On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval
UDC 519.21 We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512319294930944 |
|---|---|
| author | Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. |
| author_facet | Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. |
| author_sort | Dorogovtsev, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:02:06Z |
| description | UDC 519.21
We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, we also discuss the convergence of the corresponding approximations of the point measure associated with the Arratia flow in terms of such densities. |
| doi_str_mv | 10.37863/umzh.v72i9.6279 |
| first_indexed | 2026-03-24T03:26:54Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i9.6279
UDC 519.21
A. A. Dorogovtsev (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv,
and Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”),
M. B. Vovchanskii (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON APPROXIMATIONS OF THE POINT MEASURES
ASSOCIATED WITH THE BROWNIAN WEB
BY MEANS OF THE FRACTIONAL STEP METHOD
AND THE DISCRETIZATION OF THE INITIAL INTERVAL
ПРО АПРОКСИМАЦIЮ АСОЦIЙОВАНИХ IЗ БРОУНIВСЬКОЮ СIТКОЮ
ТОЧКОВИХ МIР ЗА ДОПОМОГОЮ МЕТОДУ ДРОБОВИХ КРОКIВ
ТА ДИСКРЕТИЗАЦIЇ ПОЧАТКОВОГО IНТЕРВАЛУ
We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein
distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities
that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval
discretized, we also discuss the convergence of the corresponding approximations of the point measure associated with the
Arratia flow in terms of such densities.
Встановлено швидкiсть слабкої збiжностi в методi дробових крокiв для потоку Арратья в термiнах вiдстанi Вассер-
штейна мiж образами мiри Лебега пiд дiєю потоку. Введено скiнченновимiрнi щiльностi, що описують послiдовностi
зiткнень в потоцi Арратья, та отримано явний вираз для них. Дослiджено збiжнiсть вiдповiдних апроксимацiй точ-
кових мiр для потоку Арратья при дискретизацiї початкового iнтервалу.
1. Introduction. In this article, we consider point measures which are constructed from the Arratia
flow and its approximations [2, 3, 8, 9]. Two types of discrete measures can be associated with a
stochastic flow \{ X(u, t) | t \geq 0, u \in \BbbR \} with coalescence on the real line: the first measure is the
image of the Lebesque measure under the action of the flow
\mu t = \lambda \circ (X(\cdot , t)) - 1 ,
and the second one is the counting measure defined by the rule
\nu t(\Delta ) =
\bigm| \bigm| X(\BbbR , t) \cap \Delta
\bigm| \bigm| , \Delta \in \scrB (\BbbR ).
Both measures are supported on the same locally finite countable set. The structure of such random
measures is studied in [5 – 7, 14, 15]. In the first part of the article the Arratia flow with drift is
considered. This flow consists of coalescing Brownian motions with diffusion 1 and drift a, where a
is a bounded Lipschitz continuous function. Such a stochastic flow was obtained in [2] by applying
the fractional step method [1, 10] to the Brownian web [8, 9] and an ordinary differential equation
driven by a. Here, the study of this approximation scheme is continued by discussing the speed of
convergence of the images of the Lebesque measure.
We start with recalling the fractional step method for the Brownian web proposed in [2]. Let
a be a bounded Lipschitz continuous function on the real line. Consider a sequence of partitions\bigl\{
0 = t
(n)
0 < . . . < t
(n)
n = 1
\bigr\}
of the interval [0; 1] with the mesh size \delta n converging to 0. Define a
family of transformations of \BbbR
c\bigcirc A. A. DOROGOVTSEV, M. B. VOVCHANSKII, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9 1179
1180 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
d\scrA s,t(u) = a
\bigl(
\scrA s,t(u)
\bigr)
dt,
\scrA s,s(u) = u, t \geq s.
Given a Brownian web
\bigl\{
\Phi s,t(u) | 0 \leq s \leq t, u \in \BbbR
\bigr\}
[8, 9] one can consider \{ \Phi s,t\} 0\leq s\leq t as random
mappings of \BbbR into itself. Put \Delta (n)
j =
\bigl[
t
(n)
j ; t
(n)
j+1
\bigr)
, j = 0, n - 1, and define, for u \in \BbbR , t \in \Delta
(n)
j ,
\Phi
(n)
t (u) = \Phi
t
(n)
j ,t
\circ \scrA
t
(n)
j ,t
(n)
j+1
\Bigl(
j - 1
\circ
l=0
(\Phi
t
(n)
l ,t
(n)
l+1
\circ \scrA
t
(n)
l ,t
(n)
l+1
)(u)
\Bigr)
,
\Phi
(n)
1 (u) = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 1 -
\Phi
(n)
t (u).
The sign \circ stands for the composition of functions: f \circ g = f(g). The main result of [2] states that
given u1, . . . , um \in \BbbR \bigl(
\Phi (n)(u1), . . . ,\Phi
(n)(um)
\bigr)
\Rightarrow
n\rightarrow \infty
\bigl(
\Phi a(u1), . . . ,\Phi
a(um)
\bigr)
(1.1)
in the Skorokhod space
\bigl(
D([0; 1])
\bigr) m
, with \{ \Phi a
s(u) | s \geq 0, u \in \BbbR \} being an Arratia flow with drift
a [3] (\S 7.3). It was proved in [2] (Proposition 1.5) that the sequence in the left-hand side of (1.1)
converges only weakly in contrast to the application of the fractional step method to ordinary SDEs
[1, 10].
Let \lambda be the Lebesque measure on [0; 1]. One can define images of \lambda under the mappings \Phi a
t ,
\Phi
(n)
t :
\mu t = \lambda \circ
\bigl(
\Phi a
t
\bigr) - 1
, \mu
(n)
t = \lambda \circ
\bigl(
\Phi
(n)
t
\bigr) - 1
, n \in \BbbN .
Such random measures along with associated point processes are central objects of the present paper,
in the first part of which an estimate on the speed of the convergence of the laws of
\bigl\{
\mu
(n)
t
\bigr\}
n\geq 1
to
the law of \mu t, for fixed t, is established in terms of an appropriate Wasserstein distance.
Our approach is based on ideas from [4]. Recall a definition of the Wasserstein distance between
two probability measures. Let X be a separable complete metric space with metric d and the
corresponding Borel \sigma -field. The set \scrM p(X) of all probability measures \mu on X such that for some
(and, therefore, for an arbitrary) point u
\int
X
d(u, v)p\mu (dv) < +\infty is a separable metric space [16]
(Theorem 6.18) w.r.t. the distance
Wp(\mu 1, \mu 2) =
\left( \mathrm{i}\mathrm{n}\mathrm{f}
\varkappa \in \Pi (\mu 1,\mu 2)
\int
X2
d(u, v)p\varkappa (du, dv)
\right) 1/p, p \geq 1,
where \Pi (\mu 1, \mu 2) is the set of all probability measures on X2 having marginals \mu 1 and \mu 2.
