On a Brownian motion conditioned to stay in an open set
UDC 519.21 Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied t...
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512321048150016 |
|---|---|
| author | Riabov, G. V. Рябов, Георгий Валентинович Riabov , G. V. |
| author_facet | Riabov, G. V. Рябов, Георгий Валентинович Riabov , G. V. |
| author_sort | Riabov, G. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:02:07Z |
| description | UDC 519.21
Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on $\mathbb{R}.$ |
| doi_str_mv | 10.37863/umzh.v72i9.6281 |
| first_indexed | 2026-03-24T03:26:55Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i9.6281
UDC 519.21
G. V. Riabov (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET
ПРО УМОВНИЙ РОЗПОДIЛ БРОУНIВСЬКОГО РУХУ,
ЩО НЕ ВИХОДИТЬ З ВIДКРИТОЇ МНОЖИНИ
Distribution of a Brownian motion conditioned to start from the boundary of an open set G and to stay in G for a finite
period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations
are obtained. Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on \BbbR .
Дослiджується розподiл броунiвського руху, що стартував iз граничної точки вiдкритої множини G i залишається в
G протягом скiнченного iнтервалу часу. Отримано характеризацiю таких розподiлiв у термiнах певних сингулярних
стохастичних диференцiальних рiвнянь. Отриманi результати застосовано до вивчення меж кластерiв у деяких
стохастичних потоках зi склеюванням на \BbbR .
1. Introduction. Let B = \{ B(t)\} t\in [0,T ] be a standard \BbbR d-valued Brownian motion. Given an open
set G \subset \BbbR d denote by \tau G = \mathrm{i}\mathrm{n}\mathrm{f}\{ t > 0 : B(t) \not \in G\} the first exit time of B from the set G. In this
paper we study the distribution of B conditioned on the event \{ \tau G > T\} , where T > 0 is a fixed
positive time. Denote this distribution by \nu x,T (\cdot ;G), where B(0) = x is the starting point. Let \scrC dT
be the space of continuous functions w : [0, T ] \rightarrow \BbbR d endowed with the sup-norm and a Borelian
\sigma -field \scrB (\scrC dT ). Then
\nu x,T (\Delta ;G) = \BbbP
\bigl(
B \in \Delta | B(0) = x, \tau G > T
\bigr)
, \Delta \in \scrB (\scrC dT ).
The measure \nu x,T is not well-defined when x \not \in G, as the event \{ B(0) = x, \tau G > T\} can be of
probability zero. However, if the set G is sufficiently regular and x is a boundary point of G, the
measure \nu x,T is well-defined as a weak limit [1] (Theorem 4.1)
\nu x,T (\cdot ;G) = \mathrm{l}\mathrm{i}\mathrm{m}
y\rightarrow x,y\in G
\nu y,T (\cdot ;G).
In the paper, we characterize the measure \nu x,T (\cdot ;G) in terms of a singular SDE. Precisely, introduce
the function
\gamma G(t, y) = \BbbP
\bigl(
\tau G > t | B(0) = y
\bigr)
, t > 0, y \in G, (1.1)
and consider the following problem:
dY (t) = \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma G
\bigl(
T - t, Y (t)
\bigr)
dt+ dW (t),
Y (0) = x,
Y (t) \in G for a.a. t \in (0, T ),
(1.2)
where W is a standard Brownian motion in \BbbR d. The main result of the paper is the following
theorem.
Theorem 1.1. Let G \subset \BbbR d be an open convex set, x \in \partial G, and the boundary of G is C2 in
the neighborhood of x. Then the problem (1.2) has a unique strong solution. The distribution of this
solution coincides with \nu x,T (\cdot , G).
c\bigcirc G. V. RIABOV, 2020
1286 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1287
The result was motivated by the study of coalescing stochastic flows on the real line. By a
coalescing stochastic flow on the real line we understand a family \{ \psi s,t : - \infty < s \leq t < \infty \} of
measurable random mappings of \BbbR such that:
1) for all r \leq s \leq t, x \in \BbbR , \omega \in \Omega
\psi s,t
\bigl(
\omega , \psi r,s(\omega , x)
\bigr)
= \psi r,t(\omega , x)
and \psi s,s(\omega , x) = x;
2) for all t1 \leq . . . \leq tn, x1, . . . , xm \in \BbbR random vectors\bigl(
\psi t1,t2(x1), . . . , \psi t1,t2(xm)
\bigr)
, . . . ,
\bigl(
\psi tn - 1,tn(x1), . . . , \psi tn - 1,tn(xm)
\bigr)
are independent;
3) for all s \leq t, h \in \BbbR , x1, . . . , xm \in \BbbR random vectors\bigl(
\psi s,t(x1), . . . , \psi s,t(xm)
\bigr)
and
\bigl(
\psi s+h,t+h(x1), . . . , \psi s+h,t+h(xm)
\bigr)
are equally distributed;
4) for all s, x \in \BbbR , \omega \in \Omega , functions
t\rightarrow \psi s,t(x, \omega ), t \geq s,
are continuous;
5) there exist x \not = y such that
\BbbP
\bigl(
\exists t > 0 : \psi 0,t(x) = \psi 0,t(y)
\bigr)
> 0.
With a stochastic flow \psi we associate the family of \sigma -fields
\scrF \psi
s,t = \sigma
\bigl( \bigl\{
\psi u,v(x) : s \leq u \leq v \leq t, x \in \BbbR
\bigr\} \bigr)
, s \leq t.
For general properties of stochastis flows we refer to [2]. In our previous works [3 – 5] properties of
clusters in certain coalescing stochastic flows were investigated. To illustrate the results and related
questions, let us consider the Arratia flow on \BbbR . A stochastic flow \{ \psi s,t : - \infty < s \leq t < \infty \} is
called the Arratia flow, if for all s \in \BbbR , n \geq 1 and x = (x1, . . . , xn) \in \BbbR n processes
Wj(t) = \psi s,s+t(xj), t \geq 0, 1 \leq j \leq n,
are (\scrF \psi
s,s+t)t\geq 0-Brownian motions with joint quadratic variation given by
\langle Wi,Wj\rangle (t) = (t - \tau ij)+, \tau ij = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : Wi(t) =Wj(t)
\bigr\}
.
Informally, the Arratia flow describes the joint motion of a continuum family of stochastic processes
that start at every moment of time from every point of the real line, each process is a standard
Brownian motion, every two trajectories move independently before they meet each other, at the
meeting time trajectories coalesce into one Brownian motion. For the existence of the Arratia flow
and its properties we refer to [2, 3, 6 – 8]. For fixed s < t consider the random mapping \psi s,t :
\BbbR \times \Omega \rightarrow \BbbR from the Arratia flow. With probability 1 it is an increasing piecewise constant function
[6]. The distribution of its range \psi s,t(\BbbR ) as a point process on the real line was described in [9].
