Support theorem on stochastic flows with interaction
We prove an analogue of the Stroock–Varadhan theorem for stochastic flows describing a motion of interacting particles in a random media. A version of the Itˆo lemma for functions on a measure-valued process is obtained.
Gespeichert in:
| Datum: | 2006 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2006
|
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/4448 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Support theorem on stochastic flows with interaction / A.Yu. Pilipenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 127–141. — Бібліогр.: 13 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4448 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-44482009-11-11T12:00:39Z Support theorem on stochastic flows with interaction Pilipenko, A.Yu. We prove an analogue of the Stroock–Varadhan theorem for stochastic flows describing a motion of interacting particles in a random media. A version of the Itˆo lemma for functions on a measure-valued process is obtained. 2006 Article Support theorem on stochastic flows with interaction / A.Yu. Pilipenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 127–141. — Бібліогр.: 13 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4448 519.21 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We prove an analogue of the Stroock–Varadhan theorem for stochastic flows describing
a motion of interacting particles in a random media. A version of the Itˆo lemma
for functions on a measure-valued process is obtained. |
| format |
Article |
| author |
Pilipenko, A.Yu. |
| spellingShingle |
Pilipenko, A.Yu. Support theorem on stochastic flows with interaction |
| author_facet |
Pilipenko, A.Yu. |
| author_sort |
Pilipenko, A.Yu. |
| title |
Support theorem on stochastic flows with interaction |
| title_short |
Support theorem on stochastic flows with interaction |
| title_full |
Support theorem on stochastic flows with interaction |
| title_fullStr |
Support theorem on stochastic flows with interaction |
| title_full_unstemmed |
Support theorem on stochastic flows with interaction |
| title_sort |
support theorem on stochastic flows with interaction |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4448 |
| citation_txt |
Support theorem on stochastic flows with interaction / A.Yu. Pilipenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 127–141. — Бібліогр.: 13 назв.— англ. |
| work_keys_str_mv |
AT pilipenkoayu supporttheoremonstochasticflowswithinteraction |
| first_indexed |
2025-07-02T07:41:28Z |
| last_indexed |
2025-07-02T07:41:28Z |
| _version_ |
1836520139376295936 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 127–141
UDC 519.21
A. YU. PILIPENKO
SUPPORT THEOREM ON STOCHASTIC
FLOWS WITH INTERACTION
We prove an analogue of the Stroock–Varadhan theorem for stochastic flows describ-
ing a motion of interacting particles in a random media. A version of the Itô lemma
for functions on a measure-valued process is obtained.
Introduction
Let us consider a flow of interacting particles in a random media. Denote by xt(u) a
position at an instant of time t of a particle starting from a point u ∈ R
d. Let μt be a
distribution of the mass of particles at the moment t. Suppose that each particle does
not change its mass in time. So μt is the image of the measure μ0 under the mapping
xt, i.e. μt = μ0 ◦ x−1
t .
Assume that the motion of each particle depends not only on its position at the current
time moment but also on the distribution of the total mass of particles and satisfies the
following system of stochastic equations:
(0.1)
⎧⎪⎨⎪⎩
dxt(u) = a(xt(u), μt)dt +
∑m
k=1 bk(xt(u), μt)dwk(t),
x0(u) = u, u ∈ R
d,
μt = μ0 ◦ x−1
t , t ∈ [0, T ],
where a, bk : R
d × P → R
d, μ0 ∈ P ; {wk(t)}k=1,m are independent one-dimensional
Wiener processes. Here, P is the space of probability measures on R
d with a topology of
weak convergence.
Systems of stochastic equations of the type (0.1) for flows of interacting particles were
introduced by A.A. Dorogovtsev and P. Kotelenez [1]. Measure-valued processes which
are solutions of (0.1) were initially obtained in [2] as some weak limits of the finite number
of systems of interacting particles. However, there was not considered any flow describing
the individual behavior of particles for the limit process.
The theorem on uniqueness and existence can be proved under some natural conditions
on the coefficients of (0.1) [3]. Moreover, if a, bk are smooth enough, then a process xt(u)
possesses a modification which is differentiable with respect to u (the corresponding
number of times), and its derivatives ∂kxt(u)
∂uk are continuous in (t, u). So, we can consider
the process xt(·), t ≥ 0 as a continuous stochastic process with values in Ck(Rd, Rd).
Here, a space Ck(Rd, Rd) of k times continuously differentiable functions is provided
by the topology of uniform convergence of functions and their derivatives on compact
sets. On the other hand, x = xt(u) can be considered as a random element in the space
Ck = C([0, T ]; Ck(Rd, Rd)). The aim of this article is a characterization of the support of
a stochastic flow xt(u).
2000 AMS Mathematics Subject Classification. Primary 60K35, 60G57.
Key words and phrases. Support of stochastic flow, systems of interacting particles.
127
128 A. YU. PILIPENKO
The corresponding result for the usual stochastic differential equation
dξt = a(ξt)dt +
m∑
k=1
bk(ξt) ◦ dwk(t)
is given by the well-known Stroock—Varadhan theorem [4]. It asserts that, under some
smoothness assumptions on the coefficients of Eq. (0.1), the support of ξ(t) distribution
is equal to the closure in C([0, T ], Rd) of the set
{xψ, ψ is piecewise smooth},
where xψ is a solution of the non-random equation
dxψ(t) = a(xψ(t))dt +
m∑
k=1
bk(xψ(t))dψ(t).
A similar result for the stochastic flow xt(u) generated by s.d.e. depending on the
initial condition
(0.2)
{
dxt(u) = a(xt(u))dt +
∑
k bk(xt(u)) ◦ dwk(t),
x0(u) = u.
was obtained by H.Kunita [5].
Stochastic flows generated by the equation
(0.3) dxt(u) =
∫
Rd
α(xt(u), xt(v))μ(dv)dt +
m∑
k=1
(∫
Rd
βk(xt(u), xt(v))μ(dv)
)
◦ dwk(t)
with interaction were considered in [6, 7].