The measures \mu t, \mu
(n)
t , n \in \BbbN , are random elements in \scrM p(\BbbR ) for any p \geq 1. Let Lt and L
(n)
t
be the laws of \mu t and \mu
(n)
t in \scrM 1
\bigl(
\scrM p(\BbbR )
\bigr)
, respectively. For fixed p, the corresponding Wasserstein
distance between probability measures L\prime , L\prime \prime \in \scrM 1
\bigl(
\scrM p(\BbbR )
\bigr)
is defined via
W1(L
\prime , L\prime \prime ) = \mathrm{i}\mathrm{n}\mathrm{f} \mathrm{E}Wp(\mu
\prime , \mu \prime \prime ),
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1181
where the infinum is taken over the set of pairs of \scrM p(\BbbR )-valued random elements \mu \prime , \mu \prime \prime satisfying
\mathrm{L}\mathrm{a}\mathrm{w}(\mu \prime ) = L\prime , \mathrm{L}\mathrm{a}\mathrm{w}(\mu \prime \prime ) = L\prime \prime . To indicate a specific value of p being used, we write W1,p for the
distance on \scrM 1
\bigl(
\scrM p(\BbbR )
\bigr)
. The main result of the second section is the following theorem (cf. [17]
(Theorem 1), [4] (Theorem 1.3)).
Theorem 1.1. Assume that the sequence \{ n\delta n\} n\in \BbbN is bounded. Then for every p \geq 2 there
exist a positive constant C and a number N \in \BbbN such that for all n \geq N
W1,p
\bigl(
Lt, L
(n)
t
\bigr)
\leq C
\bigl(
\mathrm{l}\mathrm{o}\mathrm{g} \delta - 1
n
\bigr) - 1/p
.
Section 3 is devoted to the counting measure associated with the Arratia flow. We discuss the
speed of convergence of such measures when one approximates the segment of the real line by its
finite subsets. For that, we introduce the multidimensional densities which correspond to different
sequences of collisions in the n-point motion of the Arratia flow.
Given an Arratia flow \{ X(u, t) | t \geq 0, u \in [0; 1]\} with zero drift put \Delta n = \{ u1 < . . .
. . . < un\} , n \in \BbbN , and Xt = \{ X(u, t) | u \in [0; 1]\} . The next definition is taken from [14]
(Appendix B) (see also [7, 15]) and is adjusted to reflect that the Arratia flow now starts from [0; 1]
instead of the whole real line.
Definition 1.1. The n-point density pnt is a measurable function such that for any bounded
nonnegative measurable f : \BbbR n \rightarrow \BbbR \int
\BbbR n
f(x)pnt (x) dx = \mathrm{E}
\sum
u1,...,un\in Xt
all distinct
f(u1, . . . , un). (1.2)
Recall that given u = (u1, . . . , un) the processes X(u1), . . . , X(un) are coalescing Brownian
motions. To describe all possible sequences of collisions in this system, the following notation is
used. Define \scrX n \in
\bigl(
C([0; 1])
\bigr) n
by setting \scrX n
j (\cdot ) = X(uj , \cdot ), j = 1, n. Let k be the number of
distinct values in the set
\bigl\{
X(u1, t), . . . , X(un, t)
\bigr\}
. Supposing k < n let \tau 1 be the moment of the
first collision on [0; t]. Put j1 = \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
i | \exists j \not = i \scrX n
i (\tau 1) = \scrX n
j (\tau 1)
\bigr\}
. Define \scrX n - 1 \in
\bigl(
C([0; 1])
\bigr) n - 1
by excluding the j1 th coordinate from \scrX n . If there exists a moment \tau 2 \leq t such that for some
i, j \in \{ 1, . . . , n - 1\} \scrX n - 1
i (\tau 2) = \scrX n - 1
j (\tau 2) put j2 to be equal to the smallest such number.
Repeating the procedure n - k times one obtains a random collection Jt(u) = (j1, . . . , jn - k),
ji \in
\bigl\{
1, . . . , n - i
\bigr\}
, i = 1, n - k. In the case k = n we set Jt(u) = \varnothing by definition. The set of all
possible such collections consisting of l numbers is denoted by \scrJ n,l.
Definition 1.2. The random collection Jt(u) defined via the recursive procedure described
above is called the coalescence scheme corresponding to the start points u1, . . . , un.
Definition 1.3. Given x = (x1, . . . , xn) \in \Delta n the k-point density pJ,kt (x; \cdot ) corresponding to
the coalescence scheme J \in \scrJ n,n - l, k \leq l, and the start points x1, . . . , xn is a measurable function
such that for any bounded nonnegative measurable f : \BbbR k \rightarrow \BbbR \int
\BbbR k
pJ,kt (x; y)f(y) dy = \mathrm{E}
\sum
u1,...,uk\in \{ X(x1,t),...,X(xn,t)\}
all distinct
f(u1, . . . , uk)\times 1(Jt(x) = J). (1.3)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1182 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
The integral representation is obtained for such densities (Theorem 3.1). The result on conver-
gence of the multidimensional densities given in Theorem 3.2 is motivated by the discrete approxi-
mations of Section 2.
Consider the vectors U (n) =
\bigl(
u
(n)
1 , . . . , u
(n)
n
\bigr)
\in \Delta n, such that u(n)1 = 0, u
(n)
n = 1, n \in \BbbN ,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\mathrm{m}\mathrm{a}\mathrm{x}
j=0,n - 1
\Bigl(
u
(n)
j+1 - u
(n)
j
\Bigr)
= 0,
and \bigl\{
u
(n)
1 , . . . , u(n)n
\bigr\}
\subset
\Bigl\{
u
(n+1)
1 , . . . , u
(n+1)
n+1
\Bigr\}
, n \in \BbbN .
Define
pkt (U
(n); \cdot ) =
n\sum
i=k
\sum
J\in \scrJ n,n - i
pJ,kt (U (n); \cdot ), k = 1, n, n \in \BbbN . (1.4)
Theorem 1.2. There exists an absolute positive constant C such that
0 \leq p1t (y) - p1t
\bigl(
U (n); y
\bigr)
\leq C \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n - 1
\Bigl(
u
(n)
j+1 - u
(n)
j
\Bigr) 2
for almost all y.
2. The Wasserstein distance between \bfitL \bfitt and \bfitL
(\bfitn )
\bfitt . We approximate the measures \mu t and \mu
(n)
t
with point measures
\mu
(n),m
t = m - 1
m - 1\sum
j=0
\delta
\Phi
(n)
t (j/m)
,
\mu m
t = m - 1
m - 1\sum
j=0
\delta \Phi a
t (j/m), n,m \in \BbbN .
We begin with Lp-estimates on the divergence between two solutions of a one-dimensional SDE
in terms of the difference of the initial points and estimates of the same type for their approximations
via the fractional step method.
Let a be a bounded function satisfying the Lipschitz condition with constant Ca. Put Ma =
= \mathrm{s}\mathrm{u}\mathrm{p}\BbbR | a| . Given a standard Brownian motion w and a point u \in \BbbR the equation
dx(t) = a(x(t))dt+ dw(t),
x(0) = u, t \in [0; 1],
has the unique strong solution x. Consider, for t \in \Delta
(n)
j , j = 0, n - 1,
y(n)(t) = u+
t
(n)
j+1\int
0
a
\bigl(
z(n)(s)
\bigr)
ds+ w(t),
z(n)(t) = u+
t\int
0
a
\bigl(
z(n)(s)
\bigr)
ds+ w(t
(n)
j ).
