. In order to prove the theorem, we will use Proposition 3.2. We fix a function \psi \in \mathrm{C} (\BbbR )
such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\psi \subset [ - 1, 1], 0 \leq \psi (x) \leq 1, x \in \BbbR , and \psi (0) = 1. Define \psi \varepsilon (x) := \psi
\Bigl( x
\varepsilon
\Bigr)
, x \in \BbbR ,
and
F\varepsilon (x) :=
x\int
- \infty
\left( y\int
- \infty
\psi \varepsilon (r)dr
\right) dy, x \in \BbbR .
Then 0 \leq F \prime
\varepsilon (x) \leq 2\varepsilon , x \in \BbbR , and F \prime \prime
\varepsilon (x) \rightarrow \BbbI \{ 0\} (x) as \varepsilon \rightarrow 0+ for all x \in \BbbR .
Let a nonnegative function h \in \mathrm{C}1[0, 1] be fixed. By Proposition 3.2,
\langle F\varepsilon (Zt), h\rangle = \langle F\varepsilon (Z0), h\rangle -
1
2
t\int
0
\Bigl\langle \bigl(
F \prime
\varepsilon (Zs)h
\bigr) \prime
, \.Zs
\Bigr\rangle
ds+
t\int
0
\bigl\langle
F \prime
\varepsilon (Zs)h, as
\bigr\rangle
ds+
+
1
2
t\int
0
\bigl\langle
Ls
\bigl[
F \prime \prime
\varepsilon (Zs)h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds+\scrM F\varepsilon ,h(t), t \in [0, T ],
and the quadratic variation of the local (\scrF t)-martingale \scrM F\varepsilon ,h equals
[\scrM F\varepsilon ,h]t =
t\int
0
\bigm\| \bigm\| LsF
\prime
\varepsilon (Zs)h
\bigm\| \bigm\| 2 ds, t \in [0, T ].
Making \varepsilon \rightarrow 0+, we can immediately conclude that for every t \in [0, T ]
| \langle F\varepsilon (Zt), h\rangle - \langle F\varepsilon (Z0), h\rangle | \leq 2\varepsilon \| Zt - Z0\| \| h\| \rightarrow 0 a.s.
and \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\langle F \prime
\varepsilon (Zs)h, as\rangle ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 2\varepsilon \| h\|
t\int
0
\| as\| ds\rightarrow 0 a.s.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1222 V. KONAROVSKYI
Similarly to the proof of Lemma 3.1, using the localization sequence, one can show that MF\varepsilon ,h(t) \rightarrow
\rightarrow 0 in probability. By the dominated convergence theorem and Lemma A.4,
t\int
0
\bigl\langle
Ls
\bigl[
F \prime \prime
\varepsilon (Zs)h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds\rightarrow
t\int
0
\bigl\langle
Ls
\bigl[
\BbbI \{ Zs=0\} h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds a.s.
Again, by the dominated convergence theorem and Lemma A.6, we have
t\int
0
\Bigl\langle
(F \prime
\varepsilon (Zs)h)
\prime , \.Zs
\Bigr\rangle
ds =
t\int
0
\Bigl\langle
F \prime \prime
\varepsilon (Zs) \.Zsh, \.Zs
\Bigr\rangle
ds+
+
t\int
0
\Bigl\langle
F \prime (Zs)h
\prime , \.Zs
\Bigr\rangle
ds\rightarrow
t\int
0
\bigm\| \bigm\| \bigm\| \BbbI \{ Zs=0\} \.Zs
\surd
h
\bigm\| \bigm\| \bigm\| 2 ds = 0 a.s.
We have for every t \in [0, T ]
t\int
0
\bigl\langle
Ls
\bigl[
\BbbI \{ Zs=0\} h\cdot
\bigr]
, Ls
\bigr\rangle
\scrL 2
ds = 0 a.s.
Then taking h = 1 and applying Lemma A.3, it is easy to see that
T\int
0
\bigm\| \bigm\| Ls
\bigl[
\BbbI \{ Zs=0\} \cdot
\bigr] \bigm\| \bigm\| 2
\scrL 2
ds = 0.
Theorem 3.1 is proved.
3.3. Proof of the existence theorem. In this section, we will consider the random element \mathrm{X}n
defined in Subsection 3.1. According to Proposition 3.1, there exists a subsequence N \subset \BbbN such that
\mathrm{X}n =
\Bigl(
\~Xn, Xn, \lambda n\BbbI \{ Xn=0\} , \BbbI \{ Xn>0\} ,\Gamma
n
\Bigr)
\rightarrow
\Bigl(
\~X,X, a, \sigma ,\Gamma
\Bigr)
in \scrW L2
in distribution along N. As before, without loss of generality, we may assume that N = \BbbN . By the
Skorokhod representation theorem, we can assume that this sequence converges a.s. Since \~Xn \rightarrow \~X
in \mathrm{C} ([0, T ],\mathrm{C}[0, 1]) a.s., and a.s. for all t \in [0, T ] the quality \~Xt = Xt in \mathrm{L}2 holds, the inequality
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\bigm\| \bigm\| \~Xn
t - Xn
t
\bigm\| \bigm\| \leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,T ]
\mathrm{s}\mathrm{u}\mathrm{p}
0\leq \delta \leq 1
n
\mathrm{m}\mathrm{a}\mathrm{x}
| u - u\prime | \leq \delta
\bigm| \bigm| \~Xn
t (u) - \~Xn
t (u
\prime )
\bigm| \bigm|
implies that
\BbbP [\forall t \in [0, T ], Xn
t \rightarrow Xt a.e.] = 1. (3.13)
I. We will first show that \Gamma =
\bigl[
\BbbI \{ X\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ X\cdot >0\} \cdot
\bigr]
a.s.