Note that Eq. (0.3) is a partial case of Eq. (0.1) with a(u, ν) =
∫
Rd α(u, v)ν(dv),
bk(u, ν) =
∫
Rd βk(u, v)ν(dv).
Usually, it is convenient to formulate a Stroock—Varadhan-type theorem for stochastic
equations written in the Stratonovich form. Moreover, it is always supposed that the
coefficients of equations are differentiable more than one time. The coefficients of (0.1)
depends on the measure-valued process μt. So we need some version of the Itô formula
for functions depending on a measure-valued process. We prove the corresponding result
in §2.
The rest of the proof of the Stroock—Varadhan theorem for a stochastic flow xt(u) is
quite standard.
We estimate the “small-balls” probability in §3 and prove that, for each ε > 0 and a
piecewise smooth function ψ, the conditional probability
P
(
sup
|u|<R
sup
t∈[0,T ]
|∇(xt(u) − xψ
t (u))|/ sup
t∈[0;T ]
|w(t) − ψ(t)|
)
converges to zero as δ → 0 + . This gives us the inclusion
{xψ, ψ is piecewise smooth} ⊂ supp Px.
The support is a closed set, so it contains the closure of {xψ, ψ is piecewise smooth},
Section 4 is devoted to the proof of the approximation theorem. Particularly, we prove
that if a sequence of processes {ξε
k(t), ε > 0} is such that wε
k(t) =
∫ t
0 ξε
k(s)ds converges
as ε → 0 to a Wiener process wk(t) in some sense, then a solution of Eq. (0.1) with wε(t)
instead of w(t) converges to the solution of (0.1), but may be with corrected coefficients.
It is worth to mention one principal difference between Eqs. (0.1) and (0.2).
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 129
While studying Eq. (0.2), we can consider different starting points independently.
Then the differentiability with respect to u is just a differentiability with respect to a
parameter. However, all the equations in (0.1) are coupled, and (0.1) should be consid-
ered as an infinite (even uncountable) system of stochastic equations or as a stochastic
equation in the functional space.
1. Preliminaries
Let P be a set of probability measures on R
d with a topology of weak convergence.
Definition 1.1. A pair (xt, μt) is said to be a solution to Eq. (0.1) if
1) mapping x = xt(u, ω) : [0, T ] × R
d × Ω → R
d is measurable in a triple of its
arguments;
2) processes xt(u), μt are adapted to a filtration generated by the Wiener processes
wk(t), k = 1, m;
3) for each u ∈ R
d with probability one, the equality
xt(u) = u +
∫ t
0
a(xs(u), μs)ds +
m∑
k=1
∫ t
0
bk(xs(u), μs)dwk(s), t ∈ [0, T ],
where μt = μ0 ◦ x−1
t is the image of the initial measure μ0 under the random mapping
xt, is satisfied.
Let us introduce the Wasserstein metric on the space P :
γ(μ1, μ2) := inf
κ∈Q(μ1,μ2)
∫
Rd
∫
Rd
|u − v| ∧ 1κ(du, dv),
where Q(μ1, μ2) is a set of probability measures on R
d × R
d with marginals μ1 and μ2.
It is well known that the metric space (P , γ) is complete and separable [8], and the
convergence in metric γ is equivalent to the weak convergence of measures.
Let us formulate the theorem of existence and uniqueness for the solution of Eq. (0.1).
Theorem 1.1 [3]. Assume that
∃L > 0 ∀u1, u2 ∈ R
d ∀μ1, μ2 ∈ P :
(1.1)
|a(u1, μ1) − a(u2, μ2)| +
m∑
k=1
|bk(u1, μ1) − bk(u2, μ2)| ≤
≤ L(|u1 − u2| + γ(μ1, μ2)).
Then there exists a unique solution for Eq. (0.1).
Further, we always assume that the conditions of Theorem 1.1 are satisfied.
Put ã(x, t) := a(x, μt(ω)), b̃k(x, t) := bk(x, μt(ω)). Observe that if xt, μt satisfy (0.1)
then xt satisfies the following Itô equation:
(1.2)
{
dxt(u) = ã(xt(u), t)dt +
∑m
k=1 b̃k(xt(u), t)dwk(t), t ≥ 0,
x0(u) = u.
The functions ã, b̃k are continuous and satisfy the Lipschitz condition in x. So, the flow
xt(u) has a modification which is continuous in (t, u) (cf.[5]). Moreover, if the coefficients
a, bk are (n + 1) times continuously differentiable with respect to u and have bounded
derivatives, then the partial derivatives ∂jxt(u)
∂uj , j = 1, n exist and are continuous in (t, u).
Therefore, we can consider the process xt(·) as a continuous process with values in
Cn(Rd, Rd) or as a random element in Cn = C([0, T ], Cn(Rd, Rd)), where the space of
130 A. YU. PILIPENKO
functions Cn(Rd, Rd) differentiable n-times has a topology of uniform convergence of the
functions and their derivatives on compact sets.
Usually, to study the support of a solution for some stochastic equation, one needs a
stronger condition of smoothness than the Lipschitz condition. That’s why we need the
following definition of a derivative for functions depending on a measure-valued argument
[9].
Let M be a space of finite measures on R
d with a topology of weak convergence, and
let F be a continuous function from M to R.
Definition 1.2. Assume that, for each μ ∈ M, u ∈ R
d, there exists the limit
(1.3)
δF (μ)
δμ
(u) := lim
ε→0+
F (μ + εδu) − F (μ)
ε
which is continuous in (μ, u), where δu is a unit measure concentrated at the point u ∈ R
d.
Then function F is said to be continuous differentiable, and δF
δμ is its derivative.
Higher order derivatives δnF
δμn (u1, . . . , un) are defined iteratively.