(2.1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1183
We will encode such a relation between x, y(n), z(n) and w, u by writing x = D(w, u), (y(n), z(n)) =
= S(n)(w, u). The next result is a straightforward generalization of [1] (Corollary 4.2).
Lemma 2.1. For any p \geq 1 there exists C > 0 such that
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
\bigm| \bigm| x(s) - y(n)(s)
\bigm| \bigm| p \leq C\delta p/2n ,
\mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
\mathrm{E}
\bigm| \bigm| \bigm| x(s) - z(n)(s)
\bigm| \bigm| \bigm| p \leq C\delta p/2n .
Lemma 2.2. Suppose u1, u2 \in \BbbR , and w1, w2 are independent Brownian motions. Let xk =
= D(wk, uk), k = 1, 2. Then for any p \geq 1 there exists C > 0 such that
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
\bigm| \bigm| x1(s \wedge \theta ) - x2(s \wedge \theta )
\bigm| \bigm| p \leq C
\bigl(
| u1 - u2| + | u1 - u2| p
\bigr)
, p \geq 2,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
| x1(s \wedge \theta ) - x2(s \wedge \theta )| p \leq C
\bigl(
| u1 - u2| p/2 + | u1 - u2| p
\bigr)
, p \in [1; 2),
where \theta = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | x1(s) = x2(s)
\bigr\}
.
Proof. Denote \Delta u = u2 - u1,\Delta x = x2 - x1. Assume u2 > u1. Consider the SDE
d\eta (t) = Ca\eta (t)dt+ dw2(t) - dw1(t),
\eta (0) = \Delta u,
with the unique strong solution
\eta (t) = eCat\Delta u+
\surd
2eCat
t\int
0
e - Cas dw(s), (2.2)
where w =
w2 - w1\surd
2
. We have
\eta (t) - \Delta x(t) = Ca
t\int
0
\bigl(
\eta (s) - \Delta x(s)
\bigr)
ds+
t\int
0
\bigl(
Ca\Delta x(s) - a(x2(s)) + a(x1(s))
\bigr)
ds a.s.,
therefore a.s.
\eta (t) - \Delta x(t) = eCat
t\int
0
e - Cas
\bigl(
Ca\Delta x(s) - a(x2(s)) + a(x1(s))
\bigr)
ds \geq 0, t \in [0; \theta ]. (2.3)
Applying the Knight theorem [13] (Proposition 18.8) to the stochastic integral in (2.2), we get
\eta (t) = eCat\Delta u+
\surd
2eCat\beta
\left( t\int
0
e - 2Casds
\right) ,
where \beta is some Brownian motion. Then (2.3) implies
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1184 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
\theta \leq \varkappa = \mathrm{i}\mathrm{n}\mathrm{f} \{ 1; s | \eta (s) = 0\} = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ 1; s
\bigm| \bigm| \bigm| \bigm| \bigm| \beta
\left( t\int
0
e - 2Casds
\right) =
- \Delta u\surd
2
\right\} .
Thus,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
t\leq \theta
| \Delta x(t)| p \leq \mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
t\leq \varkappa
| \eta (t)| p \leq 2p - 1epCa\Delta up + 23p/2 - 1epCa\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
t\leq \varkappa
| \beta (t)| p,
since
\int t
0
e - 2Casds < t, t \geq 0. The same reason implies that the random moment \varkappa is a stopping
time w.r.t. the filtration generated by
\bigl\{
\beta (t) | t \in [0; 1]
\bigr\}
, therefore, by the Burkholder – Davis – Gundy
inequality,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
t\leq \varkappa
| \beta (t)| p \leq Cp\mathrm{E}\varkappa p/2, p \geq 2,
for positive constants Cp. The distribution of \varkappa is given via
\sansP (\varkappa \geq t) =
\sqrt{}
2
\pi
a(t)\int
0
e - y2/2dy, a(t) =
C
1/2
a (u2 - u1)
(1 - e - 2t)1/2
,
hence, for fixed p \geq 2,
\mathrm{E}\varkappa p/2 =
p
2
1\int
0
t
p
2
- 1
\left( \sqrt{} 2
\pi
a(t)\int
0
e - y2/2dy
\right) dt \leq p\surd
2\pi
1\int
0
a(t)tp/2 - 1dt \leq C(u2 - u1) (2.4)
for some C. To handle the case p \in [1; 2) one uses the Lyapunov inequality and the foregoing
estimates.
Lemma 2.2 is proved.
We consider a modification of (2.1): on every \Delta
(n)
j , j = 0, n - 1,
y(n)(t) = uy +
t
(n)
j+1\int
0
a
\bigl(
z(n)(s)
\bigr)
ds+ w(t),
z(n)(t) = uz +
t\int
0
a
\bigl(
z(n)(s)
\bigr)
ds+ w(t
(n)
j ), t \in \Delta
(n)
j ,
where nonrandom uy and uz are not necessarily equal. The pair (y(n), z(n)) is denoted by
S(n)(w, uy, uz).
Lemma 2.3. Assume that the sequence \{ n\delta n\} n\in \BbbN is bounded. Let uy1 , uy2 , uz1 , uz2 \in \BbbR ,
and let w1, w2 be independent standard Brownian motions. Put
\bigl(
y
(n)
k , z
(n)
k
\bigr)
= S(n)(wk, uyk , uzk),
k = 1, 2. Then, for any p \geq 2 and for any \varepsilon \in
\biggl(
0;
1
2
\biggr)
there exist C > 0 and N \in \BbbN such that for
all n \geq N
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1185
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
\bigm| \bigm| \bigm| y(n)1 (s \wedge \theta (n)) - y
(n)
2 (s \wedge \theta (n))
\bigm| \bigm| \bigm| p \leq
\leq C
\Biggl(
\delta 1/2 - \varepsilon
n +
2\sum
l=1
\bigl(
| uzl - uyl | + | uzl - uyl |
p
\bigr)
+ | uy2 - uy1 | p + | uy2 - uy1 |
\Biggr)
,
where \theta (n) = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | y(n)2 (s) = y
(n)
1 (s)
\bigr\}
.
