On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients
UDC 519.21 In this paper we solve a selection problem for multidimensional SDE $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$It is assumed that $X^{\epsilon}(0)...
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| author | Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. |
| author_facet | Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. |
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| description | UDC 519.21
In this paper we solve a selection problem for multidimensional SDE $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$It is assumed that $X^{\epsilon}(0)=x^0\in H,$ the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H,$ and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution.We show that if the drift pushes the solution away from $H,$ then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to $H,$ then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new. |
| doi_str_mv | 10.37863/umzh.v72i9.6292 |
| first_indexed | 2026-03-24T03:27:00Z |
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DOI: 10.37863/umzh.v72i9.6292
UDC 519.21
A. Kulik (Wroclaw Univ. Sci. and Technology, Poland),
A. Pilipenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv,
and Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”)
ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES
WITH NON-LIPSCHITZ COEFFICIENTS*
ПРО РЕГУЛЯРИЗАЦIЮ МАЛИМ ШУМОМ БАГАТОВИМIРНИХ ЗВИЧАЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ З НЕЛIПШИЦЕВИМИ КОЕФIЦIЄНТАМИ
In this paper we solve a selection problem for multidimensional SDE dX\varepsilon (t) = a(X\varepsilon (t)) dt+ \varepsilon \sigma (X\varepsilon (t)) dW (t), where
the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane H. It is assumed that X\varepsilon (0) = x0 \in H,
the drift a(x) has a Hoelder asymptotics as x approaches H, and the limit ODE dX(t) = a(X(t)) dt does not have a
unique solution. We show that if the drift pushes the solution away from H, then the limit process with certain probabilities
selects some extremal solutions to the limit ODE. If the drift attracts the solution to H, then the limit process satisfies an
ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general
and new.
Статтю присвячено знаходженню границi розв’язкiв багатовимiрних стохастичних диференцiальних рiвнянь
dX\varepsilon (t) = a(X\varepsilon (t)) dt + \varepsilon \sigma (X\varepsilon (t)) dW (t) при \varepsilon \rightarrow 0, де коефiцiєнти переносу та дифузiї є локально лiпшице-
вими функцiями ззовнi фiксованої гiперплощини H. Припускається, що X\varepsilon (0) = x0 \in H, коефiцiєнт переносу
a(x) має гельдерову асимптотику, коли x наближається до H, i граничне звичайне диференцiальне рiвняння
dX(t) = a(X(t)) dt може не мати єдиного розв’язку. Доведено, що якщо перенос вiдштовхує розв’язок вiд гiпер-
площини H, то граничний процес iз певними ймовiрностями вибирає деякi екстремальнi розв’язки граничного
диференцiального рiвняння. Якщо перенос притягує розв’язок до H, то граничний процес задовольняє звичай-
не диференцiальне рiвняння з усередненими коефiцiєнтами. Для доведення останнього результату сформульовано
новий достатньо загальний принцип усереднення.
1. Introduction. Consider an ODE
du(t)
dt
= a(u(t)),
u(0) = 0,
(1.1)
where a is a continuous function of linear growth that satisfies a local Lipschitz condition everywhere
except of the point u = 0. Then uniqueness of the solution to (1.1) may fail; e.g., for a(u) =
=
\sqrt{}
| u| \mathrm{s}\mathrm{g}\mathrm{n}(u), the ODE (1.1) has multiple solutions \pm t2/4, t \geq 0.
Consider a perturbation of (1.1) by a small noise:
du\varepsilon (t) = a(u\varepsilon (t))dt+ \varepsilon dW (t),
u\varepsilon (0) = 0,
(1.2)
* The work of A. Kulik was supported by the Polish National Science Center (grant 2019/33/B/ST1/02923). Research
of A. Pilipenko was partially supported by Norway-Ukrainian cooperation in mathematical education Eurasia 2016-Long-
term CPEA-LT-2016/10139, DFG project “Stochastic dynamics with interfaces” (no. 452119141) and by the Alexander
von Humboldt Foundation within the Research Group Linkage Programme “Singular diffusions: analytic and stochastic
approaches” between the University of Potsdam and the Institute of Mathematics of the National Academy of Sciences of
Ukraine.
c\bigcirc A. KULIK, A. PILIPENKO, 2020
1254 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1255
where W is a Wiener process. Equation (1.2) has a unique strong solution due to the Zvonkin –
Veretennikov theorem [22]. It easy to see that a family of distributions of \{ u\varepsilon \} is weakly relatively
compact because a has a linear growth. Moreover, any limit point of \{ u\varepsilon \} as \varepsilon \rightarrow 0 satisfies
equation (1.1) because a is continuous. Hence, if the limit \mathrm{l}\mathrm{i}\mathrm{m}\varepsilon \rightarrow 0 u\varepsilon (in distribution) exists, then
this limit may be considered as a natural selection of a solution to (1.1).
The corresponding problem was originated in papers by Bafico and Baldi [2, 3], who considered
the one-dimensional case; other generalizations see, for example, in [4 – 9, 12, 15, 18 – 21] and
references therein. Investigations in multidimensional case are much complicated than in the one-
dimensional one. There are still no simple sufficient conditions that ensure existence of a limit
\mathrm{l}\mathrm{i}\mathrm{m}\varepsilon \rightarrow 0 u\varepsilon and a characterization of this limit. One of the reason for this is the absence of the linear
ordering in the multidimensional case. Indeed, in the one-dimensional situation the are only two
ways to exit from the point 0: one way to the right and another to the left. The probability of
going left or right can be easily obtained since there are explicit formulas for hitting probabilities for
one-dimensional diffusions. The equation for the limit process outside of 0 must satisfy the original
ODE because a is Lipschitz continuous there.