Consider a point \zeta \in \psi 0,T (\BbbR ). At every time t \in [0, T ] there exists a segment of points that have
coalesced into \zeta at time T :
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1288 G. V. RIABOV
K\zeta (t) =
\bigl\{
x \in \BbbR : \psi T - t,T (x) = \zeta
\bigr\}
, 0 \leq t \leq T.
We refer to the set K\zeta = \cup t\in [0,T ](\{ T - t\} \times K\zeta (t)) as to the cluster with the vertex \zeta . For fixed
t \in [0, T ] the family \{ K\zeta (t) : \zeta \in \psi 0,T (\BbbR )\} is a partition of \BbbR . Given a segment [a, b] let Nt(a, b)
denote the number of nonempty intersections K\zeta (t) \cap [a, b]. The distribution of Nt(a, b) was found
in [10]. We are interested in the distribution of boundary processes
\alpha \zeta (t) = \mathrm{i}\mathrm{n}\mathrm{f}K\zeta (t), \beta \zeta (t) = \mathrm{s}\mathrm{u}\mathrm{p}K\zeta (t).
In different terms,
\bigl(
\alpha \zeta (t), \beta \zeta (t)
\bigr)
is the largest open interval, where \psi T - t,T (x) = \zeta . Hence the
distribution of boundary processes is needed in order to describe the distribution of a random mapping
\psi s,t completely. Our method allows to characterize the distribution of the pair (\alpha \zeta , \beta \zeta ) as is given
in the end of Subsection 4.1.
The conditional distribution of boundary processes needs to be defined rigorously, as the event
\{ \zeta = x\} is of probability zero. This is done in Section 4 using duality theory for the Arratia flow.
Also in Section 4, we consider Arratia flows with drift. Let a : \BbbR \rightarrow \BbbR be a Lipschitz function.
The Arratia flow with drift a is a stochastic flow \psi such that each trajectory t \rightarrow \psi s,t(x) is a weak
solution of the SDE
d\psi s,t(x) = a
\bigl(
\psi s,t(x)
\bigr)
dt+ dws,x(t),
every two trajectories move independently before they meet each other, at the meeting time tra-
jectories coalesce (see Subsection 4.2 for the precise definition). In [5] it was proved that if
a\prime (x) \leq - \lambda < 0 a.s., then there exists a unique stationary process \{ \eta t\} t\in \BbbR such that, for all s \leq t,
\psi s,t(\eta s) = \eta t. At every moment t \geq 0 there exists an interval of points that have coalesced into \eta 0
at time 0:
K0(t) =
\bigl\{
x \in \BbbR : \psi - t,0(x) = \eta 0
\bigr\}
, t \geq 0.
The set K0 = \cup t\geq 0
\bigl(
\{ - t\} \times K0(t)
\bigr)
will be called the infinite cluster with the vertex \eta 0. The
Theorem 4.2 (Subsection 4.2) describes the conditional distribution of processes \alpha 0(t) = \mathrm{i}\mathrm{n}\mathrm{f}K0(t),
\beta 0(t) = \mathrm{s}\mathrm{u}\mathrm{p}K0(t) conditioned on the event \{ \eta 0 = x\} .
The paper is organized as follows. Our approach is based on a carefull analysis of a Brownian
meander — a particular case of Theorem 1.1, that corresponds to d = 1, G = (0,\infty ), x = 0. As
a corollary, we recover the result of [11] on the mutual equivalence between the distribution of the
Brownian meander and the distribution of the three-dimensional Bessel process. In Section 3, we
prove Theorem 1.1 in full generality, by adapting the approach of [1]. Finally, in Section 4, we apply
the result to the distribution of boundaries of clusters in the Arratia flow, and obtain analogous results
for an unbounded cluster in the Arratia flow with drift [5].
2. Brownian meander. Let Px be the Wiener measure on \scrC 1
T , i.e., the distribution of an
\BbbR -valued Brownian motion B = \{ B(t)\} t\in [0,T ] conditioned to start from x \in \BbbR . Expectation with
respect to the measure Px will be denoted by Ex. Denote \BbbR + = (0,\infty ). By the distribution of the
Brownian meander we understand the measure \nu 0,T (\cdot ,\BbbR +). Informally, it is the restriction of the
Wiener measure P0 to the set of trajectories
A =
\bigl\{
w \in \scrC 1
T : w(t) > 0, 0 < t \leq T
\bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1289
As it was mentioned in the Introduction, \nu 0,T (\cdot ,\BbbR +) is rigorously defined as a weak limit [12]
(Theorem 2.1)
\nu 0,T (\cdot ,\BbbR +) = \mathrm{l}\mathrm{i}\mathrm{m}
y\rightarrow 0+
\nu y,T (\cdot ,\BbbR +),
where now \nu y,T (\Delta ,\BbbR +) =
Py(\Delta \cap A)
Py(A)
. Introduce the function
\gamma \BbbR +(t, y) = Py
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
s\in [0,t]
w(s) > 0
\Bigr)
, t > 0, y > 0.
Precisely,
\gamma \BbbR +(t, y) =
\sqrt{}
2
\pi
y\surd
t\int
0
e -
z2
2 dz. (2.1)
Consider the following problem:
dY (t) = \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - t, Y (t)
\bigr)
dt+ dW (t),
Y (0) = 0,
Y (t) > 0 for a.a. t \in (0, T ),
(2.2)
where W is a standard \BbbR -valued Brownian motion.
Theorem 2.1. The problem (2.2) has a unique strong solution. The distribution of this solution
coincides with the distribution of the Brownian meander \nu 0,T (\cdot ,\BbbR +).
Proof. For a fixed y > 0 the measure \nu y,T (\cdot ,\BbbR +) is absolutely continuous with respect to the
Wiener measure Py. The corresponding Radon – Nikodym density is
d\nu y,T (\cdot ,\BbbR +)
dPy
=
1mint\in [0,T ] w(t)>0
\gamma \BbbR +(T, y)
.
We will apply the Girsanov theorem to the measure \nu y,T (\cdot ,\BbbR +). Let (\scrF t)t\in [0,T ] be the canonical
filtration on the space \scrC 1
T . We introduce the martingale associated with the Radon – Nikodym density
d\nu y,T (\cdot ,\BbbR +)
dPy
:
\rho t = Ey
\biggl(
d\nu y,T (\cdot ,\BbbR +)
dPy
\bigm| \bigm| \bigm| \bigm| \scrF t\biggr) .
By the Markov property,
\rho t =
Py
\bigl(
\mathrm{m}\mathrm{i}\mathrm{n}s\in [0,T ]w(s) > 0 | \scrF t
\bigr)
\gamma \BbbR +(T, y)
=
1mins\in [0,t] w(s)>0\gamma \BbbR +
\bigl(
T - t, w(t)
\bigr)
\gamma \BbbR +(T, y)
Py-a.s.