Proof. We extend the proof of Lemma 2.2. Suppose uz2 - uz1 \geq 0, uy2 - uy1 \geq 0. Denote
\Delta uy = uy2 - uy1 , \Delta uz = uz2 - uz1 , and let \eta be defined as in (2.2) with \Delta u = \Delta uy. Then, for
t \leq \theta (n), t \in \Delta
(n)
j for some j, and for \Delta y = y2 - y1,
\Delta y(t) - \eta (t) = Ca
t\int
0
(\Delta y(s) - \eta (s)) ds+
+
t\int
0
\Bigl(
a(z
(n)
2 (s)) - a(z
(n)
1 (s)) - Ca\Delta y(s)
\Bigr)
ds+
+
t
(n)
j+1\int
t
\Bigl(
a(z
(n)
2 (s)) - a(z
(n)
1 (s))
\Bigr)
ds \leq
\leq Ca
t\int
0
(\Delta y(s) - \eta (s)) ds+ Ca
t\int
0
2\sum
l=1
( - 1)l
\bigl(
z
(n)
l (s) - y
(n)
l (s)
\bigr)
ds+ 2\delta nMa,
since z
(n)
2 \geq z
(n)
1 on [0; \theta (n)]. For s \in \Delta
(n)
i , i \leq j,\bigm| \bigm| \bigm| z(n)k (s) - y
(n)
k (s) - wk(t
(n)
i ) + wk(s)
\bigm| \bigm| \bigm| \leq
\leq
t
(n)
i+1\int
s
\bigm| \bigm| a\bigl( z(n)k (s)
\bigr) \bigm| \bigm| ds+ | uzk - uyk | \leq
\leq
\bigl(
t
(n)
i+1 - s
\bigr)
Ma + | uzk - uyk | ,
so it follows, for t \in \Delta
(n)
j , t \leq \theta (n), that
\Delta y(t) - \eta (t) \leq Ca
t\int
0
(\Delta y(s) - \eta (s)) ds+ Ca
j - 1\sum
k=0
\int
\Delta
(n)
k
2\sum
l=1
( - 1)l
\bigl(
wl(t
(n)
k+1) - wl(s)
\bigr)
ds+
+2CaMa
j - 1\sum
k=0
\int
\Delta
(n)
k
(t
(n)
k+1 - s) ds+ 2\delta nMa +
2\sum
l=1
| uzl - uyl | .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1186 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
Since
2CaMa
j - 1\sum
k=0
\int
\Delta
(n)
k
\bigl(
t
(n)
k+1 - s
\bigr)
ds \leq CaMa\delta n,
the Gronwall – Bellman inequality implies
\Delta y(t) \leq \eta (t) + eCaMa(Ca + 2)\delta n + eCa
2\sum
l=1
| uzl - uyl | + eCaCa \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n
| \xi j | ,
where
\xi j =
2\sum
l=1
j - 1\sum
k=0
\int
\Delta
(n)
k
( - 1)l
\bigl(
wl(t
(n)
k+1) - wl(s)
\bigr)
ds, j = 1, n.
Thus,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq \theta (n)
| \Delta y(s)| p \leq 4p - 1
\Biggl(
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq \theta (n)
| \eta (s)| p + epCaMp
a (Ca + 2)p\delta pn+
+epCa
\Biggl(
2\sum
l=1
| uzl - uyl |
\Biggr) p
+ epCaCp
a \mathrm{E} \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n
| \xi j | p
\Biggr)
. (2.5)
The random variables \xi j+1 - \xi j , j = 1, n - 1, are independent centered Gaussian variables; \mathrm{V}\mathrm{a}\mathrm{r}(\xi n) \leq
\leq 2n\delta 2n. Therefore, by the Levy inequality, there exists a constant C such that
\mathrm{E} \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n
| \xi j | p \leq 2\mathrm{E}| \xi n| p \leq Cnp\delta 2pn (2.6)
and, for any xn > 0,
\sansP
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
k=1,n
| \xi k| \geq xn
\biggr)
\leq 2\sansP
\Biggl( \bigm| \bigm| \BbbN (0, 1)\bigm| \bigm| \geq xn
(\mathrm{V}\mathrm{a}\mathrm{r}(\xi n))
1/2
\Biggr)
\leq C
n1/2\delta n
xn
e
- x2n
4n\delta 2n . (2.7)
At the same time, proceeding exactly as in the proof of Lemma 2.2 we obtain
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
t\leq \theta (n)
\bigm| \bigm| \eta (t)\bigm| \bigm| p \leq 2p - 1epCa\Delta upy + 23p/2 - 1epCaCp\mathrm{E}(\theta
(n))p/2. (2.8)
However, at time \theta (n)
\eta (\theta (n)) \geq - eCaMa(Ca + 2)\delta n - eCa
2\sum
l=1
| uzl - uyl | - eCaCa \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n
| \xi (j)| ,
so, for fixed xn > 0,
\mathrm{E}(\theta (n))p/2 \leq \sansP
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
k=1,n
| \xi k| \geq xn
\biggr)
+ \mathrm{E}\tau p/2n ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1187
where
\tau n = \mathrm{i}\mathrm{n}\mathrm{f}
\Biggl\{
1; s
\bigm| \bigm| \bigm| \bigm| \eta (s) = - eCaMa(Ca + 2)\delta n - eCa
2\sum
l=1
| uzl - uyl | - eCaCaxn
\Biggr\}
.
Put K = \mathrm{s}\mathrm{u}\mathrm{p}k\in \BbbN k\delta k. Reasoning leading to (2.4), when combined with (2.7), implies that
\mathrm{E}(\theta (n))p/2 \leq C
\Biggl(
\Delta uy + \delta n +
2\sum
l=1
| uzl - uyl | + xn +K1/2 \delta
1/2
n
xn
e -
x2n
4K\delta n
\Biggr)
(2.9)
for the redefined constant C. Choosing xn = \delta
1/2 - \varepsilon
n , for any fixed \varepsilon \in
\biggl(
0;
1
2
\biggr)
, and substituting
(2.6), (2.8) and (2.9) into (2.5) finishes the proof.
Let us recall the definitions of the measures considered. For the random elements in \scrM p(\BbbR )
\mu t = \lambda \circ
\bigl(
\Phi a
t
\bigr) - 1
, \mu m
t =
\left( 1
m
m\sum
j=1
\delta j/m
\right) \circ
\bigl(
\Phi a
t
\bigr) - 1
,
\mu
(n)
t = \lambda \circ
\bigl(
\Phi
(n)
t
\bigr) - 1
, \mu
(n),m
t =
\left( 1
m
m\sum
j=1
\delta j/m
\right) \circ
\bigl(
\Phi
(n)
t
\bigr) - 1
, n,m \in \BbbN ,
we consider their distributions as elements of \scrM 1
\bigl(
\scrM p(\BbbR )
\bigr)
:
Lt = \mathrm{L}\mathrm{a}\mathrm{w}(\mu t), Lm
t = \mathrm{L}\mathrm{a}\mathrm{w}
\bigl(
\mu m
t
\bigr)
,
L
(n)
t = \mathrm{L}\mathrm{a}\mathrm{w}
\bigl(
\mu
(n)
t
\bigr)
, L
(n),m
t = \mathrm{L}\mathrm{a}\mathrm{w}
\bigl(
\mu
(n),m
t
\bigr)
, n,m \in \BbbN .
Analogously to [4] (Theorem 2.1), we have the following lemma.
Lemma 2.4. For any p \geq 2 there exists C > 0 such that
W1,p
\bigl(
Lt, L
m
t
\bigr)
\leq Cm - 1/p,
and, if additionally \{ n\delta n\} n\in \BbbN is bounded,
W1,p(L
(n)
t , L
(n),m
t ) \leq C
\bigl(
m - 1 + \delta 1/2 - \varepsilon
n
\bigr) 1/p
.