Using Proposition 3.1 (ii), there exists a random element L in \scrL T
2 such that \Gamma = L2 a.s. Next,
by Proposition 3.1 and Theorem 3.1, L\cdot \BbbI \{ X\cdot >0\} = L a.s. Therefore, using the convergence of
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1223
\Gamma n = \mathrm{p}\mathrm{r}n \BbbI \{ Xn
\cdot >0\} Q
2\BbbI \{ Xn
\cdot >0\} \mathrm{p}\mathrm{r}
n to \Gamma in \mathrm{B} (\scrL 2) a.s., we obtain that for every t \in [0, T ] \cap \BbbQ and
k, l \geq 1 a.s.
t\int
0
\langle \Gamma s, ek \odot el\rangle \scrL 2ds =
t\int
0
\langle \Gamma sel, ek\rangle ds =
t\int
0
\langle Lsel, Lsek\rangle ds =
=
t\int
0
\bigl\langle
Ls\BbbI \{ Xs>0\} el, Ls\BbbI \{ Xs>0\} ek
\bigr\rangle
ds =
=
t\int
0
\bigl\langle
\Gamma s\BbbI \{ Xs>0\} el, \BbbI \{ Xs>0\} ek
\bigr\rangle
ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\bigl\langle
\Gamma n
s \BbbI \{ Xs>0\} el, \BbbI \{ Xs>0\} ek
\bigr\rangle
ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\bigl\langle
Q\BbbI \{ Xn
s >0\} \mathrm{p}\mathrm{r}
n
\bigl(
\BbbI \{ Xs>0\} el
\bigr)
, Q\BbbI \{ Xn
s >0\} \mathrm{p}\mathrm{r}
n
\bigl(
\BbbI \{ Xs>0\} ek
\bigr) \bigr\rangle
ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\bigl\langle
Q\mathrm{p}\mathrm{r}n
\bigl(
\BbbI \{ Xn
s >0\} \BbbI \{ Xs>0\} el
\bigr)
, Q\mathrm{p}\mathrm{r}n
\bigl(
\BbbI \{ Xn
s >0\} \BbbI \{ Xs>0\} ek
\bigr) \bigr\rangle
ds =
=
t\int
0
\bigl\langle
Q\BbbI \{ Xs>0\} el, Q\BbbI \{ Xs>0\} ek
\bigr\rangle
ds =
=
t\int
0
\bigl\langle
\BbbI \{ Xs>0\} Q
2\BbbI \{ Xs>0\} , ek \odot el
\bigr\rangle
\scrL 2
ds.
In the last equality, we have used the fact that \BbbI (0,+\infty )(xn)\BbbI (0,+\infty )(x) \rightarrow \BbbI (0,+\infty )(x) as xn \rightarrow x in
\BbbR , convergence (3.13) and the dominated convergence theorem. Since the family \{ \BbbI [0,t]ek \odot el, t \in
\in [0, T ] \cap \BbbQ , k, l \geq 1\} is countable and its linear span is dense in \scrL T
2 , we trivially get that
\Gamma = \BbbI \{ X\cdot >0\} Q
2\BbbI \{ X\cdot >0\} a.s. (3.14)
II. Let \chi 2 be defined by (1.6). We next want to show that
\BbbI \{ \chi >0\} \sigma = \BbbI \{ \chi >0\} \BbbI \{ X>0\} in \mathrm{L}T
2 a.s. (3.15)
But this will directly follow from the following lemma.
Lemma 3.2. Let Zn
t , t \in [0, T ], n \geq 1, be a sequence of \mathrm{L}2-valued measurable functions such
that Zn
t \geq 0 for all t \in [0, T ] and n \geq 1, and
\mathrm{L}\mathrm{e}\mathrm{b}T \otimes \mathrm{L}\mathrm{e}\mathrm{b}1 \{ (t, u) \in [0, T ]\times [0, 1] : Zn
t (u) \not \rightarrow Zt(u)\} = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1224 V. KONAROVSKYI
If
\mathrm{p}\mathrm{r}n
\bigl[
\BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
\mathrm{p}\mathrm{r}n \rightarrow
\bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
(3.16)
in \mathrm{B} (\scrL 2) and \BbbI \{ Zn>0\} \rightarrow \sigma in \mathrm{B} (\mathrm{L}2) as n\rightarrow \infty , then
\BbbI \{ \chi >0\} \sigma = \BbbI \{ \chi >0\} \BbbI \{ Z>0\} . (3.17)
We postpone the proof of the lemma to the end of this section.
III. Using equality (3.15), Proposition 3.1 (i) and assumption (1.7) of Theorem 1.1, we get
a = \lambda (1 - \sigma ) = \lambda \BbbI \{ \chi >0\} (1 - \sigma ) =
= \lambda \BbbI \{ \chi >0\}
\bigl(
1 - \BbbI \{ X>0\}
\bigr)
= \lambda \BbbI \{ \chi >0\} \BbbI \{ X=0\} = \lambda \BbbI \{ X=0\} . (3.18)
Hence, using Proposition 3.1 (iii) and equalities (3.14), (3.18), we have that for every \varphi \in
\in \mathrm{C}2
\alpha 0
[0, 1] a.s.