Remark 1. Let μt be a process from (0.1). Then the full mass of μt is not changing in
time, μt(Rd) = μ0(Rd) = 1. So, in general, in order to consider Eq. (0.1), it is natural to
know the coefficients in the space R
d ×P only, but not in the larger space R
d ×M. Note
also that if μ ∈ P , then μ+εδu /∈ P as ε �= 0. We can replace the definition of continuous
differentiability (1.3) remaining in a class P by using Newton—Leibnitz formula:
∃ δF
δμ ∈ C(P × R
d) ∀μ1, μ2 ∈ P :
(1.4) F (μ2) − F (μ1) =
∫ 1
0
∫
Rd
δF (τμ2 + (1 − τ)μ1)
δμ
(u)(μ2 − μ1)(du)dτ.
The proof of (1.4) can be done for discrete measures at first. Then it is not difficult
to extend (1.4) for an arbitrary measure by continuity (recall that δF
δμ is continuous in
(μ, u)). However, formula (1.4) seems to be less natural than (1.3). Moreover, if the
function F is defined on P and satisfies (1.4) for all μ1, μ2 ∈ P , then its extension
F̃ (μ) := F
(
μ
μ(Rd)
)
to the space of all finite measures M is continuous differentiable in
the sense of Definition 1.2.
The following statement will be useful in our investigations.
Lemma 1.1. Assume that F : P → R satisfies (1.4), where the function δF
δμ is bounded
and satisfies the Lipschitz condition in u :
∃L1 ∀μ ∈ P ∀u ∈ R
d :
∣∣∣∣δF (μ)
δμ
(u)
∣∣∣∣ ≤ L1;
∃L2 ∀μ ∈ P ∀u1, u2 ∈ R
d :
∣∣∣∣δF (μ)
δμ
(u1) − δF (μ)
δμ
(u2)
∣∣∣∣ ≤ L2|u1 − u2|.
Let ν ∈ P be a probability measure. Then there exists a constant L = L(F ) such that,
for all measurable functions f1, f2 : R
d → R
d, the following inequality holds:
(1.5) |F (ν2) − F (ν1)| ≤ L
∫
Rd
|f1(u) − f2(u)| ∧ 1ν(du),
where νi = ν ◦ f−1
i , i = 1, 2.
Proof. Put μτ = τν2 + (1 − τ)ν1. Then, due to (1.4), we have
|F (ν2) − F (ν1)| ≤
∫ 1
0
∣∣∣∣∫
Rd
δF (μτ )
δμ
(u)(ν2(du) − ν1(du))
∣∣∣∣ dτ =
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 131
=
∫ 1
0
∣∣∣∣∫
Rd
(
δF (μτ )
δμ
(f2(u)) − δF (μτ )
δμ
(f1(u))
)
ν(du)
∣∣∣∣ dτ ≤
≤
∫ 1
0
∫
Rd
(L2|f2(u) − f1(u)|) ∧ 2L1ν(du)dτ ≤
≤ (L2 + 2L1)
∫
Rd
|f2(u) − f1(u)| ∧ 1ν(du).
Lemma 1.1 is proved.
Remark 2. While studying carefully the proof of Theorem 1.1 (see [3]), one can notice
that it is sufficient to demand a condition of type (1.5) instead of (1.1). So, the existence
of the bounded derivatives ∂a(x,μ)
∂x , ∂
∂u
δa(x,μ)
δμ (u), ∂bk(x,μ)
∂x , and ∂
∂u
δbk(x,μ)
δμ (u) is enough
for Theorem 1.1. This condition will always be supposed in §2, §3. However, it seems to
the author that the formulation of Theorem 1.1 is methodologically more useful than a
similar theorem with the condition
∃L > 0 ∀x, u ∈ R
d ∀μ :∣∣∣∣∂a(x, μ)
∂x
∣∣∣∣ +
∣∣∣∣ ∂
∂u
δa(x, μ)
δμ
(u)
∣∣∣∣ +
m∑
k=1
(∣∣∣∣∂bk(x, μ)
∂x
∣∣∣∣ +
∣∣∣∣ ∂
∂u
δbk(x, μ)
δμ
(u)
∣∣∣∣) ≤ L
instead of (1.1).
2. Itô formula for functions of measure-valued argument
In this section, we prove a version of the Itô formula for a process F (ξt, μt), where μt
is defined in (0.1), and the process ξt has a stochastic differential
(2.1) dξt = αtdt +
m∑
j=1
βj
t dwj(t).
The main result is as follows:
Theorem 2.1. Assume that a function F : R
k × M → R is such that the following
derivatives exist, bounded, and continuous in all arguments:
∂F (x, μ)
∂x
,
∂2F (x, μ)
∂x2
,
δF (x, μ)
δμ
(u),
δ2F (x, μ)
δμ2
(u, v),
∂
∂u
δF (x, μ)
δμ
(u),
∂2
∂u2
δF (x, μ)
δμ
(u),
∂2
∂x∂u
δF (x, μ)
δμ
(u),
∂2
∂u∂v
δ2F (x, μ)
δμ2
(u, v).
Suppose that a process ξt = (ξ1
t , . . . , ξk
t ) has the stochastic differential (2.1), and the
functions a, bk satisfy the assumptions of Theorem 1.1 and are bounded.
Let us introduce the following notations:
〈f, μ〉 =
∫
fdμ, σ2
ξ (t) =
m∑
l=1
βl
t (βl
t)
∗,
σ2
ξμ(t, x, μ) =
m∑
l=1
βl
tb
∗
l (x, μ), σ2
μμ(x, y, μ) =
m∑
l=1
bl(x, μ)b∗l (y, μ),
σ2
μ(x, μ) = σ2
μμ(x, x, μ),
where ∗ is the transposition operator of a matrix.