The Clark representation for the density equals [13] (Lemma 1)
\rho T = 1 +
T\int
0
1mins\in [0,t] w(s)>0
\partial y\gamma \BbbR +
\bigl(
T - t, w(t)
\bigr)
\gamma \BbbR +(T, y)
dw(t) Py-a.s. (2.3)
Since similar results will be used several times in the paper, we give a proof of (2.3).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1290 G. V. RIABOV
Recall that the function \gamma \BbbR +(t, y) satisfies the heat equation
\partial t\gamma \BbbR +(t, y) =
1
2
\partial 2y\gamma \BbbR +(t, y), t, y > 0.
Let \sigma = \mathrm{i}\mathrm{n}\mathrm{f}\{ t \geq 0 : w(t) = 0\} . Applying the Itô formula to the process
t\rightarrow \gamma \BbbR +
\bigl(
T - t \wedge \sigma ,w(t \wedge \sigma )
\bigr)
, t \geq 0,
we get
\gamma \BbbR +
\bigl(
T - T \wedge \sigma ,w(T \wedge \sigma )
\bigr)
= \gamma \BbbR +(T, y) +
T\wedge \sigma \int
0
\partial y\gamma \BbbR +
\bigl(
T - t, w(t)
\bigr)
dw(t).
Observe that
\gamma \BbbR +
\bigl(
T - T \wedge \sigma ,w(T \wedge \sigma )
\bigr)
=
\left\{ \gamma \BbbR +(T - \sigma ,w(\sigma )) = 0, \sigma < T,
\gamma \BbbR +(0, w(T )) = 1, \sigma > T.
Consequently,
1mins\in [0,T ] w(s)>0 = 1\sigma >T = \gamma \BbbR +(T, y) +
T\wedge \sigma \int
0
\partial y\gamma \BbbR +
\bigl(
T - t, w(t)
\bigr)
dw(t).
Dividing by \gamma \BbbR +(T, y) we recover (2.3).
Let us denote ht = 1mins\in [0,t] w(s)>0
\partial y\gamma \BbbR +
\bigl(
T - t, w(t)
\bigr)
\gamma \BbbR +(T, y)
, so that \rho T = 1+
\int T
0
ht dw(t). By the
Girsanov theorem [14] (Theorem 1.12, Ch. VIII) under the measure \nu y,T (\cdot ,\BbbR +) the process
By(t) = w(t) -
t\int
0
hs
\rho s
ds, 0 \leq t \leq T,
is a Brownian motion. Observe that 1mins\in [0,T ] w(s)>0 = 1 a.s. with respect to the measure \nu y,T (\cdot ,\BbbR +).
Hence,
hs
\rho s
=
\partial y\gamma \BbbR +(T - s, w(s))
\gamma \BbbR +(T - s, w(s))
= \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, w(s)) \nu y,T (\cdot ,\BbbR +)-a.s.,
and under the measure \nu y,T (\cdot ,\BbbR +) the process
By(t) = w(t) -
t\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, w(s))ds, 0 \leq t \leq T,
is a Brownian motion. Redenoting w with Yy we can reformulate the conclusion as follows: for
every y > 0 on some probability space there is a pair of processes (Yy, By) such that
\{ By(t)\} t\in [0,T ] is a Brownian motion with the starting point By(0) = y;
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1291
the distribution of \{ Yy(t)\} t\in [0,T ] is \nu y,T (\cdot ,\BbbR +);
for all t \in [0, T ], Yy(t) > 0;
for all t \in [0, T ],
Yy(t) =
t\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yy(s)
\bigr)
ds+By(t). (2.4)
By [12] (Theorem 2.1) Yy
d - \rightarrow \nu 0,T (\cdot ,\BbbR +). Hence, the family of processes
\bigl\{
(Yy, By) : y \in (0, 1]
\bigr\}
is weakly relatively compact. Applying the Skorokhod theorem [15] (Theorem 4.30) we can construct
a sequence yn \rightarrow 0 and copies of processes
\bigl\{
(Yyn , Byn) : n \geq 1
\bigr\}
defined on the same probability
space such that
(Yyn , Byn) \rightarrow (Y0, B0) a.s. in \scrC T (\BbbR 2).
We will check that
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
\rightarrow \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Y0(s))ds in L1(\Omega \times [0, T ]).
To prove this convergence we will use Scheffé’s lemma [16]. The lemma can be applied since
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(t, y) > 0 for t, y > 0. Thus, it is enough to show
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\BbbE
T\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds = \BbbE
T\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Y0(s)
\bigr)
ds <\infty . (2.5)
Next two results allow to control the behavior of integrals in (2.5) near boundaries.
Lemma 2.1. For each t \in (0, T ),
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\BbbE
t\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds =
\surd
T
\infty \int
0
y2\gamma \BbbR +
\bigl(
T - t,
\surd
ty
\bigr)
e - y
2/2dy.
Expression on the right-hand side is a continuous function of t \in [0, T ].
Proof. We make use of the relation (2.4):
\BbbE
t\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds = \BbbE Yyn(t) - \BbbE Byn(t) = \BbbE Yyn(t) - yn.
Further,
\BbbE Yyn(t) = \BbbE
\biggl(
Byn(t)
\bigm| \bigm| \bigm| \bigm| \mathrm{m}\mathrm{i}\mathrm{n}
s\in [0,T ]
Byn(s) > 0
\biggr)
=
\BbbE Byn(t)1mins\in [0,T ]Byn (s)>0
\gamma \BbbR +(T, yn)
=
=
\BbbE Byn(t)1mins\in [0,t]Byn (s)>0\gamma \BbbR +(T - t, Byn(t))
\gamma \BbbR +(T, yn)
=
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1292 G. V. RIABOV
=
\int \infty
0
y\gamma \BbbR +(T - t, y)
1\surd
2\pi t
e -
(y - yn)2
2t (1 - e -
2yyn
t )dy\int \infty
0
1\surd
2\pi T
e -
(y - yn)2
2T
\bigl(
1 - e -
2yyn
T
\bigr)
dy
=
=
\int \infty
0
y\gamma \BbbR +(T - t, y)
1\surd
2\pi t
e -
(y - yn)2
2t
1 - e -
2yyn
t
2yn
dy\int \infty
0
1\surd
2\pi T
e -
(y - yn)2
2T
1 - e -
2yyn
T
2yn
dy
.
Hence, by the Dominated Convergence Theorem, we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\BbbE Yyn(t) =
\int \infty
0
y2\gamma \BbbR +(T - t, y)t - 3/2e -
y2
2t dy\int \infty
0
yT - 3/2e -
y2
2T dy
=
\surd
T
\infty \int
0
y2\gamma \BbbR +
\bigl(
T - t,
\surd
ty
\bigr)
e -
y2
2 dy.