\langle Xt, \varphi \rangle = \langle g, \varphi \rangle + 1
2
t\int
0
\bigl\langle
Xs, \varphi
\prime \prime \bigr\rangle ds+ t\int
0
\bigl\langle
\lambda \BbbI \{ Xs=0\} , \varphi
\bigr\rangle
ds+
+
t\int
0
\langle f(Xs), \varphi \rangle ds+ \langle Mt, \varphi \rangle , t \in [0, T ], (3.19)
and \langle Mt, \varphi \rangle , t \in [0, T ], is a continuous square-integrable (\scrF X,M
t )-martingale with quadratic varia-
tion
[\langle M\cdot , \varphi \rangle ]t =
t\int
0
\bigm\| \bigm\| Q\BbbI \{ Xs>0\} \varphi
\bigm\| \bigm\| 2 ds, t \in [0, T ].
In particular, (3.19) yields that \scrF X,M
t = \scrF X
t , t \in [0, T ].
Theorem 1.1 is proved.
Proof of Lemma 3.2. It is easily seen that convergence (3.16) is equivalent to the convergence\bigl[
\BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
\rightarrow
\bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
in \mathrm{B} (\scrL 2)
as n\rightarrow \infty . So, for every \varphi \in \mathrm{L}T
2 , we have
T\int
0
\bigm\| \bigm\| \bigm\| Q\BbbI \{ Zn
t >0\} \varphi t
\bigm\| \bigm\| \bigm\| 2 dt = \bigl\langle \bigl[ \BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Zn
\cdot >0\} \cdot
\bigr]
, \varphi \cdot \odot \varphi \cdot
\bigr\rangle
\scrL 2,T
\rightarrow
\rightarrow
\bigl\langle \bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
Q2
\bigl[
\BbbI \{ Z\cdot >0\} \cdot
\bigr]
, \varphi \cdot \odot \varphi \cdot
\bigr\rangle
\scrL 2,T
=
=
T\int
0
\bigm\| \bigm\| Q\BbbI \{ Zt>0\} \varphi t
\bigm\| \bigm\| 2 dt
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1225
as n \rightarrow \infty , where \varphi \cdot \odot \varphi \cdot is defined as \varphi t \odot \varphi t, t \in [0, T ]. Replacing \varphi by ek\BbbI \{ Z=0\} for every
k \geq 1, we obtain that
T\int
0
\bigm\| \bigm\| Q\BbbI \{ Zn
t >0\} \BbbI \{ Zt=0\} ek
\bigm\| \bigm\| 2dt\rightarrow T\int
0
\bigm\| \bigm\| Q\BbbI \{ Zt>0\} \BbbI \{ Zt=0\} ek
\bigm\| \bigm\| 2 dt = 0 (3.20)
as n\rightarrow \infty . We set \~\BbbI nt := \BbbI \{ Zn
t >0\} \BbbI \{ Zt=0\} , t \in [0, T ]. Then (3.20) and the equality
T\int
0
\bigm\| \bigm\| \bigm\| Q\~\BbbI nt ek\bigm\| \bigm\| \bigm\| 2 dt = \infty \sum
l=1
T\int
0
\mu 2l
\Bigl\langle
\~\BbbI nt ek, el
\Bigr\rangle 2
dt
imply
T\int
0
\Bigl\langle
\~\BbbI nt ek, ek
\Bigr\rangle 2
dt\rightarrow 0, n\rightarrow \infty ,
for every k \geq 1 such that \mu k > 0. So, by the Hölder inequality,\left( T\int
0
1\int
0
\~\BbbI nt (u)e2k(u)dtdu
\right) 2
\leq T
T\int
0
\Bigl\langle
\~\BbbI nt ek, ek
\Bigr\rangle 2
dt\rightarrow 0, n\rightarrow \infty .
Taking into account the equality \~\BbbI nt =
\Bigl(
\~\BbbI nt
\Bigr) 2
, t \in [0, T ], we can conclude that
\~\BbbI nek \rightarrow 0 in \mathrm{L}T
2 , n\rightarrow \infty , (3.21)
for every k \geq 1 such that \mu k > 0.
We claim that \chi \~\BbbI n, n \geq 1, converges to 0 in \mathrm{L}T
2 as n\rightarrow \infty . Indeed, by convergence (3.21) and
the dominated convergence theorem,
\bigm\| \bigm\| \bigm\| \chi \~\BbbI n\bigm\| \bigm\| \bigm\| 2
L2,T
=
\infty \sum
k=1
\mu 2k
T\int
0
1\int
0
\~\BbbI nt (u)e2k(u)dtdu\rightarrow 0, n\rightarrow \infty . (3.22)
Next, since \BbbI \{ Zn>0\} \rightarrow \sigma in the weak topology of \mathrm{L}T
2 as n \rightarrow \infty , and \BbbI \{ Z>0\} , \BbbI \{ Z=0\} are
uniformly bounded, we trivially obtain that
\BbbI \{ Zn>0\} \BbbI \{ Z>0\} \rightarrow \sigma \BbbI \{ Z>0\} , \~\BbbI n = \BbbI \{ Zn>0\} \BbbI \{ Z=0\} \rightarrow \sigma \BbbI \{ Z=0\} , (3.23)
in the weak topology of \mathrm{L}T
2 as n\rightarrow \infty . Using the fact that
\BbbI (0,+\infty )(xn)\BbbI (0,+\infty )(x) \rightarrow \BbbI (0,+\infty )(x) as xn \rightarrow x in \BbbR ,
and the uniqueness of a weak limit, we get
\sigma \BbbI \{ Z>0\} = \BbbI \{ Z>0\} . (3.24)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1226 V. KONAROVSKYI
Since \chi \in \mathrm{L}2, convergence (3.23) yields
T\int
0
1\int
0
\chi (u)\~\BbbI nt (u)dtdu\rightarrow
T\int
0
1\int
0
\chi (u)\sigma t(u)\BbbI \{ Zt=0\} (u)dtdu, n\rightarrow \infty .