In this paper we consider the multidimensional case, where the Lipschitz condition for a may
fail at a hyperplane. Let us describe the corresponding model. Consider an SDE
du\varepsilon (t) = a(u\varepsilon (t))dt+ \varepsilon \sigma (u\varepsilon (t))dW (t),
u\varepsilon (0) = x0,
(1.3)
where a : \BbbR d \rightarrow \BbbR d, \sigma : \BbbR d \rightarrow \BbbR d\times m are measurable functions, W is an m-dimensional Wiener
process.
Assume that a and \sigma are of linear growth, \sigma is continuous and satisfies the uniform ellipticity
condition. This ensures existence and uniqueness of a weak solution to (1.3) and relative compactness
for the distributions of \{ u\varepsilon \} .
Set H := \BbbR d - 1\times \{ 0\} . Suppose that the initial starting point x0 \in H and that the drift a satisfies
the local Lipschitz property in \BbbR d \setminus H.
Note that the definition of a on H is inessential because u\varepsilon spends zero time in H with proba-
bility 1 due to the nondegeneracy of the diffusion coefficient.
The case when a is globally Lipschitz continuous in the lower half-space \BbbR d
- := \BbbR d - 1\times ( - \infty , 0)
and the upper half-space \BbbR d
+ := \BbbR d - 1 \times (0,\infty ) was investigated in [19]. The result was formulated
in terms of the vertical components of a\pm (x0) := \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow x0,x\in \BbbR d
\pm
a(x). In this paper we investigate
the case when the drift has Hölder-type asymptotic in a neighborhood of H. Namely, we will assume
that
A1) ad(x) = | xd| \gamma b(x), where \gamma < 1, xd is the dth coordinate of x = (x1, . . . , xd), and b is a
globally Lipschitz continuous function in \BbbR d
+ and \BbbR d
- , b
\pm (x) \not = 0, x \in H;
A2) ak, k = 1, . . . , d - 1, are globally Lipschitz functions in \BbbR d
+ and \BbbR d
- .
This case has new features, and the proofs will be based on new ideas compared to the proofs
from [19]. To illustrate the difference, let us recall briefly results of [19], where the case \gamma = 0 was
considered, and sketch the expected results in the case \gamma \in (0, 1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1256 A. KULIK, A. PILIPENKO
Case 1 (The vector field a pushes outwards the hyperplane). Denote by \bfn = (0, . . . , 0, 1) the
normal vector to the hyperplane H. Assume that \gamma = 0 and \pm (a\pm (x),\bfn ) > 0, x \in H. Then there
are two solutions u\pm to
du(t) = a(u(t))dt (1.4)
that start at x0 \in H and exit from H immediately to the upper and the lower half-spaces, respectively.
It was proved in [19] that if \gamma = 0, then the limit process u0 immediately leaves H and moves as
u\pm with probabilities proportional to
\bigm| \bigm| (a\pm (x0),\bfn )\bigm| \bigm| . The corresponding proof was similar to the one-
dimensional situation. It used some comparison principle adapted to the multidimensional situation.
Investigations for arbitrary \gamma \in (0, 1) will be similar, but selection probabilities will be different.
Remark 1.1. It was assumed in [19] that the noise is additive, i.e., \sigma is the identity matrix and
m = d. The case of multiplicative noise is completely analogous.
Remark 1.2. If \gamma = 0 and the vector field a pushes away H from one side of H and attracts
from another side (for example, (a\pm (x),\bfn ) > 0), then there is a unique solution to (1.4) that starts at
x0 \in H. This solution exits from H immediately (to the upper half-space in our case) and the limit
process u0 equals this solution of the ODE, see [19]. If \gamma \in (0, 1), the result is similar. Assume,
for example, that b\pm (x0) > 0. Then there exists a unique solution to (1.4) that exit H immediately
(there may be other solutions that stay in H ). Moreover, this solution exits to the upper half-space
and the limit process u0 equals this solution. We do not prove this result in this paper. The proof is
similar to [19].
Case 2 (The vector field a pushes towards the hyperplane). Assume that \gamma = 0 and
\pm (a\pm (x),\bfn ) < 0, x \in H. It can be seen that any limit point of \{ u\varepsilon \} must stay at H with pro-
bability 1. It was proved in [19] that the limit process u0 satisfies an ODE on H with the drift
PH(p+(x)a
+(x) + p - (x)a
- (x)), where PH is the orthogonal projection to H and the coefficients
p\pm (x) are equal to
a\mp d (x)
a - d (x) - a+d (x)
. Note that this multidimensional result has no one-dimensional
analogues, where the limit is zero process. In multidimensional case the first d - 1 coordinates may
change while dth coordinate stays zero.
The idea of proof was to analyze the time spent by u\varepsilon in upper and lower half-spaces. It was
seen that since any limit process stays at H and u\varepsilon is close close to H for small \varepsilon , then the
times spent in upper and lower half-spaces in a neighborhood of x \in H are proportional to the dth
coordinates a - d (x) and a+d (x), respectively (they are not zero if \gamma = 0). Note that the proof in
[19] was independent of the type of a noise. The small noise might be arbitrary process that (a)
ensures existence a solution and (b) converges to 0 uniformly in probability as \varepsilon \rightarrow 0 (however, the
corresponding results were formulated for Brownian noise only).