Set
L1(t)F (x, μ) = F ′
x(x, μ)αt +
1
2
sp(F ′′
xx(x, μ)σ2
ξ,ξ(t)),
132 A. YU. PILIPENKO
L2F (x, μ) =
∫
Rd
(
∂
∂u
δF (x, μ)
δμ
(u)a(u, μ) +
1
2
sp
∂2
∂u2
δF (x, μ)
δμ
(u)σ2
μ(u, μ)
)
μ(du)+
(2.2) +
1
2
∫
Rd
∫
Rd
sp
∂2
∂u∂v
δ2F (x, μ)
δμ2
(u, v)σ2
μμ(u, v, μ)μ(du)μ(dv).
Then
dF (ξt, μt) = L1(t)F (ξt, μt)dt + F ′
1(ξt, μt)
m∑
l=1
βl
tdwl(t)+
+L2F (ξt, μt)dt +
m∑
l=1
∫
Rd
∂
∂u
δF (ξt, μt)
δμ
(u)bl(u, μt)μt(du)dwl(t)+
(2.3) +
1
2
sp
∫
Rd
∂2
∂ξ∂u
δF (ξt, μt)
δμ
(u)σ2
ξμ(t, u, μt)μ(du)dt.
Proof. Let us verify formula (2.3) for a function F (x, μ) = F (μ) depending on the second
argument only. The general case can be proved similarly but with additional routine
calculations.
Let
μn =
n∑
j=1
cj,nδuj,n , n ∈ N, cj,n ≥ 0,
∑
j
cj,n = 1
be a sequence of discrete measures which converges weakly to μ0 as n → ∞.
Denote, by xn
t (u), a solution of Eq. (0.1) with initial measure μn instead of μ0,
μn
t := μn ◦ (xn
t )−1. The plan of the proof is to obtain the Itô formula for μn
t at first, and
then to pass to a limit as n → ∞.
Remark 3. The Itô formula for superprocesses with interaction was obtained in [10].
There, a function F (μ) was approximated by polynomials in μ, and the proof is a hard
technical work. It is worth mentioning that the process μt is neither a superprocess nor
the Fleming—Viot process. This fact can be easily checked if we write down and compare
their generators [3,9].
Observe that the measure μn
t is also discrete, μn
t =
∑n
j=1 cj,nδxn
t (uj,n). To simplify
notations, we will write cj , uj instead of cj,n, uj,n.
The processes xn
t (uk), k = 1, n satisfy a finite system of stochastic equations
dxn
t (uk) = a(xn
t (uk),
n∑
j=1
cjδxn
t (uj))dt+
+
m∑
l=1
bl(xn
t (uk),
n∑
j=1
cjδxn
t (uj))dwl(t), k = 1, n.
Consider a function f : R
nd → R defined as follows:
f(v1, . . . , vn) = F (
n∑
j=1
cjδvj ).
Then F (μn
t ) = f(xn
t (u1), . . . , xn
t (un)). Let us check that f is twice continuously differ-
entiable.
Assume for simplicity that n = 2 and calculate 〈 ∂f
∂v1
, e〉Rd , where e ∈ R
d (recall that
v1 ∈ R
d).
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 133
Set ν = c1δv1 + c2δv2 , ντ = (1− τ)(c1δv1 + c2δv2) + τ(c1δv1+εe + c2δv2). Formula (1.4)
implies
lim
ε→0
f(v1 + εe, v2) − f(v1, v2)
ε
= lim
ε→0
1
ε
∫ 1
0
(∫
Rd
δF (ντ )
δμ
(u)(c1δv1+εe + c2δv2)(du)
−
∫
Rd
δF (ντ )
δμ
(u)(c1δv1 + c2δv2)(du)
)
= lim
ε→0
1
ε
∫ 1
0
c1
(
δF (ντ )
δμ
(v1 + εe) − δF (ντ )
δμ
(v1)
)
dτ
= c1
(
∂
∂v1
δF (ν)
δμ
, e
)
Rd
.
Here, while grounding the passage to the limit, we use the continuous differentiability of
δF (μ)
δμ (u) with respect to the parameter u and the boundedness of its derivative.
So, we have verified that
(2.4)
δf
∂vj
= cj∇δF (ν)
δμ
(vj),
where ∇ is the derivative with respect to the argument vj .
Analogously,
(2.5)
∂2f
∂vi∂vj
= cicj∇1∇2
δ2F (ν)
δμ2
(vi, vj) + cj∇2 δF (ν)
δμ
(vj)δij ,
where δij is the Kronecker symbol.
Let us apply the usual Itô lemma to the process
ηn
t = f(xn
t (u1), . . . , xn
t (un)) = F (
n∑
j=1
cjδxn
t (uj)) = F (μn
t )
and use (2.4), (2.5):
dηn
t =
∑
i
ci∇δF (μn
t )
δμ
(xn
t (ui))
(
a(xn
t (ui), μn
t )dt +
∑
l
bl(xn
t (ui), μn
t )dwl(t)
)
+
+
1
2
∑
i,j
cicjsp
(
∇1∇2
δ2F (μn
t )
δμ2
(xn
t (ui), xn
t (uj))σ2
μμ(xn
t (ui), xn
t (uj), μn
t )
)
dt+
(2.6) +
1
2
∑
i
cisp
(
∇2 δF (μn
t )
δμ
(xn
t (ui))σ2
μ(xn
t (ui), μn
t )
)
dt.
Observe that, for each g = g(u), h = h(u, v), we have
∑
i
cig(xn
t (ui)) =
∫
Rd
g(u)μn
t (du),
∑
i,j
cicjh(xn
t (ui), xn
t (uj)) =
∫
Rd
∫
Rd
h(u, v)μn
t (du)μn
t (dv).
134 A. YU. PILIPENKO
So, the expression in (2.6) is equal to
(2.7) dηn
t = L2F (μn
t )dt +
m∑
l=1
∫
Rd
∂
∂u
δF (μn
t )
δμ
(u)bl(u, μn
t )μn
t (du)dwl(t),
where the operator L2 is defined in (2.2).