On the other hand side, \chi \~\BbbI n \rightarrow 0 in \mathrm{L}T
2 by (3.22). Hence
\chi \sigma \BbbI \{ Z=0\} = 0.
The latter equality and (3.24) yield
\chi \sigma = \chi \sigma \BbbI \{ Z>0\} + \chi \sigma \BbbI \{ Z=0\} = \chi \BbbI \{ Z>0\} in \mathrm{L}T
2
that is equivalent to equality (3.17).
Lemma 3.2 is proved.
A. Auxiliary statements.
Lemma A.1. Let \xi k(t), t \geq 0, k \in [2], be continuous real valued semimartingales with respect
to the same filtration. Let also the quadratic variations equal
[\xi k, \xi l]t =
t\int
0
\sigma k,l(s)ds, t \geq 0, k, l \in [2].
Then for all k, l \in [2] a.s.
[\xi k, \xi l]t =
t\int
0
\sigma k,l(s)\BbbI \{ \xi k(s)\not =0\} \BbbI \{ \xi l(s)\not =0\} ds, t \geq 0.
Proof. By Theorem 22.5 [17], one has, for k \in [2] a.s.
t\int
0
\sigma k,k(s)\BbbI \{ \xi k(s)=0\} ds =
t\int
0
\BbbI \{ 0\} (\xi k(s))d[\xi k]s =
=
+\infty \int
- \infty
\BbbI \{ 0\} (x)L
k,x
t dx = 0, t \geq 0,
where Lk,x
t , t \geq 0, x \in \BbbR , is the local time of \xi k. Applying the Cauchy-type inequality [17]
(Proposition 17.9), we estimate for every t \geq 0 a.s.
t\int
0
| \sigma 1,2(s)| \BbbI \{ \xi 1(s)=0\} ds \leq
t\int
0
\sigma 1,1(s)\BbbI \{ \xi 1(s)=0\} ds
t\int
0
\sigma 2,2(s)ds = 0.
Similarly, we get
t\int
0
| \sigma 1,2(s)| \BbbI \{ \xi 2(s)=0\} ds = 0, t \geq 0, a.s.
These equalities trivially yield the statement of the lemma.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1227
Lemma A.2. Let \lambda be a nonnegative function from \mathrm{L}2, Q be a nonnegative definite self-adjoint
Hilbert – Schmidt operator on \mathrm{L}2, \chi
2 be defined by (1.6) and
\lambda n =
n\sum
k=1
n\langle \lambda , \pi nk \rangle \BbbI \{ qnk,k>0\} \pi
n
k , n \geq 1,
where qnk,k = n\| Q\pi nk\| 2. If \lambda \BbbI \{ \chi >0\} = \lambda a.e., then \lambda n \rightarrow \lambda in \mathrm{L}2 as n\rightarrow \infty .
Proof. Denote
\~\lambda n := \mathrm{p}\mathrm{r}n \lambda =
n\sum
k=1
n\langle \lambda , \pi nk \rangle \pi nk , n \geq 1.
In this proof, functions from \mathrm{L}2 will be considered as random elements on the probability space
([0, 1],\scrB ([0, 1]),\mathrm{L}\mathrm{e}\mathrm{b}1), where \scrB ([0, 1]) is the Borel \sigma -algebra on [0, 1]. We remark that \~\lambda n is the
conditional expectation \BbbE [\lambda | \scrS n] determined on that probability space, where \scrS n = \sigma \{ \pi nk , k \in [n]\} .
By Proposition 1 [1], \~\lambda n \rightarrow \lambda in \mathrm{L}2 as n \rightarrow \infty . In particular, \~\lambda n converges to \lambda in probability as
n\rightarrow \infty .
Let
qn :=
n\sum
k=1
nqnk,k\pi
n
k =
n\sum
k=1
n2\| Q\pi nk\| 2\pi nk =
n\sum
k=1
\Biggl(
n2
\infty \sum
l=1
\mu 2l \langle el, \pi nk \rangle 2
\Biggr)
\pi nk =
=
\infty \sum
l=1
\mu 2l
\Biggl(
n\sum
k=1
n2\langle el, \pi nk \rangle 2\pi nk
\Biggr)
=
\infty \sum
l=1
\mu 2l (\mathrm{p}\mathrm{r}
n el)
2 , n \geq 1.
Remark that \mathrm{p}\mathrm{r}n el \rightarrow el in probability as n\rightarrow \infty for all l \geq 1.