The proof from [19] does not work if ad(x) \rightarrow 0 as x approaches to H. The time spent in upper
and lower half-spaces might depend on the asymptotic of decay of ad in a neighborhood of H. In
this paper we prove the result when a satisfies assumptions A1, A2 with \gamma \in (0, 1), b+(x) < 0 and
b - (x) > 0 for x \in H.
It appears that if we scale the vertical coordinate \varepsilon - \delta ud,\varepsilon (t) for a special choice of \delta > 0, then a
pair
\bigl(
u1,\varepsilon (t), . . . , ud - 1,\varepsilon (t)
\bigr)
and \varepsilon - \delta ud,\varepsilon (t) can be considered as components of a Markov process
in a “slow” and “fast” time, respectively. Hence the description of the limit process for \{ u\varepsilon \} is
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1257
closely related to the averaging principle for Markov processes. We will see that the limit process
satisfies an ODE on H whose coefficients are an averaging of functions a\pm k , k = 1, . . . , d - 1, over
a stationary distribution of a scaled vertical component given the other components were frozen. The
idea to use some scaling for small-noise problem was effectively used in one-dimensional case if the
drift is a power-type function and the noise is a Levy \alpha -stable process or even more general.
Remark 1.3. The case \gamma = 1 is critical. If ad(x) \sim xdb(x), where b\pm (x) \not = 0, x \in H, then
the limit process may be non-Markov and satisfy certain equation [19] that depends somehow on a
Wiener process W (that formally should disappear in a limit equation).
The paper is organized as follows. In Section 2 we formulate the problem and the main results.
The proofs for the cases when the drift pushes outwards H and towards H are given in Sections 3
and 4, respectively.
In Subsection 2.3 we also formulate an averaging principle, which is quite general and new result.
The proof of averaging principle is postponed to Section 5.
2. Main results. Let us represent u\varepsilon (t) as a pair
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
, where Y\varepsilon is the last coordinate
of u\varepsilon and X\varepsilon consists of the first d - 1 coordinates. Below we study only the general problem for
the pair
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
, which can be easily be reformulated for u\varepsilon . For notational convenience, we
assume below that X\varepsilon is a d-dimensional process but not (d - 1)-dimensional one.
The general setup is the following. Let X\varepsilon , Y\varepsilon be stochastic processes with values in \BbbR d and \BbbR ,
respectively. Assume that the pair X\varepsilon , Y\varepsilon satisfies the following SDE:
dX\varepsilon (t) = \psi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
dt+ \varepsilon b
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
dB(t),
dY\varepsilon (t) = \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
Y \gamma
\varepsilon (t) dt+ \varepsilon \beta
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
dW (t),
X\varepsilon (0) = x0, Y\varepsilon (0) = 0,
(2.1)
where B, W are Wiener processes (multidimensional and one-dimensional), that may be dependent.
Denote
y\gamma := | y| \gamma (1y>0 - 1y<0),
H := \BbbR d \times \{ 0\} .
Assume that
B1) \psi (x, y) = \psi +(x, y)1y\geq 0 + \psi - (x, y)1y<0 and \varphi (x, y) = \varphi +(x, y)1y\geq 0 + \varphi - (x, y)1y<0,
where functions \psi \pm , \varphi \pm are bounded, continuous in x, y.
We assume that domains of \psi \pm , \varphi \pm are the whole space x \in \BbbR d, y \in \BbbR , despite we use their
values on the corresponding half-spaces only. The functions \psi , \varphi may have jump discontinuity
on H.
B2) \varphi \pm (x, 0) \not = 0 for any x \in \BbbR d.
B3) \beta (x, y) = \beta +(x, y)1y\geq 0 + \beta - (x, y)1y<0, where \beta \pm are bounded, continuous and separated
from zero function in the whole space \BbbR d\times \BbbR ; function b is bounded and continuous in (\BbbR d\times \BbbR )\setminus H.
B4) \gamma \in (0, 1).
Under assumptions B1 – B4 there exists a weak solution to (2.1).
Indeed, it follows from the standard compactness arguments that there exists a weak solution to
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1258 A. KULIK, A. PILIPENKO
d \^X\varepsilon (t) =
\psi
\beta 2
\bigl(
\^X\varepsilon (t), \^Y\varepsilon (t)
\bigr)
dt+ \varepsilon
b
\beta
\bigl(
\^X\varepsilon (t), \^Y\varepsilon (t)
\bigr)
dB(t),
d \^Y\varepsilon (t) = \varepsilon dW (t),
\^X\varepsilon (0) = x0, \^Y\varepsilon (0) = 0.
Note that all coefficients may be discontinuous in H but the processes spend zero time there with
probability 1. Any redefinition of coefficients in H does not affect the equations.
By using the transformation of time arguments (see, for example, [13]), we get a solution to
d
\^\^X\varepsilon (t) = \psi
\Bigl(
\^\^X\varepsilon (t),
\^\^Y\varepsilon (t)
\Bigr)
dt+ \varepsilon b
\Bigl(
\^\^X\varepsilon (t),
\^\^Y\varepsilon (t)
\Bigr)
dB(t),
d
\^\^Y\varepsilon (t) = \varepsilon \beta
\Bigl(
\^\^X\varepsilon (t),
\^\^Y\varepsilon (t)
\Bigr)
dW (t),
\^\^X\varepsilon (0) = x0,
\^\^Y\varepsilon (0) = 0.