Proof. Since the random measures
\bigl(
\mu t, \mu
m
t
\bigr)
is a coupling for the pair
\bigl(
Lt, L
m
t
\bigr)
, it follows from
the definition of the distance W1,p that
W1,p
\bigl(
Lt, L
m
t
\bigr)
\leq \mathrm{E}Wp
\bigl(
\mu t, \mu
m
t
\bigr)
.
Therefore, by [16] (Theorem 2.18, Remark 2.19) and Lemma 2.2, for some C,
W1,p
\bigl(
Lt, L
m
t
\bigr)
\leq
\left( m - 1\sum
j=0
(j+1)/m\int
j/m
\mathrm{E}
\bigm| \bigm| \Phi a
t (y) - \Phi a
t (j/m)
\bigm| \bigm| pdy
\right)
1/p
\leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1188 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
\leq C
\left( m - 1\sum
j=0
(j+1)/m\int
j/m
(y - j/m) dy
\right)
1/p
= Cm - 1/p
as, for x1, x2 from Lemma 2.2,\Bigl\{ \bigl(
\Phi a
t\wedge \theta 1(y),\Phi
a
t\wedge \theta 1 (j/m)
\bigr)
| t \in [0; 1]
\Bigr\}
d
=
\Bigl\{ \bigl(
x1(t \wedge \theta 2), x2(t \wedge \theta 2)
\bigr) \bigm| \bigm| t \in [0; 1]
\Bigr\}
,
\theta 1, \theta 2 being he moments of meeting for the corresponding pairs of processes. Similarly, using
Lemma 2.3 with uzk = uyk , k = 1, 2,
W1,p
\bigl(
L
(n)
t , L
(n),m
t
\bigr)
\leq C
\left( m - 1\sum
j=0
(j+1)/m\int
j/m
(y - j/m) dy + \delta 1/2 - \varepsilon
n
\right)
1/p
\leq C
\bigl(
m - 1 + \delta 1/2 - \varepsilon
n
\bigr) 1/p
for some C.
Lemma 2.4 is proved.
Now we describe appropriate couplings for
\bigl(
\mu m
t , \mu (n),m
\bigr)
, n \in \BbbN , given fixed m. Suppose
w1, . . . , wm are independent standard Brownian motions. Denoting uj = j/m, j = 0,m, put
xj = D(wj , uj),
(y
(n)
j , z
(n)
j ) = S(n)(wj , uj), n \in \BbbN ,
and define \widetilde x1 = x1, \widetilde y(n)1 = y
(n)
1 , \widetilde z(n)1 = z
(n)
1 . Proceeding recursively, put
\theta j = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | xj(s) = \widetilde xj - 1(s)
\bigr\}
,
\theta
(n)
j = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | y(n)j (s) = \widetilde y(n)j - 1(s)
\bigr\}
,
\widetilde xj(t) = xj(t)1 (t < \theta j) + \widetilde xj - 1(t)1 (t \geq \theta j) ,
\widetilde y(n)j (t) = y
(n)
j (t)1
\bigl(
t < \theta
(n)
j
\bigr)
+ \widetilde y(n)j - 1(t)1
\bigl(
t \geq \theta
(n)
j
\bigr)
, j = 2,m.
Consider a random number k(n)j such that \theta (n)j \in \Delta
(n)
j and put
\widetilde z(n)j (t) = z
(n)
j (t)1
\Bigl(
t < t
(n)
k
(n)
j +1
\Bigr)
+ \widetilde z(n)j - 1(t)1
\Bigl(
t \geq t
(n)
k
(n)
j +1
\Bigr)
, t \in [0; 1), j = 2,m.
Values at t = 1 are taken to be equal to the corresponding left limits. The processes
\widetilde w1 = w1, \widetilde w(n)
1 = w1,
\widetilde wj(t) = wj(t)1 (t < \theta j) + \widetilde wj - 1(t)1 (t \geq \theta j) ,
\widetilde w(n)
j (t) = wj(t)1
\bigl(
t < \theta
(n)
j
\bigr)
+ \widetilde w(n)
j - 1(t)1
\bigl(
t \geq \theta
(n)
j
\bigr)
, j = 2,m, n \in \BbbN ,
can be checked to be Brownian motions.
The proofs of the next two lemmas are based on the repeated application of (2.1) and are thus
omitted.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1189
Lemma 2.5. For n,m \in \BbbN and j = 1,m,
\widetilde xj = D( \widetilde wj , uj),\bigl( \widetilde y(n)j , \widetilde z(n)j
\bigr)
= S(n)( \widetilde w(n)
j , uj).
Lemma 2.6. For n,m \in \BbbN ,
(\Phi a(u1), . . . ,\Phi
a(um))
d
=
\bigl( \widetilde x1, . . . , \widetilde xm\bigr) ,\Bigl(
\Phi
(n)
0,\cdot (u1), . . . ,\Phi
(n)
0,\cdot (um)
\Bigr)
d
=
\bigl( \widetilde y(n)1 , . . . , \widetilde y(n)m
\bigr)
in
\bigl(
D([0; 1])
\bigr) m
.
Proof of Theorem 1.1. Repeating the reasoning of the proof of Lemma 2.4 and using Lemma 2.6,
we get
\bigl(
W1,p
\bigl(
Lm
t , L
(n),m
t
\bigr) \bigr) p \leq m - 1\sum
j=0
(j+1)/m\int
j/m
\mathrm{E}
\bigm| \bigm| \widetilde xj(t) - \widetilde y(n)j (t)
\bigm| \bigm| pdu = m - 1
m - 1\sum
j=0
\mathrm{E}
\bigm| \bigm| \widetilde xj(t) - \widetilde y(n)j (t)
\bigm| \bigm| p.
By Lemma 2.1, for some positive C1,
\mathrm{E}
\bigm| \bigm| \widetilde x1(t) - \widetilde y(n)1 (t)
\bigm| \bigm| p \leq \mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
s\leq 1
\bigm| \bigm| x1(s) - y
(n)
1 (s)
\bigm| \bigm| p \leq C1\delta
p/2
n .