We fix a subsequence N \subset \BbbN . Then, by Lemma 4.2 [17], there exists a subsequence N \prime \subset N
such that \~\lambda n \rightarrow \lambda a.s. along N \prime . Using Lemma 4.2 [17] again and the diagonalisation argument, we
can find a subsequence N \prime \prime \subset N \prime such that \mathrm{p}\mathrm{r}n el \rightarrow el a.s. along N \prime \prime for all l \geq 1. By Fatou’s
lemma,
\mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
qn \geq
\infty \sum
l=1
\mu 2l e
2
l = \chi 2 a.s.
This inequality and the lower semicontinuity of the map \BbbR \ni x \mapsto \rightarrow \BbbI (0,+\infty )(x) yield
\mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
\BbbI \{ qn>0\} \geq \BbbI \{ \chi 2>0\} = \BbbI \{ \chi >0\} a.s.
Consequently, using the equality
\lambda n =
n\sum
k=1
n\langle \lambda , \pi nk \rangle \BbbI \{ qnk,k>0\} \pi
n
k = \~\lambda n\BbbI \{ qn>0\} , (A.1)
and the convergence \~\lambda n \rightarrow \lambda a.s. along N \prime \prime , we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
\lambda n = \mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
\~\lambda n\BbbI \{ qn>0\} \geq \lambda \BbbI \{ \chi >0\} = \lambda a.s.
By (A.1), we also have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1228 V. KONAROVSKYI
\mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
\lambda n \leq \mathrm{l}\mathrm{i}\mathrm{m}
N \prime \prime \ni n\rightarrow \infty
\~\lambda n = \lambda a.s.
This implies the convergence \lambda n \rightarrow \lambda a.s. along N \prime \prime , and hence, \lambda n \rightarrow \lambda in probability as n \rightarrow \infty
by Lemma 4.2 [17]. We also remark that \lambda n \leq \~\lambda n, n \geq 1, and \~\lambda n \rightarrow \lambda in \mathrm{L}2. Hence, dominated
convergence Theorem 1.21 [17] implies that \| \lambda n\| \rightarrow \| \lambda \| . By Proposition 4.12 [17], \lambda n \rightarrow \lambda in \mathrm{L}2
as n\rightarrow \infty .
Lemma A.3. Let A \in \scrL 2 and Bi, i = 1, 2, be bounded operators on \mathrm{L}2. Then ABi \in \scrL 2,
i = 1, 2, and
\langle AB1, AB2\rangle \scrL 2 =
\infty \sum
n=1
\nu 2n\langle B\ast
1\varepsilon n, B
\ast
2\varepsilon n\rangle ,
where \{ \varepsilon n, n \geq 1\} and
\bigl\{
\nu 2n, n \geq 1
\bigr\}
are eigenvectors and eigenvalues of A\ast A, respectively.
Proof. Set An :=
\sum n
l=1
\nu l\varepsilon l \odot \varepsilon l, n \geq 1. Then it is easily seen that the sequence \{ An\} n\geq 1
converges to
\surd
A\ast A =
\sum \infty
l=1
\nu l\varepsilon l \odot \varepsilon l in \scrL 2. Hence
\langle AB1, AB2\rangle \scrL 2 =
\infty \sum
k=1
\langle AB1\varepsilon k, AB2\varepsilon k\rangle =
\infty \sum
k=1
\langle A\ast AB1\varepsilon k, B2\varepsilon k\rangle =
=
\Bigl\langle \surd
A\ast AB1,
\surd
A\ast AB2
\Bigr\rangle
\scrL 2
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\langle AnB1, AnB2\rangle \scrL 2
=
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
\langle AnB1\varepsilon k, AnB2\varepsilon k\rangle = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
n\sum
l=1
\nu 2l \langle B1\varepsilon k, \varepsilon l\rangle \langle B2\varepsilon k, \varepsilon l\rangle =
=
\infty \sum
l=1
\infty \sum
k=1
\nu 2l \langle \varepsilon k, B\ast
1\varepsilon l\rangle \langle \varepsilon k, B\ast
2\varepsilon l\rangle =
\infty \sum
l=1
\nu 2l \langle B\ast
1\varepsilon l, B
\ast
2\varepsilon l\rangle .
Lemma A.4. Let A \in \scrL 2 and a sequence of bounded operators Bn, n \geq 1, in \mathrm{L}2 converge
pointwise to an operator B, that is, for every \varphi \in \mathrm{L}2, Bn\varphi \rightarrow B\varphi in \mathrm{L}2 as n \rightarrow \infty . Then B is
bounded and AB\ast
n \rightarrow AB\ast in \scrL 2 as n\rightarrow \infty .
Proof. We first note that norms \| Bn\| , n \geq 1, are uniformly bounded, by the Banach – Steinhaus
theorem. Consequently, B is a bounded operator on \mathrm{L}2.
Next, we will show that \{ AB\ast
n\} n\geq 1 converges to AB\ast in the weak topology of \scrL 2. Let \{ \varepsilon n, n \geq
\geq 1\} and \{ \nu 2n, n \geq 1\} are eigenvectors and eigenvalues of A\ast A. Then for every k, l \geq 1
\langle AB\ast
n, \varepsilon k \odot \varepsilon l\rangle \scrL 2
= \langle AB\ast
n\varepsilon l, \varepsilon k\rangle = \langle \varepsilon l, BnA
\ast \varepsilon k\rangle \rightarrow
\rightarrow \langle \varepsilon l, BA\ast \varepsilon k\rangle = \langle AB\ast , \varepsilon k \odot \varepsilon l\rangle \scrL 2
as n\rightarrow \infty .