Finally, Girsanov’s theorem yields existence of a weak solution to (2.1).
Remark 2.1. If b is nondegenerate, then existence of a solution can be proved without transfor-
mation of time arguments.
2.1. Repulsion from the hyperplane. In this subsection, we assume that \varphi \pm (x, 0) > 0 for all
x \in \BbbR d. Then \mathrm{s}\mathrm{g}\mathrm{n}(y)\varphi (x, y)y\gamma > 0, y \not = 0 and the drift pushes away from the hyperplane \BbbR d\times \{ 0\} .
Suppose that assumptions B1 – B4 hold true and functions \psi \pm , \varphi \pm are locally Lipschitz conti-
nuous in (x, y) \in \BbbR d \times \BbbR .
Then there are unique solutions (X+(t), Y +(t)) and (X - (t), Y - (t)) to the unperturbed system
(i.e., \varepsilon = 0):
dX(t) = \psi
\bigl(
X(t), Y (t)
\bigr)
dt,
dY (t) = \varphi
\bigl(
X(t), Y (t)
\bigr)
Y \gamma (t) dt,
X(0) = x0, Y (0) = 0,
such that Y +(t) > 0 and Y - (t) < 0 for all t > 0.
Indeed, set \~Y (t) := Y 1 - \gamma (t). Then
X(t) = x0 +
t\int
0
\psi
\Bigl(
X(s), \~Y
1
1 - \gamma (s)
\Bigr)
ds,
\~Y (t) = (1 - \gamma )
t\int
0
\varphi
\Bigl(
X(s), \~Y
1
1 - \gamma (s)
\Bigr)
ds.
Since \gamma \geq 0, the functions (x, \~y) \rightarrow \psi \pm (x, \~y) and (x, \~y) \rightarrow \varphi \pm (x, \~y) are locally Lipschitz continu-
ous. So, equations
X\pm (t) = x0 +
t\int
0
\psi \pm
\Bigl(
X\pm (s), ( \~Y \pm (s))
1
1 - \gamma
\Bigr)
ds,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1259
\~Y \pm (t) = (1 - \gamma )
t\int
0
\varphi \pm
\Bigl(
X\pm (s), ( \~Y \pm (s))
1
1 - \gamma
\Bigr)
ds
have unique solutions (X\pm (t), \~Y \pm (t)) and these solutions are such that \~Y +(t) > 0 and \~Y - (t) <
< 0 for all t > 0. Making the inverse change of variables we get the desired functions Y \pm (t) =
=
\bigl(
\~Y \pm (t)
\bigr) 1
1 - \gamma .
The solution does not explode in a finite time because \psi \pm , \varphi \pm are bounded by assumption B1.
Theorem 2.1. The distribution of (X\varepsilon , Y\varepsilon ) in C([0, T ]) converges weakly as \varepsilon \rightarrow 0 to the
measure
p - \delta (X - ,Y - ) + p+\delta (X+,Y +),
where
p\pm =
\Biggl(
\varphi \pm (x0, 0)\bigl(
\beta \pm (x0, 0)
\bigr) 2
\Biggr) 1
\gamma +1
\biggl(
\varphi - (x0, 0)
(\beta - (x0, 0))2
\biggr) 1
\gamma +1
+
\Biggl(
\varphi +(x0, 0)\bigl(
\beta +(x0, 0)
\bigr) 2
\Biggr) 1
\gamma +1
(2.2)
and \delta (X+,Y +), \delta (X - ,Y - ) means the unit mass that concentrated on the functions (X+, Y +) and
(X - , Y - ), respectively.
The proof is given in Section 3.
Remark 2.2. If \pm \varphi \pm (x, 0) > 0 (or \pm \varphi \pm (x, 0) < 0) for all x \in \BbbR d, then the limit process is
(X+(t), Y +(t)) (respectively, (X - (t), Y - (t))) with probability 1.
Remark 2.3. If we have inequality \varphi +(x0, 0) > 0 and \varphi - (x0, 0) < 0 only at the initial point
(and hence in some neighborhood by continuity of coefficients), then the functions (X\pm (t), Y \pm (t))
are well defined up to the moment \tau \pm H := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t > 0 : Y \pm (t) = 0
\bigr\}
of the first return to H. In this
case we have the convergence in distribution for the stopped processes:\bigl(
X\varepsilon (\cdot \wedge \tau +H \wedge \tau - H ), Y\varepsilon (\cdot \wedge \tau +H \wedge \tau - H )
\bigr)
\Rightarrow
\Rightarrow p - \delta (X - (\cdot \wedge \tau +H\wedge \tau - H ),Y - (\cdot \wedge \tau +H\wedge \tau - H )) + p+\delta (X+(\cdot \wedge \tau +H\wedge \tau - H ),Y +(\cdot \wedge \tau +H\wedge \tau - H )).
The proof is essentially the same, but it involves routine localization arguments in addition.
2.2. Attraction to the hyperplane. In this subsection, we assume that \varphi \pm (x, 0) < 0 for all
x \in \BbbR d.
Suppose that assumptions B1 – B4 hold true and \psi \pm are locally Lipschitz in x for any fixed y.
Theorem 2.2. For any T > 0 we have the uniform convergence in probability
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm\| \bigm\| \bigl( X\varepsilon (t), Y\varepsilon (t)
\bigr)
- (X(t), 0)
\bigm\| \bigm\| = 0,
where X(t) is a solution to the ODE
dX(t) = \=\psi (X(t))dt, X(0) = 0,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1260 A. KULIK, A. PILIPENKO
and
\psi (x) = \psi +(x, 0)
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
+
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
+
+\psi - (x, 0)
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
+
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
. (2.3)
The proof is given in Section 4.