The initial distributions μn are weakly convergent to μ0. Therefore [3], for each t ≥ 0,
we have the convergence μn
t to μt in probability. Taking a subsequence, if it is needed,
it can be assumed without loss of generality that, for almost all ω ∈ Ω and almost all
t ≥ 0,
μn
t (ω) ⇒ μt(ω), n → ∞.
All functions in the differentiated expression in (2.7) are bounded and continuous. The
proof that we can pass to a limit under the integral sign in (2.7) follows from the next
lemma.
Lemma 2.1. Let {νn, n ≥ 1} ⊂ P be a non-random sequence of probability measures
which converges weakly to ν0. Assume that the function g : R
d × P → R is continuous
and bounded.
Then ∫
Rd
g(u, νn)νn(du) →
∫
Rd
g(u, ν0)ν0(du), n → ∞.
Proof. ∣∣∣∣∫
Rd
g(u, νn)νn(du) −
∫
Rd
g(u, ν0)ν0(du)
∣∣∣∣ ≤
≤
∫
Rd
|g(u, νn) − g(u, ν0)|νn(du) +
∣∣∣∣∫
Rd
g(u, ν0)νn(du) −
∫
Rd
g(u, ν0)ν0(du)
∣∣∣∣ .
The second item converges to zero due to the weak convergence νn ⇒ ν0, n → ∞.
To estimate the first item, observe that the set {νn, n ≥ 0} is a weak compact. By
the Prokhorov theorem, for each ε > 0, there exists a compact Kε ⊂ R
d such that
νn(Rd \ Kε) < ε for all n ≥ 0.
The function g is continuous on a compact Kε × {νn; n ≥ 0} ⊂ R
d ×P and, hence, it
is uniformly continuous. Thus,
sup
u∈Kε
|g(u, νn) − g(u, ν0)| → 0, n → ∞.
Therefore,
lim
n→∞
∫
Rd
|g(u, νn) − g(u, ν0)|νn(du) ≤
≤ lim
n→∞
∫
Kε
|g(u, νn) − g(u, ν0)|νn(du) + lim
n→∞
∫
Rd\Kε
(|g(u, νn)| + |g(u, ν0)|)νn(du) ≤
≤ 2 sup
u∈Rd
sup
ν∈P
|g(u, ν)|ε.
The number ε > 0 is arbitrary. Thus, Lemma 2.1 and also Theorem 2.1 are proved.
3. Lower estimate for support of xt(u)
Let ψ = (ψ1, . . . , ψm) : [0, T ] → R
m be a piecewise smooth function. Denote, by xψ ,
a solution of the following (deterministic) equation
(3.1)
⎧⎪⎨⎪⎩
dxψ
t (u) = ã(xψ
t (u), μψ
t )dt +
∑m
k=1 bk(xψ
t (u), μψ
t )ψk(t), u ∈ R
d,
μψ
t = μ0 ◦ (xψ
t )−1, t ∈ [0, T ],
xψ
0 (u) = u,
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 135
where
(3.2) ã(x, μ) = a(x, μ) − 1
2
m∑
k=1
(
∂bk(x, μ)
∂x
bk(x, μ) +
∫
Rd
∂
∂v
δbk(x, μ)
δμ
(v)bk(v, μ)μ(dv)
)
is a corrected coefficient.
The aim of this section is to show that the support suppx of the flow xt(u) contains
the set
S = {xψ : ψ is piecewise smooth}.
Note that if we replace the integral with respect to ψk(t) by the Stratonovich integral
w.r.t. wk(t), then we obtain Eq. (0.1).
Theorem 3.1. Assume that the coefficients a, bk are such that the derivatives
∂α
∂ūα
∂β
∂xβ
δja(x, μ)
δμj
(ū) and
∂α
∂ūα
∂β
∂xβ
δjbk(x, μ)
δμj
(ū)
are bounded and continuous in all their arguments, where j = 0, n + 3, ū = (u1, . . . , uj),
α = (α1, . . . , αj), αi ≥ 0, β = (β1, . . . , βd), βi ≥ 0, α1 + · · · + αj + β1 + · · ·+ βd ≤ n + 3.
Then the support of xt considered as a random element in C([0, T ], Cn(Rd, Rd)) contains
the set S.
Proof. We restrict ourselves only to the case n = 0 and ψ ≡ 0. The case of arbitrary ψ
can be considered with the use of the Girsanov theorem (see the reasoning of Theorem
[11]). The reasoning for arbitrary n is similar to the case n = 0 (see [6] for details).
Lemma 3.1. Assume that the conditions of Theorem 3.1 are satisfied. Then, for each
ball U ⊂ R
d and each ε > 0, δ > 0, we have the following convergence of conditional
probabilities:
lim
δ→0+
P
⎛⎜⎝ sup
u∈U
t∈[0,T ]
∣∣∣∣∫ t
0
bk(xs(u), μs) ◦ dwk(s)
∣∣∣∣ > ε/‖w‖ < δ
⎞⎟⎠ = 0,
where | · | is a norm in R
d, ‖w‖ = supt∈[0,T ] maxi=1,m |wi(t)| is a norm in C([0, T ], Rm).
Proof. Due to the Sobolev embedding theorems [12], it is enough to verify that
lim
δ→0+
P
(
sup
t∈[0,T ]
‖
∫ t
0
bk(xs(·), μs) ◦ dwk(s)‖W 1
p (U) ≥ ε/‖w‖ < δ
)
= 0,
where p > d, ‖f‖W 1
p (U) =
(∫
U
(|f |p + |∇f |p)dx
)1/p
. The proof of the corresponding fact
can be done similarly [7]. We need to use the Itô formula (2.3), boundedness of a, bk and
their derivatives, and some estimates of the moments for xt(u) and ∂xt(u)
∂u (to estimate
the moments, one can apply results in [5], §4.5, 4.6, to (1.2)).