Since \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ \varepsilon k \odot \varepsilon l, k, l \geq 1\} is dense in \scrL 2 and \| AB\ast
n\| \scrL 2
\leq \| A\| \scrL 2\| B\ast
n\| , n \geq 1, is uniformly
bounded, the sequence \{ AB\ast
n\} n\geq 1 converges to AB\ast in the weak topology of \scrL 2. By the dominated
convergence theorem, the uniform boundedness of the norms of \| Bn\| , n \geq 1, and Lemma A.3, we
obtain
\| AB\ast
n\|
2
\scrL 2
=
\infty \sum
l=1
\nu 2l \| Bn\varepsilon l\| 2 \rightarrow
\infty \sum
l=1
\nu 2l \| B\varepsilon l\| 2 = \| AB\ast \| 2\scrL 2
as n\rightarrow \infty .
This implies that \{ AB\ast
n\} n\geq 1 converges to AB\ast in the strong topology of \scrL 2.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1229
Let \scrL p,sa
2 be a closed subset of \scrL 2 consisting of nonnegative definite self-adjoint operators.
Consider
\mathrm{B} (\scrL p,sa
2 ) := \{ L \in \mathrm{B} (\scrL 2) : L \in \scrL p,sa
2 a.e.\} .
Remark that a nonnegative definite self-adjoint operator A on \mathrm{L}2 has the square root, i.e., there exists
a unique nonnegative definite self-adjoint operator
\surd
A on \mathrm{L}2 such that
\Bigl( \surd
A
\Bigr) 2
= A. This trivially
follows from the spectral theorem.
Lemma A.5. (i) The set \mathrm{B} (\scrL p,sa
2 ) is closed in \mathrm{B} (\scrL 2) .
(ii) For every r > 0 the set
Sr :=
\left\{ L \in \mathrm{B} (\scrL p,sa
2 ) :
T\int
0
\bigm\| \bigm\| \bigm\| \sqrt{} Lt
\bigm\| \bigm\| \bigm\| 2
\scrL 2
dt \leq r
\right\}
is closed in \mathrm{B} (\scrL 2) .
(iii) For every r > 0 the map \Phi r : Sr \rightarrow \mathrm{B} (\scrL p,sa
2 ) defined as
\Phi r
t (L) =
\sqrt{}
Lt, t \in [0, T ], L \in Sr,
is Borel measurable.
Proof. Let Ln, n \geq 1, be a sequence from \mathrm{B} (\scrL p,sa
2 ) which converges to L in \mathrm{B} (\scrL 2) . We take
arbitrary t \in [0, T ] and \varphi ,\psi \in \mathrm{L}2 and consider
t\int
0
\langle Ls\varphi ,\psi \rangle ds =
t\int
0
\langle Ls, \psi \odot \varphi \rangle \scrL 2
ds = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\langle Ln
s , \psi \odot \varphi \rangle \scrL 2
ds =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\langle Ln
s\varphi ,\psi \rangle ds = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
t\int
0
\langle \varphi ,Ln
s\psi \rangle ds =
t\int
0
\langle \varphi ,Ls\psi \rangle ds.
Due to the density of the set \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}
\bigl\{
\BbbI [0,t]\varphi \odot \psi , t \in [0, T ], \varphi , \psi \in \mathrm{L}2
\bigr\}
in \scrL T
2 , we obtain that L is
self-adjoint a.e. Similarly, one can show that L is nonnegative definite. Hence, \mathrm{B} (\scrL p,sa
2 ) is closed.
Next we prove (ii). Take a sequence Ln, n \geq 1, from Sr which converges to L in \mathrm{B} (\scrL 2) . We
remark that L \in \mathrm{B} (\scrL p,sa
2 ) due to (i). Then
T\int
0
\bigm\| \bigm\| \bigm\| \sqrt{} Lt
\bigm\| \bigm\| \bigm\| 2
\scrL 2
dt =
T\int
0
\Biggl[ \infty \sum
k=1
\bigm\| \bigm\| \bigm\| \sqrt{} Ltek
\bigm\| \bigm\| \bigm\| 2\Biggr] dt = \infty \sum
k=1
T\int
0
\langle Ltek, ek\rangle dt \leq
\leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
T\int
0
\langle Ln
t ek, ek\rangle dt = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
T\int
0
\bigm\| \bigm\| \bigm\| \sqrt{} Ln
t
\bigm\| \bigm\| \bigm\| 2
\scrL 2
dt \leq r,
by Fatou’s lemma and the fact that
\int T
0
\langle Ln
t ek, ek\rangle \rightarrow
\int T
0
\langle Ltek, ek\rangle , n \rightarrow \infty , for all k \geq 1. Thus,
Sr is closed.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1230 V. KONAROVSKYI
In order to check (iii), we first remark that it is enough to show that, for every t \in [0, T ] and
\varphi ,\psi \in \mathrm{L}2, the map
Sr \ni L \mapsto \rightarrow
T\int
0
\bigl\langle
\Phi r
s(L), \BbbI [0,t](s)\psi \odot \varphi
\bigr\rangle
\scrL 2
ds =
t\int
0
\langle \Phi r
s(L)\varphi ,\psi \rangle ds \in \BbbR (A.2)
is Borel measurable. By Theorem 1.2 [23], the Borel \sigma -algebra on \mathrm{B} (\scrL 2) coincides with \sigma -algebra
of all Borel measurable sets of \scrL T
2 contained in the ball \mathrm{B} (\scrL 2) . Consequently, it is enough to show
that map (A.2) is Borel measurable as a map from Sr to \BbbR , where Sr is embedded with the strong
topology of \scrL T
2 . But then map (A.2)
Sr \ni L \mapsto \rightarrow
t\int
0
\langle \Phi r
s(L)\varphi ,\psi \rangle =
t\int
0
\langle Ls\varphi ,Ls\psi \rangle ds
is continuous, and, thus, Borel measurable.