Remark 2.4. Note that
\varphi +(x, 0)
- 1
\gamma +1
\varphi +(x, 0)
- 1
\gamma +1 + \varphi - (x, 0)
- 1
\gamma +1
= \pi (x)([0,\infty )),
\varphi - (x, 0)
- 1
\gamma +1
\varphi +(x, 0)
- 1
\gamma +1 + \varphi - (x, 0)
- 1
\gamma +1
= \pi (x)(( - \infty , 0)),
where \pi (x) is the stationary distribution for the SDE
dy(x)(t) =
\bigl(
\varphi +(x, 0)1y(x)(t)>0 + \varphi - (x, 0)1y(x)(t)<0
\bigr)
(y(x)(t))\gamma dt+ \beta
\bigl(
x, 0) dW (t).
Hence,
\psi (x) = \psi +(x, 0)\pi (x)([0,\infty )) + \psi - (x, 0)\pi (x)(( - \infty , 0)),
i.e., the drift of the limit equation is the averaging of \psi \pm over the stationary distribution of an SDE
with frozen x variable. The corresponding relation between the averaging principle and averaging of
coefficients in the limit equation for the small noise perturbation problem will be seen from the proof.
In the next subsection, we formulate an averaging principle, which is applied in the proof of
Theorem 2.2. We consider more general SDEs than (2.1) because the idea of the proof is universal.
The corresponding result may be interesting by itself.
2.3. Averaging. Let for \varepsilon > 0 the processes X\varepsilon (t), Y\varepsilon (t) take values in \BbbR d,\BbbR k and have the
form
X\varepsilon (t) = X\varepsilon (0) +
t\int
0
a\varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (t)
\bigr)
ds+
t\int
0
\sigma \varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dB\varepsilon
s+
+
t\int
0
\int
\BbbR m
c\varepsilon
\bigl(
X\varepsilon (s - ), Y\varepsilon (s - ), u
\bigr) \Bigl[
N \varepsilon (du, ds) - 1| u| \leq \rho \nu
\varepsilon (du)ds
\Bigr]
+ \xi \varepsilon (t),
(2.4)
Y\varepsilon (t) = Y\varepsilon (0) + \varepsilon - 1
t\int
0
A\varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
ds+ \varepsilon - 1/2
t\int
0
\Sigma \varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW \varepsilon
s+
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1261
+
t\int
0
\int
\BbbR l
C\varepsilon
\bigl(
X\varepsilon (s - ), Y\varepsilon (s - ), z
\bigr) \Bigl[
Q\varepsilon (dz, ds) - 1| z| \leq \rho \varepsilon
- 1\mu \varepsilon (dz)ds
\Bigr]
,
where B\varepsilon
t , W
\varepsilon
t are Brownian motions and N \varepsilon (du, dt), Q\varepsilon (dz, dt) are Poisson point measures on a
common filtered probability space (\Omega \varepsilon ,\scrF \varepsilon ,\bfP \varepsilon ), and the random measures N \varepsilon (du, dt), Q\varepsilon (dz, dt)
have the intensity measures \nu \varepsilon (du)dt and \varepsilon - 1\mu \varepsilon (dz)dt, respectively. These random measures are
involved into the system in the partially compensated form, which is quite typical for the Lévy-driven
SDEs; what is a bit unusual is the choice of the cutoff functions 1| u| \leq \rho , 1| z| \leq \rho with the number \rho > 0
to be specified separately. This choice will become clear later, when we describe the limit behavior
of the Lévy measures \nu \varepsilon (du), \mu \varepsilon (dz) as \varepsilon \rightarrow 0. Note that here and below we do not assume a
uniqueness of a solution to prelimit equation (2.4).
The factor \varepsilon - 1 in the intensity measure for Q\varepsilon (dz, dt) and the factors \varepsilon - 1, \varepsilon - 1/2 at the integrals
w.r.t. ds and dW \varepsilon
s in the equation for Y\varepsilon mean that the evolution of the component Y\varepsilon happens at
the “fast” time scale \varepsilon - 1t, which we will also call the “microscopic” time scale. The component X\varepsilon
evolves at the “slow”, or “macroscopic” time scale t; its evolution involves the deterministic term,
two stochastic terms (continuous and partially compensated jump parts), and a residual term \xi \varepsilon , for
which we do not impose any structural assumptions, and only require it to be asymptotically small
in the following sense:
H0 (Negligibility of the residual term). The process \xi \varepsilon (t) is an adapted càdlàg process, and, for
any T > 0,
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
| \xi \varepsilon (t)| \rightarrow 0, \varepsilon \rightarrow 0,
in probability.
The aim of this subsection is to get the averaging principle for the “slow” component X\varepsilon . Let us
stress that the framework we adopt is quite general. In particular,
the two-scale system (2.4) is fully coupled in the sense that the coefficients of the “slow”
component depend on the “fast” one, and vice versa;
the noises for the “slow” and the “fast” component are allowed to be dependent;
the coefficients of the “slow” component can be discontinuous.
Let us introduce further assumptions on the system (2.4). Note that all the assumptions listed below
are quite natural and nonrestrictive.