Let us show now that x0 ∈ suppx. To verify this, it is sufficient to check that, for
every ball U ⊂ R
d and ε > 0,
P
(
sup
t∈[0,T ]
sup
u∈U
|xt(u) − x0
t (u)| < ε
)
> 0.
136 A. YU. PILIPENKO
Denote
∑m
k=1
∫ t
0 bk(xs(u), μs) ◦ dwk(s) by ot(u). Then
xt(u) = u +
∫ t
0
ã(xs(u), μs)ds + ot(u),
x0
t (u) = u +
∫ t
0
ã(x0
s(u), μ0
s)ds,
where ã is defined in (3.2), x0
t (u) is a solution of (3.1) with ψ ≡ 0, μ0
t = μ0 ◦ (x0
t )
−1.
Let us use inequality (1.5) to estimate the difference |xt(u) − x0
t (u)|, u ∈ U :
|xt(u) − x0
t (u)| ≤ L
∫ t
0
(
|xs(u) − x0
s(u)| +
∫
Rd
|xs(v) − x0
s(v)| ∧ 1μ0(dv)
)
ds + |ot(u)| ≤
≤ 2L
∫ t
0
(sup
v∈U
|xs(v) − x0
s(v)| + μ(Rd \ U))ds + sup
s∈[0,t]
v∈U
|os(v)|,
where L > 0 is a constant.
So, by the Gronwall lemma, we get the estimate
sup
v∈U
|xt(v) − x0
t (v)| ≤
⎛⎜⎝ sup
v∈U
s∈[0,t]
|os(v)| + μ(Rd \ U)
⎞⎟⎠ e2LT .
Let ε > 0, and let U ⊂ R
d be fixed. Choose a ball Ũ ⊃ U such that μ(Rd \ Ũ)e2LT < ε
2 .
Then we choose δ > 0 such that
P
⎛⎜⎝ sup
u∈U
t∈[0,T ]
|ot(u)| <
e−2LT ε
2
/
‖w‖ < δ
⎞⎟⎠ > 0.
Therefore,
P
⎛⎜⎝ sup
u∈U
t∈[0,T ]
|xt(u) − x0
t (u)| ≤ ε
⎞⎟⎠ ≥ P
⎛⎜⎝ sup
u∈U
t∈[0,T ]
|xt(u) − x0
t (u)| ≤ ε
⎞⎟⎠ ≥
≥ P
⎛⎜⎝( sup
u∈U
t∈[0,T ]
|ot(u)| + μ(Rd \ Ũ))e2LT ≤ ε
/ ‖w‖ < δ
⎞⎟⎠ · P (‖w‖ < δ) > 0.
Moreover, it is easy to verify that
P
⎛⎜⎝ sup
u∈U
t∈[0,T ]
|xt(u) − x0
t (u)| ≤ ε
/ ‖w‖ < δ
⎞⎟⎠ → 0, δ → 0 + .
Theorem 3.1 is proved.
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 137
4. Approximation theorem.
Let us consider the system⎧⎪⎨⎪⎩
d
dtx
ε
t (u) = a(xε
t (u), με
t ) +
∑m
k=1 bk(xε
t (u), με
t )ξ
ε
k(t),
xε
0(u) = u, u ∈ R
d,
με
t = μ0 ◦ (xε
t )
−1, t ∈ [0, T ],
where the stochastic processes ξε
k(t) are such that wε
k(t) =
∫ t
0 ξε
k(s)ds converges as ε → 0
to a Wiener process wk(t) in some sense.
In this section, we give some sufficient conditions that ensure the convergence of
(xε
t , μ
ε
t ) to (xt, μt), where (xt, μt) is a solution of (0.1).
Theorem 4.1. Assume that the functions a, bk, k = 1, m satisfy the conditions of The-
orem 3.1 and
A1. The collection of processes
Kε(s) =
N∑
l,j=1
(∫ T
s
|E (ξε
l (r)/Gε
s) |dr
) (
1 + |ξε
j (s)|) , ε > 0
is uniformly Lp-bounded for each p ≥ 1:
∀p ≥ 1 sup
ε
sup
s
EKp
ε (s) = Kp < ∞
and is uniformly exponentially bounded in mean:
∀λ > 0 sup
ε
E exp{λ
∫ T
0
Kε(s) ds} < ∞,
where
Gε
t := σ(ξε
k(z) : 0 ≤ z ≤ t, k = 1, . . . , N).
A2. For each s ∈ [0, T ], the following convergence holds in L2 :∫ T
s
|E(
ξε
l (z)/Gε
s
)| dz → 0, ε → 0.
A3. There exist the deterministic bounded functions σlm(t), 1 ≤ l, m ≤ N such that,
for all s < t, we have the following convergence in L1 :
E
(∫ t
s
ξε
l (τ)dτ
∫ τ
s
ξε
m(v)dv/Gε
s
)
−→
ε→0
∫ t
s
σlm(z)dz.
Then the distribution in the space
Cn × C([0; T ], Rm) × C([0; T ],P)
of the triple (xε
t , w
ε(t), με
t ) converges weakly as ε → 0 to a limit measure such that:
1) the distribution of the second coordinate w(t) = (w1(t), . . . , wm(t)) is a Wiener
process with the covariation matrix ‖ ∫ t
0
(σij(s) + σji(s))ds‖;
138 A. YU. PILIPENKO
2) the processes xt(u), w(t) and μt are connected by the system
(4.1)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dxt(u) =
[
a(xt(u), μt) +
∑m
j,k=1
(
∂bk(xt(u),μt)
∂x bj(xt(u), μt)+
+
∫
Rd
∂
∂v
δbk(xt(u),μt)
δμ (v)bj(v, μt)μt(dv)
)]
σkj(t)dt +
∑m
k=1 bk(xt(u), μt)dwk(t),
x0(u) = u, u ∈ R
d,
μt = μ0 ◦ x−1
t , t ∈ [0, T ].