Lemma A.5 is proved.
Let the basis \{ \~ek, k \geq 1\} in \mathrm{L}2 be defines as in Subsection 3.2, that is, \~e1(u) = 1, u \in [0, 1],
and \~ek(u) =
\surd
2 \mathrm{c}\mathrm{o}\mathrm{s}\pi (k - 1)u, u \in [0, 1], k \geq 2. For h \in \mathrm{L}2, we define
\.h =
\infty \sum
n=1
\langle h, \~en\rangle \~e\prime n,
if the series converges in \mathrm{L}2. We remark that \langle \.h, \varphi \rangle = - \langle h, \varphi \prime \rangle for every \varphi \in \mathrm{C}1[0, 1] with
\varphi (0) = \varphi (1) = 0.
Lemma A.6. Let h \in \mathrm{L}2 be nonnegative and \.h exist. Then \.h\BbbI \{ 0\} (h) = 0 a.e.
Proof. We consider, for every \varepsilon > 0, the function \psi \varepsilon (x) =
\surd
x2 + \varepsilon 2 - \varepsilon , x \in \BbbR . Then \psi \varepsilon
is continuously differentiable, \psi \varepsilon (0) = 0 and \psi \varepsilon (x) \rightarrow | x| as \varepsilon \rightarrow 0+ for all x \in \BbbR . Moreover,
| \psi \prime
\varepsilon (x)| \leq 1 and \psi \prime
\varepsilon (x) \rightarrow \mathrm{s}\mathrm{g}\mathrm{n}(x) for all x \in \BbbR . Take any function \varphi \in \mathrm{C}[0, 1] satisfying \varphi (0) =
= \varphi (1) = 0. By the dominated convergence theorem, it is easily seen that
\langle \psi \varepsilon (h), \varphi
\prime \rangle = - \langle \psi \prime
\varepsilon (h)
\.h, \varphi \rangle .
Making \varepsilon \rightarrow 0+, and using the nonnegativity of h, we have
- \langle \.h, \varphi \rangle = \langle h, \varphi \prime \rangle = - \langle \BbbI (0,+\infty )(h) \.h, \varphi \rangle .
Since \varphi was arbitrary, we can conclude that \.h = \.h\BbbI (0,+\infty )(h) a.e.
Lemma A.6 is proved.
Remark A.1. The same statement of Lemma A.6 remains true if the “cos” basis is replaced by
the “sin” basis \~ek =
\surd
2 \mathrm{s}\mathrm{i}\mathrm{n}\pi ku, u \in [0, 1], k \geq 1.
Acknowledgement. The author is very grateful to Prof. A. Dorogovtsev for valuable discussions
during the work on this paper.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
STICKY-REFLECTED STOCHASTIC HEAT EQUATION . . . 1231
References
1. A. Alonso, F. Brambila-Paz, Lp -continuity of conditional expectations, J. Math. Anal. and Appl., 221, No. 1, 161–176
(1998).
2. D. Aldous, Stopping times and tightness, Ann. Probab., 6, No. 2, 335 – 340 (1978).
3. S. A. Cherny, H.-J. Engelbert, Singular stochastic differential equations, Lect. Notes Math., vol. 1858, Springer-Verlag,
Berlin (2005).
4. R. Chitashvili, On the nonexistence of a strong solution in the boundary problem for a sticky Brownian motion, Proc.
A. Razmadze Math. Inst., 115, 17 – 31 (1997).
5. N. Dunford, J. T. Schwartz, Linear operators, Part I, General theory, John Wiley & Sons, Inc., New York (1988).
6. S. N. Ethier, T. G. Kurtz, Markov processes, Wiley Ser. Probab. and Math. Statist.: Probab. and Math. Statist., John
Wiley & Sons, Inc., New York (1986).
7. H.-J. Engelbert, G. Peskir, Stochastic differential equations for sticky Brownian motion, Stochastics, 86, No. 6,
993 – 1021 (2014).
8. T. Fattler, M. Grothaus, R. Vobhall, Construction and analysis of a sticky reflected distorted Brownian motion, Ann.
Inst. Henri Poincaré Probab. Stat., 52, No. 2, 735 – 762 (2016).
9. T. Funaki, S. Olla, Fluctuations for \delta \phi interface model on a wall, Stoch. Process. and Appl., 94, No. 1, 1 – 27 (2001).
10. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, extended ed., De Gruyter
Stud. Math., vol. 19, Walter de Gruyter & Co., Berlin (2011).
11. T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89, 129–193
(1983).
12. M. Grothaus, R. Vobhall, Stochastic differential equations with sticky reflection and boundary diffusion, Electron. J.
Probab., 22, Paper No. 7 (2017).
13. M. Grothaus, R. Vobhall, Strong Feller property of sticky reflected distorted Brownian motion, J. Theoret. Probab.,
31, No. 2, 827 – 852 (2018).