Continuing, for j = 2,
\mathrm{E}
\bigm| \bigm| \widetilde x2(t) - \widetilde y(n)2 (t)
\bigm| \bigm| p = \mathrm{E}
\bigm| \bigm| \widetilde x2(t) - \widetilde y(n)2 (t)
\bigm| \bigm| p \times \Bigl[ 1\bigl( t \geq \theta 1 \wedge \theta
(n)
1
\bigr)
+ 1
\bigl(
\theta
(n)
1 \leq t < \theta 1
\bigr)
+
+1
\bigl(
\theta 1 \leq t < \theta
(n)
1
\bigr)
+ 1
\bigl(
t < \theta
(n)
1 \wedge \theta 1
\bigr) \Bigr]
\leq
\leq \mathrm{E}
\bigm| \bigm| \widetilde x1(t) - \widetilde y(n)1 (t)
\bigm| \bigm| p1\bigl( t \geq \theta 1 \wedge \theta
(n)
1
\bigr)
+ \mathrm{E}
\bigm| \bigm| \bigm| x2(t) - y
(n)
2 (t)
\bigm| \bigm| \bigm| p 1\bigl( t < \theta
(n)
1 \wedge \theta 1
\bigr)
+
+2p - 1\mathrm{E}
\Bigl[ \bigm| \bigm| x2(t) - \widetilde x1(t)\bigm| \bigm| p + \bigm| \bigm| \widetilde x1(t) - \widetilde y(n)1 (t)
\bigm| \bigm| p\Bigr] 1\bigl( \theta (n)1 \leq t < \theta 1
\bigr)
+
+2p - 1\mathrm{E}
\Bigl[ \bigm| \bigm| \widetilde x1(t) - \widetilde y(n)1 (t)
\bigm| \bigm| p + \bigm| \bigm| \widetilde y(n)1 (t) - y
(n)
2 (t)
\bigm| \bigm| p\Bigr] 1\bigl( \theta 1 \leq t < \theta
(n)
1
\bigr)
\leq
\leq 2p - 1\mathrm{E}
\bigm| \bigm| \widetilde x1(t) - \widetilde y(n)1 (t)
\bigm| \bigm| p + 2p - 1\mathrm{E}
\bigm| \bigm| \widetilde x1(t) - x2(t)
\bigm| \bigm| p1\bigl( \theta (n)1 \leq t < \theta 1
\bigr)
+
+2p - 1\mathrm{E}
\bigm| \bigm| \widetilde y(n)1 (t) - y
(n)
2 (t)
\bigm| \bigm| p1\bigl( \theta 1 \leq t < \theta
(n)
1
\bigr)
+ \mathrm{E}
\bigm| \bigm| x2(t) - y
(n)
2 (t)
\bigm| \bigm| p.
By using Lemma 2.1 again, we obtain
\mathrm{E}
\bigm| \bigm| \widetilde x2(t) - \widetilde y(n)2 (t)
\bigm| \bigm| p \leq (2p - 1 + 1)C1\delta
p/2
n + 2p - 1\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta
(n)
1 \leq s\leq \theta 1
\bigm| \bigm| \widetilde x1(s) - x2(s)
\bigm| \bigm| p1\bigl( \theta (n)1 \leq \theta 1
\bigr)
+
+2p - 1\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta 1\leq s\leq \theta
(n)
1
\bigm| \bigm| \widetilde y(n)1 (s) - y
(n)
2 (s)
\bigm| \bigm| p1\bigl( \theta 1 \leq \theta
(n)
1
\bigr)
. (2.10)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1190 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
Consider the last two summands in (2.10) separately. Note that \widetilde x1 and x2 are independent and such
are \widetilde y1 and y2, whence one can deduce, using the Markov property, that
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta
(n)
1 \leq s\leq \theta 1
\bigm| \bigm| \widetilde x1(s) - x2(s)
\bigm| \bigm| p1\bigl( \theta (n)1 \leq \theta 1
\bigr)
\leq \mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq \tau 1
\bigm| \bigm| \eta 1(s) - \eta 2(s)
\bigm| \bigm| p, (2.11)
where \eta k = D(\beta k, vk), k = 1, 2, with \beta 1, \beta 2 being independent Brownian motions, also independent
of w1, w2 (and, therefore, of \widetilde x1, x2), and
v1 = \widetilde x1\bigl( \theta (n)1
\bigr)
, v2 = x2
\bigl(
\theta
(n)
1
\bigr)
,
\tau 1 = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | \eta 1(s) = \eta 2(s)
\bigr\}
.
Thus, by the first inequality of Lemma 2.1, for any q \geq 1,
\mathrm{E} | v1 - v2| q \leq \mathrm{E}
\bigm| \bigm| \bigm| \widetilde x1\bigl( \theta (n)1
\bigr)
- \widetilde y(n)1
\bigl(
\theta
(n)
1
\bigr) \bigm| \bigm| \bigm| q + \mathrm{E}
\bigm| \bigm| \bigm| x2\bigl( \theta (n)1
\bigr)
- y
(n)
2
\bigl(
\theta
(n)
1
\bigr) \bigm| \bigm| \bigm| q \leq 2C1\delta
q/2
n ,
so after taking the conditional expectation in (2.11) and averaging over v1, v2 one gets due to
Lemma 2.2
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta
(n)
1 \leq s\leq \theta 1
\bigm| \bigm| \widetilde x1(s) - x2(s)
\bigm| \bigm| p1\bigl( \theta (n)1 \leq \theta 1
\bigr)
\leq C2\delta
1/2
n (2.12)
for some C2. Similarly,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta 1\leq s\leq \theta
(n)
1
\bigm| \bigm| \widetilde y(n)1 (s) - y
(n)
2 (s)
\bigm| \bigm| p1\bigl( \theta 1 \leq \theta
(n)
1
\bigr)
\leq \mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
0\leq s\leq \tau 2
\bigm| \bigm| \xi 1(s) - \xi 2(s)
\bigm| \bigm| p,
where \xi k =
\bigl(
S(n)(\beta k, vk1, vk2)
\bigr)
1
, k = 1, 2, and
v11 = \widetilde y(n)1 (\theta 1), v12 = \widetilde z(n)1 (\theta 1), v21 = y
(n)
2 (\theta 1), v22 = z
(n)
2 (\theta 1),
\tau 2 = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
1; s | \xi 1(s) = \xi 2(s)
\bigr\}
.
Using both inequalities of Lemma 2.1, applying Lemma 2.3 with uy1 = v11, uy2 = v21, uz1 = v12,
uz2 = v22 and taking expectation one can show that, for some positive C3,
\mathrm{E} \mathrm{s}\mathrm{u}\mathrm{p}
\theta 1\leq s\leq \theta
(n)
1
\bigm| \bigm| \widetilde y(n)1 (s) - y2(s)
\bigm| \bigm| p1\bigl( \theta 1 \leq \theta
(n)
1
\bigr)
\leq C3\delta
1/2 - \varepsilon
n . (2.13)
Substituting (2.12) and (2.13) into (2.11) gives, for some C4,
\mathrm{E}
\bigm| \bigm| \widetilde x2(t) - \widetilde y(n)2 (t)
\bigm| \bigm| p \leq C4\delta
1/2 - \varepsilon
n ,
starting from some N independent of m. Using such an estimate recursively for j = 3, . . . ,m one
finally concludes that
m - 1\sum
j=0
\mathrm{E}
\bigm| \bigm| \widetilde xj(t) - \widetilde y(n)j (t)
\bigm| \bigm| p \leq m - 1\sum
j=1
Cj
4\delta
1/2 - \varepsilon
n .