Lemma 2.1 is proved.
Applying Dini’s theorem, we deduce the corollary from the Lemma 2.1.
Corollary 2.1. Functions fn(t) = \BbbE
\int t
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds, 0 \leq t \leq T, are equicon-
tinuous on [0, T ]. In particular,
\mathrm{l}\mathrm{i}\mathrm{m}
\delta \rightarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 1
\left( \BbbE
\delta \int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds+ \BbbE
T\int
T - \delta
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds
\right) = 0.
Now we return to the proof of the Theorem 2.1. By Corollary 2.1 it is enough to check the
convergence
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\BbbE
T - \delta \int
\delta
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr)
ds = \BbbE
T - \delta \int
\delta
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Y0(s)
\bigr)
ds
for any \delta \in (0, T ). This in turn will follow from the uniform integrability condition [15] (Ch. 4)
\mathrm{s}\mathrm{u}\mathrm{p}
n\geq 1
\BbbE
T - \delta \int
\delta
\bigl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr) \bigr) 3/2
ds <\infty . (2.6)
In order to verify (2.6) we make use of the estimate
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(t, y) =
e -
y2
2t
\surd
t
\int y/
\surd
t
0
e - u
2/2du
\leq 1
y
, y > 0, t > 0.
We get following inequalities:
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ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1293
\BbbE
T - \delta \int
\delta
\bigl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Yyn(s)
\bigr) \bigr) 3/2
ds \leq
T - \delta \int
\delta
\BbbE (Yyn(s)) - 3/2ds =
=
T - \delta \int
\delta
\BbbE (Byn(s)) - 3/21minr\in [0,s]Byn (r)>0\gamma \BbbR +(T - s,Byn(s))
\gamma \BbbR +(T, yn)
ds =
=
T - \delta \int
\delta
\int \infty
0
y - 3/2\gamma \BbbR +(T - s, y)
1\surd
2\pi s
e -
(y - yn)2
2s (1 - e -
2yyn
s )dy\int \infty
0
1\surd
2\pi T
e -
(y - yn)2
2T
\bigl(
1 - e -
2yyn
T
\bigr)
dy
ds \leq
\leq (T - 2\delta )
\sqrt{}
T
\delta
\int \infty
0
y - 3/2e
- (y - yn)2
2(T - \delta ) (1 - e -
2yyn
\delta )dy\int \infty
0
e -
(y - yn)2
2T
\bigl(
1 - e -
2yyn
T
\bigr)
dy
- - - \rightarrow
n\rightarrow \infty
- - - \rightarrow
n\rightarrow \infty
(T - 2\delta )
\biggl(
T
\delta
\biggr) 3/2
\int \infty
0
y - 1/2e
- y2
2(T - \delta )dy\int \infty
0
y e -
y2
2T
dy
.
This proves (2.6). Passing to the limit in (2.4), we get the relation
Y0(t) =
t\int
0
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - s, Y0(s)
\bigr)
ds+B0(t).
The weak existence for the problem (2.2) is proved. We prove the existence and uniqueness of the
strong solution using the Yamada – Watanabe theorem [14] (Theorem 1.7, Ch. IX). Let Y and \~Y
solve (2.2). Then, for almost all t \in (0, T ),
1
2
\partial t(Y (t) - \~Y (t))2 =
\Bigl(
Y (t) - \~Y (t)
\Bigr) \Bigl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - t, Y (t)
\bigr)
- \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - t, \~Y (t))
\Bigr)
\leq 0,
since the function y \rightarrow \gamma \BbbR +(T - t, y) is log-concave. It follows that Y (t) = \~Y (t) for all t \in [0, T ].
The pathwise uniqueness of the problem (2.2) is proved.
Theorem 2.1 is proved.
Next we derive two corollaries of the theorem. The first one is a straightforward generalization
to the multidimensional case.
Corollary 2.2. Let x \in \BbbR d be arbitrary, e \in \BbbR d be a unit vector, and H =
\bigl\{
y \in \BbbR d : (y - x)\cdot e >
> 0
\bigr\}
. The statement of the Theorem 1.1 holds for G = H and x.
In the next corollary we give a new proof of the known theorem on the equivalence between the
distribution \nu 0,T (\cdot ,\BbbR +) of the Brownian meander and the distribution Q of the three-dimensional
Bessel process. We recall that the three-dimensional Bessel process is defined as the process t \rightarrow
\rightarrow
\sqrt{}
B2
1(t) +B2
2(t) +B2
3(t), where B1, B2, B3 are independent \BbbR -valued Brownian motions
started at zero. Consider the problem
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1294 G. V. RIABOV
dZ(t) =
1
Z(t)
dt+ dW (t),
Z(0) = 0,
Z(t) > 0, t > 0,
(2.7)
where W is a standard \BbbR -valued Brownian motion. This problem has a unique strong solution [17],
and its distribution coincides with Q. By QT we denote the distribution of the process \{ Z(t)\} t\in [0,T ]
in \scrC 1
T .
Corollary 2.3 [11]. The measure \nu 0,T (\cdot ,\BbbR +) is equivalent to the distribution QT of the three-
dimensional Bessel process started at 0. The Radon – Nikodym density is given by
d\nu 0,T (\cdot ,\BbbR +)
dQt
(Z) =
\surd
\pi T\surd
2Z(T )
.
Proof. The idea of the proof is to change the underlying probability measure QT in order to
convert the problem (2.7) to the problem (2.2). A natural candidate for the density is given by the
Girsanov theorem:
\rho = \mathrm{e}\mathrm{x}\mathrm{p}
\left( T\int
0
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Z(s)) - 1
Z(s)
\biggr)
dW (s) -
- 1
2
T\int
0
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Z(s)) - 1
Z(s)
\biggr) 2
ds
\right) .
Because of singularities as s\rightarrow 0 and s\rightarrow T it is not obvious that \rho is well-defined and is a density.
From (2.1), we have
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(s, y) =
e -
y2
2s
\surd
s
\int y\surd
s
0
e -
u2
2 du
.
Elementary inequalities
0 \leq 1 - ye -
y2
2\int y
0
e -
u2
2 du
\leq y2
2
imply that the process
X(t) = \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - t, Z(t)
\bigr)
- 1
Z(t)
, 0 \leq t < T,
satisfies
| X(t)| = 1
Z(t)
\left( 1 -
Z(t)\surd
T - t
e
- Z(t)2
2(T - t)
\int Z(t)\surd
T - t
0
e -
u2
2 du
\right) \leq
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ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1295
\leq 1
Z(t)
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl(
1,
Z(t)2
2(T - t)
\biggr)
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1,
1
2(T - t)
\biggr)
, 0 \leq t < T.