14. U. G. Haussmann, É. Pardoux, Stochastic variational inequalities of parabolic type, Appl. Math. and Optim., 20,
No. 2, 163 – 192 (1989).
15. N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, second ed., North-Holland Math.
Library, vol. 24, North-Holland Publ. Co., Amsterdam; Kodansha, Ltd., Tokyo (1989).
16. J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, second ed., Grundlehren Math. Wiss., vol. 288,
Springer-Verlag, Berlin (2003).
17. O. Kallenberg, Foundations of modern probability, second ed., Probab. and Appl. (N.Y.), Springer-Verlag, New York
(2002).
18. V. Konarovskyi, Coalescing-fragmentating Wasserstein dynamics: particle approach, arXiv:1711.03011 (2017).
19. I. Karatzas, A. N. Shiryaev, M. Shkolnikov, On the one-sided Tanaka equation with drift, Electron. Commun. Probab.,
16, 664 – 677 (2011).
20. R. S. Liptser, A. N. Shiryaev, Statistics of random processes, I, General theory, Appl. Math. (N.Y.), vol. 5, Springer-
Verlag, Berlin (2001).
21. Zhi Ming Ma, M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, Berlin
(1992).
22. D. Nualart, É. Pardoux, White noise driven quasilinear SPDEs with reflection, Probab. Theory and Related Fields,
93, No. 1, 77 – 89 (1992).
23. N. N. Vakhania, V. I. Tarieladze, S. A. Chobanyan, Probability distributions on Banach spaces, Math. and Appl.
(Sov. Ser.), vol. 14, D. Reidel Publ. Co., Dordrecht (1987).
24. L. Zambotti, A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge,
J. Funct. Anal., 180, No. 1, 195 – 209 (2001).
Received 21.08.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
|
| id | umjimathkievua-article-6282 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:26:55Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/78/ff122a2cdc77b844bcf071d50b430a78.pdf |
| spelling | umjimathkievua-article-62822022-03-26T11:02:08Z Sticky-reflected stochastic heat equation driven by colored noise Sticky-reflected stochastic heat equation driven by colored noise Sticky-reflected stochastic heat equation driven by colored noise Konarovskyi, V. Konarovskyi, V. Konarovskyi, V. stochastic heat equation colored noise Q-Wiener process discontinuous coefficients Sticky-reflected Brownian motion stochastic heat equation colored noise Q-Wiener process discontinuous coefficients Sticky-reflected Brownian motion UDC 519.21 We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval $[0,1]$ driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients. Мы доказываем существование решения стохастического уравнения теплопроводности на отрезке [0,1] с липким отражением, управляемое цветным шумом. Данный процесс можно интерпретировать как бесконечномерный аналог броуновского движения на действительной оси с липким отражением, но теперь решение подчиняется обычному уравнению теплопроводности за исключением точек, где он достигает нуля. В нуле шум не влияет на решение, а снос отталкивает его так, чтобы решение оставалось положительным. Доказательство основывается на новом подходе, который можно применить для других типов СДУ с частными производными и разрывными коэффициентами. УДК 519.21 Стохастичне рівняння теплопровідності з липким відбиттям, кероване кольоровим шумом Доведено існування розв'язку стохастичного рівняння теплопровідності на відрізку $[0,1]$ з липким відбиттям, керованого кольоровим шумом. Даний процес може бути інтерпретований як нескінченновимірний аналог броунівського руху на дійсній прямій із липким відбиттям, але тепер розв'язок підпорядковується звичайному стохастичному рівнянню теплопровідності за винятком точок, в яких він досягає нуля. В нулі шум не впливає на розв'язок, а перенос штовхає його так, щоб розв'язок залишався додатним. Доведення грунтується на новому підході, який може бути застосований до інших типів стохастичних диференціальних рівнянь з частинними похідними із розривними коефіцієнтами. Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6282 10.37863/umzh.v72i9.6282 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1195-1231 Український математичний журнал; Том 72 № 9 (2020); 1195-1231 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6282/8752 Copyright (c) 2020 Віталій Конаровський |
| spellingShingle | Konarovskyi, V. Konarovskyi, V. Konarovskyi, V. Sticky-reflected stochastic heat equation driven by colored noise |
| title | Sticky-reflected stochastic heat equation driven by colored noise |
| title_alt | Sticky-reflected stochastic heat equation driven by colored noise Sticky-reflected stochastic heat equation driven by colored noise |
| title_full | Sticky-reflected stochastic heat equation driven by colored noise |
| title_fullStr | Sticky-reflected stochastic heat equation driven by colored noise |
| title_full_unstemmed | Sticky-reflected stochastic heat equation driven by colored noise |
| title_short | Sticky-reflected stochastic heat equation driven by colored noise |
| title_sort | sticky-reflected stochastic heat equation driven by colored noise |
| topic_facet | stochastic heat equation colored noise Q-Wiener process discontinuous coefficients Sticky-reflected Brownian motion stochastic heat equation colored noise Q-Wiener process discontinuous coefficients Sticky-reflected Brownian motion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6282 |
| work_keys_str_mv | AT konarovskyiv stickyreflectedstochasticheatequationdrivenbycolorednoise AT konarovskyiv stickyreflectedstochasticheatequationdrivenbycolorednoise AT konarovskyiv stickyreflectedstochasticheatequationdrivenbycolorednoise |