By Lemma 2.4 there exist positive C5 and a number N \prime \geq N such that, for any n \geq N \prime ,
W1,p
\bigl(
Lt, L
(n)
t
\bigr)
\leq C5
\bigl(
m - 1 + \delta 1/2 - \varepsilon
n
\bigr) 1/p
+ C5
\bigl(
Cm
4 \delta 1/2 - \varepsilon
n
\bigr) 1/p
,
therefore, choosing m = m(n) in such a way that m(n) =
\biggl(
1
4
- \varepsilon
2
\biggr)
\mathrm{l}\mathrm{o}\mathrm{g} \delta - 1
n
\mathrm{l}\mathrm{o}\mathrm{g}C4
concludes the proof.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1191
3. On counting measures associated with the Arratia flow. Recall that \Delta n = \{ u1 < . . .
. . . < un\} , n \in \BbbN , and \{ X(u, t) | t \geq 0, u \in [0; 1]\} is an Arratia flow with zero drift. Denote the
density of a standard m-dimensional Brownian motion killed upon exiting \Delta m by pm0,t. This density
is given via the Karlin – McGregor determinant
pm0,t(x; y) = \mathrm{d}\mathrm{e}\mathrm{t}
\bigm\| \bigm\| gt(xi - yj)
\bigm\| \bigm\|
i,j=1,m
, x, y \in \Delta m,
where gt(a) =
1\surd
2\pi t
e - a2/2t.
Any J = (j1, . . . , jn - k) \in \scrJ n,n - k can be associated with a partition of the set \{ 1, . . . , n\} by the
following procedure. Starting from the partition consisting of singletons, at each step i = 1, . . . , n - k
proceed by merging two subsequent blocks in the current partition with the numbers ji and ji + 1,
the blocks being listed in order of appearance w.r.t. the usual ordering of \BbbN . The resulting partition
will be denoted by \pi (J); the blocks of \pi (J), by \pi 1(J), . . . , \pi k(J). Note that\bigl\{
Jt(u) = J
\bigr\}
=
\Bigl\{
\forall j \in \pi i(J) X(xj , t) = X(xmin\pi i(J), t), i = 1, k
\Bigr\}
.
Lemma 3.1. For all t \in [0; 1], x \in \Delta n, k \in \{ 1, . . . , n\} and J = (j1, . . . , jn - m) \in \scrJ n,n - m,
m \geq k, the density pJ,kt (x; \cdot ) exists. Moreover, pJ,kt (x; \cdot ) \leq pk0,t(x; \cdot ) a.e. if m = k.
Proof. Suppose k = m. Let A be a Borel subset of \Delta k. Define a mapping T : \Delta n \mapsto \rightarrow \Delta k by
the rule T (u)l = umin\pi l(J), l = 1, k. Then
\mathrm{E}
\sum
u1,...,uk\in \{ X(x1,t),...,X(xn,t)\}
1A(u1, . . . , uk)\times 1(Jt(x) = J) \leq
\leq \mathrm{E}1
\bigl(
T
\bigl(
X(x1, t), . . . , X(xn, t)) \in A
\bigr)
=
=
\int
A
pk0,t(x; y) dy.
The Radon – Nikodym theorem yields the claim of the lemma. The cases when A is not a subset of
\Delta k and m \not = k are treated similarly.
Lemma 3.1 is proved.
It is possible to derive an explicit expression for pJ,kt . Denote the boundary of \Delta n by \partial \Delta n.
Additionally, define
\partial \Delta n,j =
\Bigl\{
(u1, . . . , un)
\bigm| \bigm| \bigm| u1 < . . . < uj = uj+1 < . . . < un
\Bigr\}
, j = 1, n - 1.
Let w = (w1, . . . , wn) be a standard Brownian motion. Define \Delta n(a) = \{ u1 < . . . < un \leq a\} ,
n \in \BbbN .
Theorem 3.1. For all t \in [0; 1] and J = (j1, . . . , jn - k) \in \scrJ n,n - k and x \in \Delta n a.e.
pJ,kt (x; y) =
\int
\Delta n - k(t)
dt1 . . . dtn - k
\int
\partial \Delta n,j1
m(dz1)
\int
\partial \Delta n,j2
m(dz2) . . .
\int
\partial \Delta k+1,jn - k
m(dzn - k)( - 1)k2 - k\times
\times \partial
\partial \nu z1
pn0,t1(x, z1)
\partial
\partial \nu z2
pn - 1
0,t2 - t1
\bigl(
Sn
j1z1, z2
\bigr)
. . .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1192 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
. . .
\partial
\partial \nu zn - k
pk+1
0,tn - k - tn - k - 1
\bigl(
Sk+2
jn - k - 1
zn - k - 1, zn - k
\bigr)
pk0,t - tn - k
\bigl(
Sk+1
jn - k
zn - k, y
\bigr)
,
where m is the surface measure on
\bigcup n - 1
j=1 \partial \Delta n, j, the operator
\partial
\partial \nu a
is the outward normal derivative
w.r.t. the a-variables, and the mapping Sm
j : \partial \Delta m,j \rightarrow \Delta m - 1 is given via
Sm
j (u1, . . . , uj , uj+1, uj+2, . . . , um) = (u1, . . . , uj , uj+2, . . . , um), j = 1,m - 1, m \in \BbbN .
The proof is standard and follows the ideas from [11] (Section 3) (see also [12] (Section VII.5)).
Recalling (1.4) note that each pkt (U
(n); \cdot ) satisfies (1.2) with Xt replaced with XU(n)
t =
=
\bigl\{
X
\bigl(
u
(n)
1 , t
\bigr)
, . . . , X
\bigl(
u
(n)
n , t
\bigr) \bigr\}
, n \in \BbbN .
Theorem 3.2. For all k \in \BbbN pkt (U
(n); \cdot ) \nearrow pkt , n \rightarrow \infty , a.e.
Proof. The restrictions imposed on \{ U (n)\} n\geq 1 imply that a.e.
pkt
\bigl(
U (n); \cdot
\bigr)
\leq pkt
\bigl(
U (n+1); \cdot
\bigr)
< pkt , n \in \BbbN .
Put q(y) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty pkt (U
(n); y) a.e. Given a bounded continuous f the dominated convergence
theorem implies \int
\BbbR k
q(y)f(y) dy = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\BbbR k
pkt (U
(n); y)f(y) dy =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{E}
\sum
u1,...,uk\in XU(n)
t
all distinct
f(u1, . . . , uk)
n\sum
i=k
\sum
J\in \scrJ n,n - i
1
\bigl(
Jt
\bigl(
U (n)
\bigr)
= J
\bigr)
=
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{E}
\sum
u1,...,uk\in XU(n)
t
all distinct
f(u1, . . . , uk)1
\bigl( \bigm| \bigm| XU(n)
t
\bigm| \bigm| \geq k
\bigr)
=
=
\int
\BbbR k
pkt (y)f(y) dy,
which proves the assertion.
Theorem 3.1 can be used to study the speed of convergence in Theorem 3.2.
Proof of Theorem 1.2. Let A\varepsilon = [x;x+ \varepsilon ] for some x \in \BbbR and any \varepsilon \ll 1. Consider
0 \leq
\int
A\varepsilon
\bigl(
p1t (y) - p1t
\bigl(
U (n); y
\bigr) \bigr)
dy = \mathrm{E}
\sum
u\in Xt
1A\varepsilon (u) - \mathrm{E}
\sum
u\in XU(n)
t
1A\varepsilon (u).