In particular, there is no singularity as s \rightarrow 0 in the definition of \rho . To deal with the singularity as
s\rightarrow T we consider the process
\rho t = \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Z(s)) - 1
Z(s)
\biggr)
dW (s) -
- 1
2
t\int
0
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Z(s)) - 1
Z(s)
\biggr) 2
ds
\right) .
Since Novikov’s condition [14] (Proposition 1.15, Ch. VIII) holds for the process X, the process
(\rho t)0\leq t<T is a martingale. Let us show that (\rho t)0\leq t<T is a uniformly integrable martingale, with
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow T
\rho t =
\surd
\pi T\surd
2Z(T )
.
To this end consider the function
b(t, y) = \mathrm{l}\mathrm{o}\mathrm{g}
\left(
y\surd
T - t\int
0
e -
u2
2 du
\right) - \mathrm{l}\mathrm{o}\mathrm{g}(y).
It has the following limit values:
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0,y\rightarrow 0
b(t, y) = - \mathrm{l}\mathrm{o}\mathrm{g}
\surd
T , \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow T,y\rightarrow z
b(t, y) = \mathrm{l}\mathrm{o}\mathrm{g}
\sqrt{}
\pi
2
- \mathrm{l}\mathrm{o}\mathrm{g} z, (2.8)
where z > 0 is arbitrary. Further, we have
\partial tb(t, y) =
ye
- y2
2(T - t)
2(T - t)
3
2
\int y\surd
T - t
e -
u2
2 du
,
\partial yb(t, y) =
e
- y2
2(T - t)
\surd
T - t
\int y\surd
T - t
0
e -
u2
2 du
- 1
y
,
\partial 2yb(t, y) = - ye
- y2
2(T - t)
(T - t)
3
2
\int y\surd
T - t
0
e -
u2
2 du
- e -
y2
T - t
(T - t)
\Biggl( \int y\surd
T - t
0
e -
u2
2 du
\Biggr) 2 +
1
y2
.
By the Itô formula, we obtain
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1296 G. V. RIABOV
db (t, Z(t)) =
Z(t)e
- Z(t)2
2(T - t)
2(T - t)
3
2
\int Z(t)\surd
T - t
e -
u2
2 du
dt+
+
\left( e
- Z(t)2
2(T - t)
\surd
T - t
\int Z(t)\surd
T - t
0
e -
u2
2 du
- 1
Z(t)
\right)
\biggl(
1
Z(t)
dt+ dW (t)
\biggr)
+
+
1
2
\left( - Z(t)e
- Z(t)2
2(T - t)
(T - t)
3
2
\int Z(t)\surd
T - t
0
e -
u2
2 du
- e -
Z(t)2
T - t
(T - t)
\Biggl( \int Z(t)\surd
T - t
0
e -
u2
2 du
\Biggr) 2 +
1
Z(t)2
\right) dt =
=
\left( e
- Z(t)2
2(T - t)
\surd
T - t
\int Z(t)\surd
T - t
0
e -
u2
2 du
- 1
Z(t)
\right) dW (t) - 1
2
\left( e
- Z(t)2
2(T - t)
\surd
T - t
\int Z(t)\surd
T - t
0
e -
u2
2 du
- 1
Z(t)
\right)
2
dt.
By (2.8),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0
b (t, Z(t)) = - \mathrm{l}\mathrm{o}\mathrm{g}
\surd
T , \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow T
b (t, Z(t)) = \mathrm{l}\mathrm{o}\mathrm{g}
\sqrt{}
\pi
2
- \mathrm{l}\mathrm{o}\mathrm{g}Z(T ).
Hence,
\rho t = \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl(
b(t, Z(t)) + \mathrm{l}\mathrm{o}\mathrm{g}
\surd
T
\Bigr)
\rightarrow
\surd
\pi T\surd
2Z(T )
, t\rightarrow T.
By the Girsanov theorem, under the measure \rho dQT the process
\~W (t) =W (t) -
t\int
0
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +(T - s, Z(s)) - 1
Z(s)
\biggr)
ds, 0 \leq t < T,
is a Brownian motion. Hence, under the measure \rho dQT , the process \{ Z(t)\} 0\leq t\leq T is a solution of
the SDE
dZ(t) =
1
Z(t)
dt+ d \~W (t) +
\biggl(
\partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - t, Z(t)
\bigr)
- 1
Z(t)
\biggr)
dt =
= \partial y \mathrm{l}\mathrm{o}\mathrm{g} \gamma \BbbR +
\bigl(
T - t, Z(t)
\bigr)
dt+ d \~W (t),
and, thus, is a Brownian meander.
Corollary 2.3 is proved.
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ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1297
3. Proof of Theorem 1.1. Given an open set A \subset \BbbR d and a continuous function f \in \scrC dT we
will denote by \tau A(f) the first exit time
\tau A(f) = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t > 0 : f(t) \not \in A
\bigr\}
.
We recall that the set G is assumed to be convex with a C2 boundary in the neighborhood of its
boundary point x. Let us choose a unit vector e \in \BbbR d and r > 0 such that B(x + re, r) \subset G.
Consider the half-space
H =
\bigl\{
y \in \BbbR d : (y - x) \cdot e > 0
\bigr\}
,
so that
B(x+ re, r) \subset G \subset H.
Consider an auxiliary measure \nu x,T (\cdot ;H) (see Corollary 2.3). The corresponding process can be
described as follows. Choose an orthonormal basis
\bigl\{
e1, . . . , ed
\bigr\}
in \BbbR d, such that e1 = e. Let
\{ \~Y1(t)\} 0\leq t\leq T be a Brownian meander, and
\bigl\{
( \~W2(t), . . . , \~Wd(t))
\bigr\}
0\leq t\leq T be a \BbbR d - 1-valued Brownian
motion independent from \~Y1 . Then \nu x,T (\cdot ;H) is the distribution of the process
\Bigl\{
x + \~Y1(t)e1 +
+
\sum d
i=2
\~Wi(t)ei
\Bigr\}
0\leq t\leq T
.
By the Corollary 2.2 \nu x,T (\cdot ;H) is the distribution of the solution of the problem
dY (t) = \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma H
\bigl(
T - t, Y (t)
\bigr)
dt+ dW (t),
Y (0) = x,
Y (t) \in H for a.a. t \in (0, T ),
(3.1)
where W is an \BbbR d-valued Brownian motion. By Corollary 2.3 the measure \nu x,T (\cdot ;H) is equiva-
lent to the distribution of the process
\Bigl\{
x + \~Z1(t)e1 +
\sum d
i=2
\~Wi(t)ei
\Bigr\}
0\leq t\leq T
, where \{ \~Z1(t)\} t\geq 0 is
a three-dimensional Bessel process independent from
\bigl\{
( \~W2(t), . . . , \~Wd(t))
\bigr\}
0\leq t\leq T . Applying [18]
(Theorem 3.4), we deduce
\nu x,T
\bigl(
\tau B(x+re,r)(Y ) > 0;H
\bigr)
= 1.