If, in addition, the processes wε(t) and w(t) are given on the same probability space and
wε(t) → w(t), ε → 0 in probability for all t ∈ [0; T ], then
(4.2) ∀p ≥ 1 ∀R > 0 E sup
|u|≤R
sup
t∈[0;T ]
n∑
k=0
|∇k(xε
t (u) − xt(u))|p → 0, ε → 0.
An example of the sequence wε(t) with σij =
{
0, i �= j
0.5, i = j
is a polygonal approxi-
mation:
ξj
ε(t) = ε−1(wj((k + 1)ε) − wj(kε))
if t ∈ [kε; (k + 1)ε).
Theorem 4.1 and this example imply that the support of the xt(u) distribution in the
space Cn is contained in a set
{xψ : ψ is a piecewise linear function, xψ satisfies (3.1)}.
It is easy to show that if {ψn} is a sequence of piecewise smooth functions such that
sup
t∈[0;T ]
|ψn(t) − ψ0(t)| → 0, n → ∞
ess supt∈[0;T ]|ψ′
n(t) − ψ′
0(t)| → 0, n → ∞,
then we have the convergence xψn → xψ0 , n → ∞ in Cn.
So, the closure of
{xψ : ψ is piecewise linear function, xψ satisfies (3.1)}.
contains a set
{xψ : ψ is piecewise smooth function, xψ satisfies (3.1)}.
Therefore, we have obtained the following support theorem.
Theorem 4.2. Assume that the conditions of Theorems 3.1, 4.1 are satisfied. Then the
support of the xt(u) distribution in Cn is equal to the closure in Cn of the set
{xψ : ψ is a piecewise smooth function, xψ satisfies (3.1)}.
Proof of the Theorem 4.1. We follow the ideas of [5] Ch.5, see also [7] for some details
related to the equations with interaction. Note, at first, that, under the conditions of
Theorem 4.1, the processes wε converge weakly to a Wiener process w with required
covariation [5] §5.7.
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 139
Lemma 4.1.
∀m ∈ N ∀R > 0 ∃C = CR,m > 0 : sup
ε
E‖xε(·, t)‖2m
W k
2m(BR)
≤ C,
∀t′, t′′ : sup
ε
E‖xε(·, t′) − xε(·, t′′)‖2m
W k
2m(BR)
≤ C|t′ − t′′|2− 1
m ,
where BR is a ball in R
d with the center at zero and with radius R.
The course of the proof is similar to [5] §5.2 (see also [7] for the equations with inter-
action), but we have to use Theorem 2.1 instead of the formula of the usual integration
by parts.
It is well known that the Sobolev space Wn+1
p,loc is compactly embedded into the space
Cn(Rd, Rd) if p > d. Thus, by Theorem 1.4.7 [5], we have the weak relative compactness
of the family xε, wε in Cn × C([0; T ], Rm).
The next lemma implies that if a sequence of couples (xεk , wεk) converges weakly
to some limit (x, w), then a sequence of triples (xεk
. (·), wεk (·), μεk
. ) converges weakly to
(x.(·), w(·), μ.), where μt = μ0 ◦ (xt)−1.
Lemma 4.2. Assume that a sequence of random elements yn = yn
t (u) converges weakly
to y0 = y0
t (u) in C([0; T ], C(Rd, Rd)). Then, for each probability measure μ, we have the
weak convergence of measure-valued processes μn
t = μ ◦ (yn
t )−1 → μ0
t = μ ◦ (y0
t )−1 in the
space C([0; T ],P).
Proof. By the Skorokhod theorem [13], we can assume that all the elements yn, n ≥ 0
are given on the same probability space and with probability one:
∀R > 0 : sup
t∈[0;T ]
sup
|u|<R
|yn
t (u) − y0
t (u)| → 0, n → ∞.
Let ω be from the corresponding set of full probability. Denote, by κ̃t, the image of the
measure μ w.r.t. mapping (yn
t , y0
t ). Then
sup
t∈[0;T ]
γ(μn
t , μ0
t ) = sup
t∈[0;T ]
sup
κ∈Q(μn
t ;μ0
t )
∫ ∫
|u − v| ∧ 1κ(du, dv) ≤
(4.3) ≤ sup
t∈[0;T ]
∫ ∫
|u − v| ∧ 1κ̃t(du, dv) = sup
t∈[0;T ]
∫
|yn
t (u) − y0
t (u)| ∧ 1μ(du).
Let ε > 0 be fixed. Choose R > 0, n0 ≥ 1 such that μ(u : |u| ≥ R) < ε and
∀n ≥ n0 : sup
t∈[0;T ]
sup
|u|<R
|yn
t (u) − y0
t (u)| < ε.
Then the right-hand side of (4.3) is less than 2ε. Lemma 4.2 is proved.
If we verify that the limit triple (xt, wt, μt) satisfies Eq. (4.1), then, by uniqueness of
the solution, we get the desired convergence (xε
t , w
ε
t , μ
ε
t ) → (xt, wt, μt).
To check (4.1) for a limit, it is sufficient to verify that
M(u, t) = x(u, t) − u −
∫ t
0
[
a(xz(u), μz) +
m∑
j,k=1
(∂bk(xz(u), μz)
∂x
bj(xz(u), μz)+
(4.4) +
∫
Rd
∂
∂v
δbk(xz(u), μt)
δμ
(v)bj(v, μz)μz(dv)
)
σkj(z)
]
dz
140 A. YU. PILIPENKO
is a continuous L2-martingale with respect to Ft = σ
(
w(s), xs(u), s ∈ [0, t], u ∈ R
d
)
, t ∈
[0, T ], with the square characteristics
(4.5)
〈M (i)(u, t), M (j)(v, t)〉 =
∑
k,l
∫ t
0
∫
b
(i)
k (x(u, s), μs)b
(j)
l (x(v, s), μs))(σkl(s) + σlk(s))ds
(4.6) 〈M (i)(u, t), wj(t)〉 =
∑
k
∫ t
0
∫
b
(i)
k (xs(u)μs)(σkj(s) + σjk(s))ds.