H1 (Bounds for the coefficients). There exists a constant C such that
| a\varepsilon (x, y)| \leq C, | \sigma \varepsilon (x, y)| \leq C, | \Sigma \varepsilon (x, y)| \leq C, | c\varepsilon (x, y, u)| \leq C| u| , | C\varepsilon (x, y, z)| \leq C| z|
for all values of x, y, u, z.
In addition, for any R > 0 there exists a constant CR such that
| A\varepsilon (x, y)| \leq CR, x \in \BbbR d, | y| \leq R.
H2 (Bounds for the Lévy measures). There exist constants C and p > 0 such that\int
\BbbR m
(| u2| \wedge 1)\nu \varepsilon (du) \leq C,
\int
\BbbR l
(| z2| 1| z| \leq 1 + | z| p1| z| >1)\mu
\varepsilon (dz) <\infty .
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1262 A. KULIK, A. PILIPENKO
H3 (The coefficients of the fast component are convergent). There exist continuous functions
A(x, y), \Sigma (x, y), C(x, y, z) such that
A\varepsilon (x, y) \rightarrow A(x, y), \Sigma \varepsilon (x, y) \rightarrow \Sigma (x, y), and C\varepsilon (x, y, z) \rightarrow C(x, y, z) as \varepsilon \rightarrow 0
uniformly on every compact set in \BbbR d \times \BbbR k, \BbbR d \times \BbbR k, and \BbbR d \times \BbbR k \times
\bigl(
\BbbR l \setminus \{ 0\}
\bigr)
, respectively.
To introduce the next condition, let us define the weak convergence of a family of Lévy measures
on \BbbR m in the following way: \nu \varepsilon (du) =\Rightarrow \nu (du) if for every continuous function \varphi with a support
compactly embedded into \BbbR m \setminus \{ 0\} ,\int
\BbbR m
\varphi (z) \nu \varepsilon (dz) \rightarrow
\int
\BbbR m
\varphi (z) \nu (dz), \varepsilon \rightarrow 0.
H4 (The Lévy measures of the noises are weakly convergent). There exist Lévy measures \nu (du),
\mu (dz) on \BbbR m,\BbbR l respectively, such that
\nu \varepsilon (du) =\Rightarrow \nu (du) and \mu \varepsilon (dz) =\Rightarrow \mu (dz) as \varepsilon \rightarrow 0.
In addition,
\nu
\bigl(
\{ u : | u| = \rho \}
\bigr)
= 0, \mu
\bigl(
\{ z : | z| = \rho \}
\bigr)
= 0. (2.5)
Condition (2.5) yield that the cutoff functions 1| u| \leq \rho , 1| z| \leq \rho used in (2.4) are a.s. continuous w.r.t.
the measures \nu (du), \mu (dz), respectively. Note that there exists at most countable set of levels \rho such
that (2.5) fails, hence one can always choose \rho to satisfy this condition. Of course, changing the
cutoff level would change the drift coefficients respectively.
Next, assume that the drift of the fast component performs an attraction to origin.
H5 (The drift condition for the microscopic dynamics). There exist \kappa > 0 and c, r > 0 such
that
A\varepsilon (x, y) \cdot y \leq - c| y| \kappa +1, | y| \geq r. (2.6)
In addition, the balance condition holds:
\kappa + p > 1, (2.7)
where p is introduced in the assumption H2 .
Consider a family of “frozen microscopic equations”
dy(t) = A
\bigl(
x, y(t)
\bigr)
dt+\Sigma
\bigl(
x, y(t - )
\bigr)
dWt+
+
\int
\BbbR l
C
\bigl(
x, y(t - ), z
\bigr) \Bigl[
Q(dz, ds) - 1| z| \leq 1\mu (dz)ds
\Bigr]
, (2.8)
where W is a Wiener process and Q(dz, dt) is an independent Poisson point measure with the
intensity measure \mu (dz)dt. For the corresponding “frozen dynamics” we introduce a separate family
of assumptions.
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1263
F0 (The “frozen microscopic dynamics” is well defined and Feller). For any x and any initial
value y(0) = y, the SDE (2.8) has a unique weak solution, which is a Markov process. Furthermore
we denote the corresponding family of Markov processes by y(x), x \in \BbbR d, and write P (x)
t (y, dy\prime )
for the corresponding family of transition probabilities.
We also denote
P frozen
t f(x, y) =
\int
\BbbR k
f(x, y\prime )P
(x)
t (y, dy\prime ), t \geq 0,
the semigroup of operators corresponding to the two-component process (x, y(x)) in which the first
component is constant and the second one is the Markov process specified above. We assume that
this semigroup is Feller.
For this family, we assume the following mixing property, which is actually the local Dobrushin
condition, uniform in parameter x; see [16] (Section 2).
F1 (The “frozen microscopic dynamics” is locally mixing). There exists h > 0 such that, for
any R > 0, there exists \rho = \rho R > 0 such that, for any x, y1, y2 with | x| \leq R, | y1| \leq R, | y2| \leq R\bigm\| \bigm\| P (x)
h (y1, dy
\prime ) - P
(x)
h (y2, dy
\prime )
\bigm\| \bigm\|
TV
\leq 1 - \rho ,
where P
(x)
t (y, dy\prime ) denotes the transition probability of the process y(x), and the total variation
distance between probability measures is defined as
\| \lambda 1 - \lambda 2\| TV = \mathrm{s}\mathrm{u}\mathrm{p}
A
(\lambda 1(A) - \lambda 2(A)).
We note that assumptions F1 , H5 ensure that, for each x \in \BbbR d, the laws of y(x)t converge to the
invariant probability measure (IPM) \pi (x)(dy) with an explicitly rate; see Proposition 5.1 below.