Consequently,
\nu x,T
\bigl( \bigl\{
\tau G(Y ) > T
\bigr\}
;H
\bigr)
> 0,
and we can represent the measure \nu x,T (\cdot ;G) via the density with respect to the measure \nu x,T (\cdot ;H)
(see [1] for the details):
d\nu x,T (\cdot ;G)
d\nu x,T (\cdot ;H)
=
1\tau G(Y )>T
\nu x,T
\bigl(
\{ \tau G(Y ) > T\} ;H
\bigr) .
Let us apply the Girsanov theorem to this density. Introduce the function
\theta (t, y) = \BbbP
\bigl(
\forall r \in [t, T ] Y (r) \in G | Y (t) = y
\bigr)
=
\gamma G(T - t, y)
\gamma H(T - t, y)
, y \in G, 0 \leq t < T.
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1298 G. V. RIABOV
As in the proof of Theorem 2.1, an application of the Itô formula implies the Clark representation
1\tau G(Y )>T = \theta (0, x) +
T\int
0
1\tau G(Y )>s
\bigl(
\nabla y\theta (s, Y (s)), dW (s)
\bigr)
.
By the Markov property, we have
\BbbE [1\tau G(Y )>T | \scrF s] = 1\tau G(Y )>s\theta (s, Y (s)).
Repeating arguments of the Theorem 2.1, under the measure \nu x,T (\cdot ;G) the process
\~W (t) =W (t) -
t\int
0
\nabla y \mathrm{l}\mathrm{o}\mathrm{g} \theta (s, Y (s)) ds, 0 \leq t \leq T,
is a Brownian motion. From (3.1), we deduce that under the measure \nu x(\cdot ;G) the process Y satisfies
the equation
dY (t) = \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma H
\bigl(
T - t, Y (t)
\bigr)
dt+\nabla y \mathrm{l}\mathrm{o}\mathrm{g} \theta
\bigl(
t, Y (t)
\bigr)
dt+ d \~W (t) =
= \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma G
\bigl(
T - t, Y (t)
\bigr)
dt+ \~W (t).
It remains to check pathwise uniqueness for the problem (1.2). Let Y and \~Y solve (1.2). Then
1
2
\partial t
\bigm| \bigm| Y (t) - \~Y (t)
\bigm| \bigm| 2 = \Bigl( Y (t) - \~Y (t),\nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma G
\bigl(
T - t, Y (t)
\bigr)
- \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma G(T - t, \~Y (t))
\Bigr)
\leq 0,
where the last inequality follows from log-concavity of the function y \rightarrow \gamma G(T - t, y) [19].
Theorem 1.1 is proved.
4. Clusters in coalescing stochastic flows. 4.1. Arratia flow. In this section, we will use
duality theory for coalescing stochastic flows on the real line developed in [4]. By a backward
stochastic flow we will understand a family
\bigl\{
\phi t,s : - \infty < s \leq t < \infty
\bigr\}
of measurable random
mappings of \BbbR , such that the family
\bigl\{
\^\phi s,t = \phi - s, - t : - \infty < s \leq t < \infty
\bigr\}
is a stochastic flow. Let
\psi =
\bigl\{
\psi s,t : - \infty < s \leq t < \infty
\bigr\}
be the Arratia flow. A dual flow \~\psi =
\bigl\{
\~\psi t,s : - \infty < s \leq t < \infty
\bigr\}
is defined as a backward stochastic flow whose trajectories do not cross trajectories of the flow \psi ,
i.e., for all s \leq t, x, y \in \BbbR and \omega \in \Omega \bigl(
\psi s,t(\omega , x) - y
\bigr) \bigl(
x - \~\psi t,s(\omega , y)
\bigr)
\geq 0.
For the needed properties of the Arratia flow as well as for existence and properties of its dual we
refer to [3, 4]. In particular, we recall that the dual \~\psi of the Arratia flow \psi is itself the Arratia flow
(with time reversed). As it was mentioned in the Introduction, the image \psi 0,T (\BbbR ) is a locally finite
subset of \BbbR unbounded from below and from above. Let us fix \omega for a while. With every point
\zeta \in \psi 0,T (\omega ,\BbbR ) we associate a cluster
K\zeta = \cup t\in [0,T ]
\bigl\{
(T - t, x) : \psi T - t,T (\omega , x) = \zeta
\bigr\}
.
By \alpha \zeta and \beta \zeta we denote the lower and the upper boundaries of the cluster K\zeta :
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ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1299
\alpha \zeta (t) = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
x \in \BbbR : (T - t, x) \in K\zeta
\bigr\}
, \beta \zeta (t) = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
x \in \BbbR : (T - t, x) \in K\zeta
\bigr\}
.
This natural definition of \alpha \zeta , \beta \zeta is not a rigorous definition of a stochastic processes, as the choice
of the random quantity \zeta is not specified. In the following lemma we overcome this issue and
simultaneously define the conditional distribution of boundary processes conditioned on the event
\{ \zeta = x\} .
Lemma 4.1. With probability 1, for all \zeta \in \psi 0,T (\BbbR ),
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \zeta +
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm| \bigm| \~\psi T,T - t(x) - \beta \zeta (t)
\bigm| \bigm| = 0
and
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \zeta -
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm| \bigm| \~\psi T,T - t(x) - \alpha \zeta (t)
\bigm| \bigm| = 0.
Proof. For continuity of \alpha \zeta , \beta \zeta we refer to [4]. Let x > \zeta and t \in [0, T ]. If \~\psi T,T - t(x) < \beta \zeta (t),
then there exists y > \~\psi T,T - t(x) such that
\psi T - t,T (y) = \zeta < x,
which contradicts duality. So, for all x > \zeta and all t \in [0, T ],
\beta \zeta (t) \leq \~\psi T,T - t(x).
It remains to check that, for all t \in [0, T ],
\mathrm{i}\mathrm{n}\mathrm{f}
x>\zeta
\~\psi T,T - t(x) = \beta \zeta (t).
Assume that \mathrm{i}\mathrm{n}\mathrm{f}x>\zeta \~\psi T,T - t(x) > \beta \zeta (t) and let y \in
\bigl(
\beta \zeta (t), \mathrm{i}\mathrm{n}\mathrm{f}x>\zeta \~\psi T,T - t(x)
\bigr)
. For every x > \zeta
duality implies that \psi T - t,T (y) \leq x. Hence, \psi T - t,T (y) \leq \zeta . But the latter contradicts y > \beta \zeta (t).
The proof for \alpha \zeta is similar.
Lemma 4.1 is proved.