The process bk(xε
t (u), με
t ) is differentiable with respect to t by Theorem 2.1 and
∂
∂t
(bk(xε
t (u), με
t )) =
=
∂
∂x
(bk(xε
t )(u), με
t ))
⎛⎝a(xε
t (u), με
t ) +
m∑
j=1
bj(xε
t (u), με
t )ξ
ε
j (t)
⎞⎠ +
+
∫
Rd
∂
∂v
δbk(xε
t (u), με
t )
δμ
(v)
⎛⎝a(v, με
t ) +
m∑
j=1
bj(v, με
t )ξ
ε
j (t)
⎞⎠ με
t (dv).
Then
xε
t (u) − xε
s(u) −
∫ t
s
a(xε
z(u), με
z)dz =
m∑
k=1
∫ t
s
bk(xε
z(u), με
z)ξ
ε
k(z)dz =
=
m∑
k=1
bk(xε
s(u), με
s)
∫ t
s
ξk
z dz +
m∑
k=1
∫ t
s
∫ z
s
∂
∂r
(bk(xε
r(u), με
r)) drξε
k(z)dz =
=
m∑
k=1
bk(xε
s(u), με
s)
∫ t
s
ξε
k(z)dz+
+
m∑
k=1
∫ t
s
∫ z
s
[
∂
∂x
(bk(xε
r)(u), μr))
⎛⎝a(xε
r(u), με
r) +
m∑
j=1
bj(xε
r(u), με
r)ξ
ε
j (r)
⎞⎠ +
+
∫
Rd
∂
∂v
δbk(xε
r(u), με
r)
δμ
(v)
⎛⎝a(v, με
r) +
m∑
j=1
bj(v, με
r)ξ
ε
j (r)
⎞⎠]
με
r(dv)drξε
k(z)dz.
Let s ∈ [0; T ], sj ∈ [0; s), uj ∈ R
d, l ∈ N. Put
Φε = f(xε
s1
(u1), . . . , xε
sl
(ul), wε(s1), . . . , wε(sl)),
where f is some continuous bounded function,
Φ = f(xs1(u1), . . . , xsl
(ul), w(s1), . . . , w(sl)).
We recall that {εn} is such that the sequence of triples (xεn
. (·), wεn (·), μεn
. ) converges
weakly to some limit (x.(·), w(·), μ.)
The following statement can be proved similarly to Lemma 5 [7]; see also the reasoning
of Lemma 4.2.
SUPPORT THEOREM ON STOCHASTIC FLOWS WITH INTERACTION 141
Lemma 4.3. Let α = α(x, μ) be a bounded continuous function. Then
E
∫ t
s
α(xεn
r (u), μεn
r )drΦεn → E
∫ t
s
α(xr(u), μr)drΦ, n → ∞;
E
∫ t
s
α(xεn
r (u), μεn
r )ξεn
k (r)drΦεn → 0, n → ∞;
E
∫ t
s
∫ z
s
α(xεn
r (u), μεn
r )ξεn
k (r)ξεn
j (z)drdzΦεn → E
∫ t
s
α(xr(u), μr)σkj(r)drΦ, n → ∞.
As a corollary of Lemma 4.3, we have a fact that M(u, t) is Ft−martingale. The
reasoning for (4.5), (4.6) is similar. That is, the limit triple satisfies (4.1). So we have
proved the desired weak convergence.
The proof of (4.2) is quite classical and may be roughly formulated as follows. Let a
sequence of solutions of SDEs given on the same probability space be weakly relatively
compact, and let the uniqueness for the limit equation hold. Then the sequence converges
strongly (cf. Theorem 5.2.8 [5]).
Bibliography
1. Dorogovtsev A.A. and Kotelenez P., Smooth Stationary Solutions of Quasilinear Stochastic
Partial Differential Equations: 1. Finite Mass, Preprint No. 97-145, Cleveland (1997).
2. Kotelenez P., A class of quasilinear stochastic partial differential equations of McKean-Vlasov
type with mass conservation., Probab. Theory Related Fields 102 (1995), no. 2, 159–188.
3. Dorogovtsev A.A., Kotelenez P., Stochastic flows with interaction and random measures. (to
appear)
4. Stroock D.W., Varadhan S.R.S., On the support of diffusion processes with applications to the
strong maximum principle., Proceedings of the Sixth Berkeley Symposium on Mathematical
Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability
theory, pp. 333–359.
5. Kunita H., Stochastic Flows and Stochastic Differential Equations, vol. 24, Cambridge studies
in advanced mathematics, 1990, pp. 346.
6. Pilipenko A. Yu., Stroock-Varadhan theorem for flows generated by stochastic differential equa-
tions with interaction, Ukrainian Math. J. 54 (2002), no. 2, 227–236.
7. Pilipenko A.Yu., Approximation theorem for stochastic differential equations with interaction,
Random Oper. and Stoch. Equ. 11 (2003), no. 3,, 213–228.
8. Dudley R. M., Real Analysis and Probability, Pacific Grove, Calif.: Wodsworth & Brooks/Cole
Pub. Co., 1989.
9. Dawson D.A., Measure-Valued Markov processes. Ecole d’Ete de Probabilites de Saint-Flour
XXI—1991, Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1–260.
10. Jacka S., Tribe R., Comparisons for measure valued processes with interactions., Ann. Probab.
31 (2003), no. 3, 1679–1712.
11. Ikeda N., Watanabe S., Stochastic Differential Equations and Diffusion Processes, North-
Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Ko-
dansha, Ltd., Tokyo, 1981.
12. Adams R.A., Sobolev Spaces, Academic Press, New York, 1975.
13. Ethier S.N, Kurtz T.G., Markov Processes: Characterization and Convergence, Wiley, New
York, 1986.
E-mail : apilip@imath.kiev.ua
|