Using the reasoning of [14] (Appendix B) one shows the existence of a constant C such that\bigm| \bigm| \bigm| \bigm| \bigm| \mathrm{E} \sum
u\in Xt
1A\varepsilon (u) - \sansP (Xt \cap A\varepsilon \not = \varnothing )
\bigm| \bigm| \bigm| \bigm| \bigm| \leq C\varepsilon 2,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON APPROXIMATIONS OF THE POINT MEASURES ASSOCIATED WITH THE BROWNIAN WEB . . . 1193\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \mathrm{E}
\sum
x\in XU(n)
t
1A\varepsilon (u) - \sansP
\bigl(
XU(n)
t \cap A\varepsilon \not = \varnothing
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq C\varepsilon 2.
Therefore,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0+
\varepsilon - 1
\int
A\varepsilon
\bigl(
p1t (y) - p1t
\bigl(
U (n); y
\bigr) \bigr)
dy =
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0+
\varepsilon - 1
\Bigl(
\sansP
\bigl(
Xt \cap A\varepsilon \not = \varnothing
\bigr)
- \sansP
\bigl(
XU(n)
t \cap A\varepsilon \not = \varnothing
\bigr) \Bigr)
. (3.1)
By using the notion of the dual Brownian web
\bigl\{ \widetilde X(u, t) | u \in \BbbR , t \in [0; 1]
\bigr\}
running backwards in
time and the noncrossing property of it [15] (Section 2.2) one has:
\sansP
\bigl(
Xt \cap A\varepsilon \not = \varnothing
\bigr)
- \sansP
\bigl(
XU(n)
t \cap A\varepsilon \not = \varnothing
\bigr)
= \sansP
\Bigl(
\forall j = 1, n X(u
(n)
j , t) /\in A\varepsilon , Xt \cap A\varepsilon \not = \varnothing
\Bigr)
\leq
\leq \sansP
\Bigl( \widetilde X(x+ \varepsilon , t) \not = \widetilde X(x, t), \exists j \in \{ 1, . . . , n - 1\} :
\Bigl( \widetilde X(x, t); \widetilde X(x+ \varepsilon , t)
\Bigr)
\subset
\bigl(
u
(n)
j ;u
(n)
j+1
\bigr) \Bigr)
\leq
\leq \mathrm{E}1
\biggl(
X(x+ \varepsilon , t) - X(x, t) \leq \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n - 1
\bigl(
u
(n)
j+1 - u
(n)
j
\bigr) \biggr)
1
\bigl(
Jt(x, x+ \varepsilon ) = \varnothing
\bigr)
=
=
\int
\BbbR 2
1
\biggl(
y2 - y1 < \mathrm{m}\mathrm{a}\mathrm{x}
j=1,n - 1
\bigl(
u
(n)
j+1 - u
(n)
j
\bigr) \biggr)
p\varnothing ,2
t
\bigl(
(x, x+ \varepsilon ); (y1, y2)
\bigr)
dy1 dy2, (3.2)
since X and \widetilde X have the same distribution. Here,
p\varnothing ,2
t (a; b) = p20,t(a; b) =
1
2\pi t
e -
\| a - b\| 2
2t
\bigl(
1 - e - (b2 - b1)(a2 - a1)
\bigr)
.
Thus, there exists C > 0 such that if (y1, y2) \in \Delta 2, y2 - y1 \leq \delta n, where \delta n = \mathrm{m}\mathrm{a}\mathrm{x}j=1,n - 1
\bigl(
u
(n)
j+1 -
- u
(n)
j
\bigr)
, then
p\varnothing ,2
t
\bigl(
(x;x+ \varepsilon ); (y1, y2)
\bigr)
\leq Cgt(x - y1)\times \varepsilon \delta n.
Substituting the last estimate into (3.2) and returning to (3.1) we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0+
\varepsilon - 1
\int
A\varepsilon
\Bigl(
p1t (y) - p1t
\bigl(
U (n); y
\bigr) \Bigr)
dy \leq C
\int
\BbbR
dy1
y1+\delta n\int
y1
dy2 gt(x - y1)\delta n \leq C\delta 2n
for new C. The application of the Lebesque differentiation theorem completes the proof.
4. Acknowledgments. The authors are very grateful to V. Konarovskyi for valuable comments
and suggestions.
References
1. A. Bensoussan, R. Glowinski, A. Rascanu, Approximation of some stochastic differential equations by the splitting
up method, Appl. Math. and Optim., 25, 81 – 106 (1992).
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1194 A. A. DOROGOVTSEV, M. B. VOVCHANSKII
2. A. A. Dorogovtsev, M. B. Vovchanskii, Arratia flow with drift and Trotter formula for Brownian web, Commun.
Stoch. Anal., 12, No. 1, 89 – 105 (2018).
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66, Kiev (2007) (in Russian).
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Commun. Stoch. Anal., 11, No. 3, 301 – 312 (2017).
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Received 20.08.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
|
| id | umjimathkievua-article-6279 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:26:54Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d5/9818f9ceb033e572517823b9af37c2d5.pdf |
| spelling | umjimathkievua-article-62792022-03-26T11:02:06Z On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. Brownian web Arratia flow Fractional Step Method Splitting, Random Measure Stochastic Flow Stochastic Differential Equations Brownian web Arratia flow Fractional Step Method Splitting, Random Measure Stochastic Flow Stochastic Differential Equations UDC 519.21 We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, we also discuss the convergence of the corresponding approximations of the point measure associated with the Arratia flow in terms of such densities. Найдена скорость слабой сходимости в методе дробных шагов для потока Арратья в терминах расстояния Вассерштейна между образами меры Лебега под действием потока. Введены конечномерные плотности, описывающие последовательности столкновений в потоке Арратья, и получено явное выражение для них. Исследуется сходимость соответствующих аппроксимаций точечных мер для потока Арратья при дискретизации начального интервала. УДК 519.21 Про апроксимацiю асоцiйованих iз броунiвською сiткою точкових мiр за допомогою методу дробових крокiв та дискретизацiї початкового iнтервалу Встановлено швидкiсть слабкої збiжностi в методi дробових крокiв для потоку Арратья в термiнах вiдстанi Вассерштейна мiж образами мiри Лебега пiд дiєю потоку. Введено скiнченновимiрнi щiльностi, що описують послiдовностi зiткнень в потоцi Арратья, та отримано явний вираз для них. Дослiджено збiжнiсть вiдповiдних апроксимацiй точкових мiр для потоку Арратья при дискретизацiї початкового iнтервалу. Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6279 10.37863/umzh.v72i9.6279 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1179-1194 Український математичний журнал; Том 72 № 9 (2020); 1179-1194 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6279/8750 Copyright (c) 2020 Микола Богданович Вовчанський |
| spellingShingle | Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. Dorogovtsev, A. A. Vovchanskii, M. B. On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_alt | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_full | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_fullStr | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_full_unstemmed | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_short | On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval |
| title_sort | on approximations of the point measures associated with the brownian web by means of the fractional step method and the discretization of the initial interval |
| topic_facet | Brownian web Arratia flow Fractional Step Method Splitting Random Measure Stochastic Flow Stochastic Differential Equations Brownian web Arratia flow Fractional Step Method Splitting Random Measure Stochastic Flow Stochastic Differential Equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6279 |
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