Observe the equality of events\bigl\{
(u, v) \cap \psi 0,T (\BbbR ) \not = \varnothing
\bigr\}
=
\bigl\{
\~\psi T,0(u) < \~\psi T,0(v)
\bigr\}
,
the latter event being the event that two independent \BbbR -valued Brownian motions started at u and
v and haven’t met during the time T . Combining this consideration with results of Lemma 4.1 and
Theorem 1.1, we get the following corollary.
Corollary 4.1. Conditional distribution of the process\bigl\{ \bigl(
\~\psi T,T - t(u), \~\psi T,T - t(v)
\bigr) \bigr\}
t\in [0,T ]
conditionally on the event
\bigl\{
(u, v) \cap \psi 0,T (\BbbR ) \not = \varnothing
\bigr\}
weakly converge as u \rightarrow x - , v \rightarrow x+ to the
solution of the problem
dY (t) = \nabla y \mathrm{l}\mathrm{o}\mathrm{g} \gamma H
\bigl(
T - t, Y (t)
\bigr)
dt+ dW (t),
Y (0) = (x, x),
Y (t) \in H for a.a. t \in (0, T ),
where H =
\bigl\{
y \in \BbbR 2 : y1 < y2
\bigr\}
, W is a standard \BbbR 2-valued Brownian motion, and \gamma H is defined
in (1.1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1300 G. V. RIABOV
Direct computation gives \gamma H(t, y) =
\sqrt{}
2
\pi
E
\biggl(
y2 - y1\surd
2t
\biggr)
, where E(x) =
\int x
0
e -
u2
2 du. Conse-
quently, we can identify the conditional law of boundaries (\alpha \zeta , \beta \zeta ) given that \{ \zeta = x\} via the
problem
d\alpha \zeta (t) = - e
-
(\beta \zeta (t) - \alpha \zeta (t))
2
4(T - t)\sqrt{}
2(T - t)E
\biggl(
\beta \zeta (t) - \alpha \zeta (t)\sqrt{}
2(T - t)
\biggr) dt+ dW1(t),
d\beta \zeta (t) =
e
-
(\beta \zeta (t) - \alpha \zeta (t))
2
4(T - t)\sqrt{}
2(T - t)E
\biggl(
\beta \zeta (t) - \alpha \zeta (t)\sqrt{}
2(T - t)
\biggr) dt+ dW2(t),
\alpha \zeta (0) = \beta \zeta (0) = x,
\alpha \zeta (t) < \beta \zeta (t) for a.a. t \in (0, T ).
4.2. Arratia flow with drift. In this section the developed approach is adapted to the unbounded
cluster in the Arratia flow with drift. Let a : \BbbR \rightarrow \BbbR be a Lipschitz function. Consider the SDE
dX(t) = a
\bigl(
X(t)
\bigr)
dt+ dw(t), (4.1)
where w is a Wiener process. Informally, the Arratia flow with drift describes the joint motion of
solutions of the equation (4.1) that start from all points of the real line at every moment of time,
move independently before the meeting time and coalesce at the meeting time. Precisely, we say that
a coalescing stochastic flow \psi =
\bigl\{
\psi s,t : - \infty < s \leq t < \infty
\bigr\}
is the Arratia flow with drift a, if the
following condition is satisfied:
For any n \geq 1 and x1 < . . . < xn let
\bigl\{ \bigl(
X1(t), . . . , Xn(t)
\bigr) \bigr\}
t\geq 0
be the solution of the problem
dXi(t) = a
\bigl(
Xi(t)
\bigr)
dt+ dWi(t),
Xi(0) = xi,
1 \leq i \leq n,
where W1, . . . ,Wn are independent standard \BbbR -valued Brownian motions. Denote \sigma = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 :
\exists i \not = j Xi(t) = Xj(t)
\bigr\}
. Further, let
\bigl\{ \bigl(
\psi s,s+t(x1), . . . , \psi s,s+t(xn)
\bigr) \bigr\}
t\geq 0
be the n-point motion of
the flow started at time s from points x1, . . . , xn. Denote \tau = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : \exists i \not = j \psi s,s+t(xi) =
= \psi s,s+t(xj)
\bigr\}
. Then \BbbR n-valued processes
t\rightarrow
\bigl(
\psi s,s+t\wedge \tau (x1), . . . , \psi s,s+t\wedge \tau (xn)
\bigr)
and
t\rightarrow
\bigl(
X1(t \wedge \sigma ), . . . , Xn(t \wedge \sigma )
\bigr)
are identically distributed.
For the existence of the Arratia flow with drift we refer to [3]. When the drift a is strictly
monotone, an infinite cluster arises in the flow \psi .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1301
Theorem 4.1 [5]. Let \psi be the Arratia flow with drift a. Assume that the drift a is Lipschitz
and for some \lambda > 0 and all x, y \in \BbbR one has\bigl(
a(x) - a(y)
\bigr)
(x - y) \leq - \lambda (x - y)2.
Then there exists a unique stationary process (\eta t)t\in \BbbR such that for all s \leq t and all \omega
\psi s,t
\bigl(
\omega , \eta s(\omega )
\bigr)
= \eta t(\omega ).
Further, we assume that the drift a satisfies assumptions of the Theorem 4.1. The process (\eta t)t\geq 0
represents the motion of a stationary point in the flow. In particular, the one-dimensional distribution
of (\eta t)t\geq 0 is given by the stationary distribution of the equation (4.1):
\BbbP (\eta t \in \Delta ) = C
\int
\Delta
e2
\int x
0 a(y)dy dx,
where C =
\biggl( \int \infty
- \infty
e2
\int x
0 a(y)dy dx
\biggr) - 1
. An infinite cluster can be associated with \eta 0. Namely, at
every moment t \geq 0 there exists an interval of points that have coalesced into \eta 0 at time 0:
K0(t) =
\bigl\{
x \in \BbbR : \psi - t,0(x) = \eta 0
\bigr\}
, t \geq 0.
The set K0 = \cup t\geq 0
\bigl(
\{ - t\} \times K0(t)
\bigr)
will be called the cluster with the vertex \eta 0. Let us introduce
boundary processes
\alpha 0(t) = \mathrm{i}\mathrm{n}\mathrm{f}K0(t), \beta 0(t) = \mathrm{s}\mathrm{u}\mathrm{p}K0(t).
The Theorem 4.2 describes the conditional distribution of processes (\alpha 0(t), \beta 0(t)) conditioned on
the event \{ \eta 0 = x\} . The following analogue of the Lemma 4.1 follows from properties of the dual
flow
\bigl\{
\~\psi t,s : - \infty < s \leq t <\infty
\bigr\}
obtained in [4].
Lemma 4.2. With probability 1, for all T \geq 0,
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \eta 0+
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm| \bigm| \~\psi 0, - t(x) - \beta 0(t)
\bigm| \bigm| = 0
and
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \eta 0 -
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm| \bigm| \~\psi 0, - t(x) - \alpha 0(t)
\bigm| \bigm| = 0.