For the coefficients of the “slow” component, we assume a weaker analogue of H3 , where the
convergence and continuity of the limiting coefficients may fail on an exceptional set, which should
be negligible, in a sense.
H6 (The coefficients of the slow component are convergent). There exist functions a(x, y),
\sigma (x, y), c(x, y, u) and an open set B \subset \BbbR d \times \BbbR k such that, for any compact set K \subset B,
a\varepsilon (x, y) \rightarrow a(x, y) and \sigma \varepsilon (x, y) \rightarrow \sigma (x, y) as \varepsilon \rightarrow 0
uniformly on K, and for any R > 1
c\varepsilon (x, y, u) \rightarrow c(x, y, u), \varepsilon \rightarrow 0,
uniformly on K \times \{ u : R - 1 \leq | u| \leq R\} . The set \Delta = (\BbbR d \times \BbbR k) \setminus B satisfies
\pi (x)\{ y : (x, y) \in \Delta \} = 0 for any x \in \BbbR d.
In addition, the functions a(x, y), \sigma (x, y), and c(x, y, u) are continuous on B and B \times (\BbbR m \setminus \{ 0\} ),
respectively.
Define the averaging of the limiting drift coefficient for the macroscopic component w.r.t. the
family of IPMs for the frozen microscopic one:
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1264 A. KULIK, A. PILIPENKO
a(x) =
\int
\BbbR k
a(x, y)\pi (x)(dy).
Next, consider the limiting diffusion matrix and compensated/noncompensated jump kernels for the
macroscopic component,
b(x, y) = \sigma (x, y)\sigma (x, y)\ast ,
K(\rho )(x, y,A) = \nu
\bigl(
\{ u : | u| \leq \rho , c(x, y, u) \in A\}
\bigr)
,
K(\rho )(x, y,A) = \nu
\bigl(
\{ u : | u| > \rho , c(x, y, u) \in A\}
\bigr)
,
and introduce the corresponding averaged characteristics as
b(x) =
\int
\BbbR k
b(x, y)\pi (x)(dy),
K(\rho )(x, dv) =
\int
\BbbR k
K(\rho )(x, y, dv)\pi
(x)(dy),
K
(\rho )
(x, dv) =
\int
\BbbR k
K(\rho )(x, y, dv)\pi (x)(dy).
Finally, we introduce an auxiliary technical assumption.
A0 The averaged coefficients a(x), b(x) are continuous. The averaged Lévy kernels K(\rho )(x, dv),
K
(\rho )
(x, dv) depend on x continuously, in the sense that
K(\rho )(x
\prime , dv) =\Rightarrow K(\rho )(x, dv) and K
(\rho )
(x\prime , dv) =\Rightarrow K
(\rho )
(x, dv) as x\prime \rightarrow x.
Remark 2.5. It is easy to give a sufficient condition for A0 to holds. Namely, it is enough to
assume, in addition to H0 – H6 , F0 , F1 , that the transition probabilities P (x)
t (y, dy\prime ) are continuous
in x w.r.t. the total variation convergence for each y \in \BbbR k, t \geq t0. Then, because of the conver-
gence (5.6), the same continuity holds for the family of the IPMs \pi (x)(dy). The latter continuity,
combined with H1 , H2 , H4 , and H6 , yields the required continuity of the averaged coefficients.
Now we are ready to formulate our main statement.
Theorem 2.3. Assume H0 – H6 , F0 , F1 , and A0 to hold,
X\varepsilon (0) \rightarrow x0, \varepsilon \rightarrow 0,
in probability and \{ Y\varepsilon (0)\} be bounded in probability.
Then the family \{ X\varepsilon , \varepsilon > 0\} is weakly compact in \BbbD
\bigl(
[0,\infty ),\BbbR d
\bigr)
, and any of its weak limit point
as \varepsilon \rightarrow 0 is a solution to the martingale problem (L,C\infty
0 ) with
L\varphi (x) = \nabla \varphi (x) \cdot a(x) + 1
2
\nabla 2\varphi (x) \cdot b(x)+
+
\int
\BbbR m
\bigl(
\varphi (x+ v) - \varphi (x) - \nabla \varphi (x) \cdot v
\bigr)
K(\rho )(x, dv)+
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1265
+
\int
\BbbR m
\bigl(
\varphi (x+ v) - \varphi (x)
\bigr)
K
(\rho )
(x, dv) =
= \nabla \varphi (x) \cdot a(x) + 1
2
\nabla 2\varphi (x) \cdot b(x)+
+
\int
\BbbR k
\int
\BbbR m
\Bigl(
\varphi (x+ c(x, y, u)) - \varphi (x) - \nabla \varphi (x) \cdot c(x, y, u)1| u| \leq \rho
\Bigr)
\nu (du)\pi (x)(dy), (2.9)
where \varphi \in C\infty
0 .
If the martingale problem (2.9) is well posed, then X\varepsilon weakly converges as \varepsilon \rightarrow 0 to its unique
solution with X(0) = x0.
3. Proof of Theorem 2.1. The proof almost copying the proof of Theorem 3.1 in [19]. Thus we
only sketch the main steps of the proof.
Step 1. The sequence \{ (X\varepsilon , Y\varepsilon )\} is weakly relatively compact. The proof follows from boun-
dedness of functions \varphi , \psi , b, \beta .