In [5] it was proved that with probability 1
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\beta 0(t) = \infty , \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\alpha 0(t) = - \infty .
Hence, the following equality of events holds:
\{ u < \eta 0 < v\} =
\Bigl\{
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\~\psi 0, - t(u) = - \infty , \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\~\psi 0, - t(v) = \infty
\Bigr\}
.
Let
\theta (y1, y2) = \BbbP
\bigl(
\eta 0 \in (y1, y2)
\bigr)
=
y2\int
y1
\pi (x) dx.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1302 G. V. RIABOV
Theorem 4.2. Conditional distribution of the process\bigl\{ \bigl(
\~\psi 0, - t(u), \~\psi 0, - t(v)
\bigr) \bigr\}
t\geq 0
conditionally on the event \{ u < \eta 0 < v\} , weakly converge as u \rightarrow x - , v \rightarrow x+ to the solution of
the problem
dY1(t) =
\biggl(
- a(Y1(t)) +
\partial \mathrm{l}\mathrm{o}\mathrm{g} \theta (Y1(t), Y2(t))
\partial y1
\biggr)
dt+ dW1(t),
dY2(t) =
\biggl(
- a(Y2(t)) +
\partial \mathrm{l}\mathrm{o}\mathrm{g} \theta (Y1(t), Y2(t))
\partial y2
\biggr)
dt+ dW2(t),
Y1(0) = Y2(0) = x,
Y1(t) < Y2(t) for a.a. t > 0,
(4.2)
where W is a standard \BbbR 2-valued Brownian motion.
Proof. The dual process \~\psi is the Arratia flow with drift - a, see [4, 5]. Let
\bigl\{
(Y1(t), Y2(t))
\bigr\}
t\geq 0
be a solution of the SDE
dY1(t) = - a
\bigl(
Y1(t)
\bigr)
dt+ dW1(t),
dY2(t) = - a
\bigl(
Y2(t)
\bigr)
dt+ dW2(t),
Y1(0) = u, Y2(0) = v,
where W is a standard \BbbR 2-valued Brownian motion. The law of the process\bigl\{ \bigl(
\~\psi 0, - t(u), \~\psi 0, - t(v)
\bigr) \bigr\}
t\geq 0
conditioned on the event \{ u < \eta 0 < v\} coincides with the law of the process Y conditioned on the
event
A =
\Bigl\{
\forall t \geq 0 Y1(t) < Y2(t), \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
Y1(t) = - \infty , \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
Y2(t) = \infty
\Bigr\}
.
Let \sigma = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : Y1(t) = Y2(t)
\bigr\}
. Applying arguments from the proof of Theorem 2.1 to the
process
t\rightarrow \theta
\bigl(
Y1(t \wedge \sigma ), Y2(t \wedge \sigma )
\bigr)
,
we get the Clark representation
1A = \theta (u, v) +
\tau \int
0
\bigl(
\nabla \theta (Y (s)), dW (s)
\bigr)
.
By the Markov property,
\BbbE
\bigl[
1A | Y (s), s \leq t
\bigr]
= 1\tau >t\theta (Y (t)).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON A BROWNIAN MOTION CONDITIONED TO STAY IN AN OPEN SET 1303
The Girsanov theorem implies that with respect to the law of Y conditioned on the event A, the
process
\~W (t) =W (t) -
t\int
0
\nabla \mathrm{l}\mathrm{o}\mathrm{g} \theta
\bigl(
Y (s)
\bigr)
ds, t \geq 0,
is a Brownian motion. This implies equations (4.2) for the distribution of the process
\bigl\{ \bigl(
\~\psi 0, - t(u),
\~\psi 0, - t(v)
\bigr) \bigr\}
t\geq 0
conditioned on the event \{ u < \eta 0 < v\} .
Theorem 4.2 is proved.
References
1. R. Garbit, Brownian motion conditioned to stay in a cone, J. Math. Kyoto Univ., 49, No. 3, 573 – 592 (2009).
2. Y. Le Jan, O. Raimond, Flows, coalescence and noise, Ann. Probab., 32, No. 2, 1247 – 1315 (2004).
3. G. V. Riabov, Random dynamical systems generated by coalescing stochastic flows on R, Stoch. and Dyn., 18,
No. 04, Article 1850031 (2018).
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(2018).
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and Appl., 130, No. 8, 4910 – 4926 (2020).
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Received 20.08.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
|
| id | umjimathkievua-article-6281 |
| 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/ca/58683b8db030dfd873c2f2b5548bc7ca.pdf |
| spelling | umjimathkievua-article-62812022-03-26T11:02:07Z On a Brownian motion conditioned to stay in an open set On a Brownian motion conditioned to stay in an open set On a Brownian motion conditioned to stay in an open set Riabov, G. V. Рябов, Георгий Валентинович Riabov , G. V. UDC 519.21 Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on $\mathbb{R}.$ Исследуется распределение броуновского движение, которое стартует из граничной точки открытого множества G и остается в множестве G в течении конечного периода времени. Получено характеризацию такого распределения в терминах одного сингулярного стохастического дифференциального уравнения. Полученые результаты применяются для изучения границ кластеров в некоторых стохастических потоках сосклеиванием на R. УДК 519.21 Про умовний розподiл броунiвського руху, що не виходить з вiдкритої множини Досліджується розподіл броунівського руху, що стартував із граничної точки відкритої множини $G$ і залишається в $G$ протягом скінченного інтервалу часу. Отримано характеризацію таких розподілів у термінах певних сингулярних стохастичних диференціальних рівнянь. Отримані результати застосовано до вивчення меж кластерів у деяких стохастичних потоках зі склеюванням на $\mathbb{R}.$ Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6281 10.37863/umzh.v72i9.6281 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1286-1303 Український математичний журнал; Том 72 № 9 (2020); 1286-1303 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6281/8755 Copyright (c) 2020 Георгій Валентинович Рябов |
| spellingShingle | Riabov, G. V. Рябов, Георгий Валентинович Riabov , G. V. On a Brownian motion conditioned to stay in an open set |
| title | On a Brownian motion conditioned to stay in an open set |
| title_alt | On a Brownian motion conditioned to stay in an open set On a Brownian motion conditioned to stay in an open set |
| title_full | On a Brownian motion conditioned to stay in an open set |
| title_fullStr | On a Brownian motion conditioned to stay in an open set |
| title_full_unstemmed | On a Brownian motion conditioned to stay in an open set |
| title_short | On a Brownian motion conditioned to stay in an open set |
| title_sort | on a brownian motion conditioned to stay in an open set |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6281 |
| work_keys_str_mv | AT riabovgv onabrownianmotionconditionedtostayinanopenset AT râbovgeorgijvalentinovič onabrownianmotionconditionedtostayinanopenset AT riabovgv onabrownianmotionconditionedtostayinanopenset |