Therefore, to prove the theorem it suffices to verify that any subsequence \{ (X\varepsilon n , Y\varepsilon n)\} contains
sub-subsequence \{ (X\varepsilon nk
, Y\varepsilon nk
)\} that converges to the desired limit. Without loss of generality we
will assume that \{ (X\varepsilon , Y\varepsilon )\} is weakly convergent by itself.
Step 2. Estimate for the time spent by Y\varepsilon in a neighborhood of 0.
We will use the following general statement.
Lemma 3.1. Assume that processes \{ \eta \varepsilon (t)\} satisfy the SDE
d\eta \varepsilon (t) = a\varepsilon (t)\eta
\gamma
\varepsilon (t)dt+ \varepsilon b\varepsilon (t)dW (t),
\eta \varepsilon (0) = 0,
where | \gamma | < 1, and a\varepsilon (t), b\varepsilon (t) are \scrF t-adapted processes such that
a\varepsilon (t) \geq A > 0, 0 < C1 \leq b\varepsilon (t) \leq C2
for all \omega , t, \varepsilon .
Set
\tau \varepsilon (\delta ) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : | \eta \varepsilon (t)| \geq \delta
\bigr\}
.
Then there is a constant K = K(A,C1, C2) such that
\forall \delta > 0 \exists \varepsilon 0 > 0 \forall \varepsilon \in (0, \varepsilon 0) : \bfE \tau \varepsilon (\delta ) \leq K\delta 1 - \gamma .
The proof of the lemma is quite standard. We postpone it to the Appendix.
Without loss of generality we will assume that
\psi \pm (x, 0) \geq c1 > 0 and 0 < c2 \leq \beta (x) \leq c3 for all x \in \BbbR d, (3.1)
where c1,2,3 are some positive constants. This assumption does not restrict generality, since the
general case can be considered using a localization. Under this additional assumption, Lemma 3.1
applied to
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1266 A. KULIK, A. PILIPENKO
\tau \varepsilon (\delta ) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : | Y\varepsilon (t)| \geq \delta
\bigr\}
and the Chebyshev inequality yield
\forall \delta > 0 \exists \varepsilon 0 > 0 \forall \varepsilon \in (0, \varepsilon 0) : \bfP
\bigl(
\tau \varepsilon (\delta ) \geq \delta
1 - \gamma
2
\bigr)
\leq K\delta
1 - \gamma
2 . (3.2)
Remark 3.1. It can be seen from the construction of Y \pm that the inequality (3.2) is valid for
\tau \pm (\delta ) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : | Y \pm (t)| \geq \delta
\bigr\}
also.
Step 3. We see from (3.2) that with high probability the random variable \tau \varepsilon (\delta ) is dominated by
\delta
1 - \gamma
2 . It follows from the standard estimates for moments of SDEs that for small t we have
\bfE \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,t]
\bigm| \bigm| X\varepsilon (s) - x0
\bigm| \bigm| 2 \leq Ct,
where constant C can be selected independently of \varepsilon \in [0, 1].
So, we have the following estimates:
\exists C1 > 0 \forall \delta > 0 \exists \varepsilon 0 > 0 \forall \varepsilon \in (0, \varepsilon 0) : \bfP
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\tau \varepsilon (\delta )]
| X\varepsilon (t) - x0| \geq \delta
1 - \gamma
6
\biggr)
\leq C1\delta
1 - \gamma
6 , (3.3)
\bfP
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\tau \varepsilon (\delta )]
| X\pm (t) - x0| + | Y \pm (t)| \geq 2\delta
1 - \gamma
6
\biggr)
\leq C1\delta
1 - \gamma
6 . (3.4)
Verify, for example, (3.3):
\bfP
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\tau \varepsilon (\delta )]
| X\varepsilon (t) - x0| \geq \delta
1 - \gamma
6
\biggr)
\leq
\leq \bfP
\bigl(
\tau \varepsilon (\delta ) > \delta
1 - \gamma
2
\bigr)
+\bfP
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
t\in
\bigl[
0,\delta
1 - \gamma
2
\bigr] | X\varepsilon (t) - x0| \geq \delta
1 - \gamma
6
\Biggr)
\leq
\leq \delta
1 - \gamma
2 +
\bfE \mathrm{s}\mathrm{u}\mathrm{p}
t\in
\bigl[
0,\delta
1 - \gamma
2
\bigr] | X\varepsilon (t) - x0| 2
\delta
1 - \gamma
3
\leq
\leq \delta
1 - \gamma
2 +
C\delta
1 - \gamma
2
\delta
1 - \gamma
3
\leq C1\delta
1 - \gamma
6 .
Note also that
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\tau \varepsilon (\delta )]
| Y\varepsilon (t)| =
\bigm| \bigm| Y\varepsilon (\tau \varepsilon (\delta ))\bigm| \bigm| = \delta a.s. (3.5)
by the definition of \tau \varepsilon (\delta ).
Step 4. We denote by
\bigl(
Xx,y(t), Y x,y(t)
\bigr)
a solution to the corresponding ODE that starts from
x \in \BbbR d, y \not = 0. This solution never hits \BbbR d\times \{ 0\} , recall (3.1). We have correctness of the definition
of
\bigl(
Xx,y(t), Y x,y(t)
\bigr)
because in all other points coefficients satisfy the local Lipschitz condition.
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1267
If we wish to highlight that y > 0 (or y < 0), then the corresponding solution is denoted by\bigl(
X+,x,y(t), Y +,x,y(t)
\bigr) \bigl(
or
\bigl(
X - ,x,y(t), Y - ,x,y(t)
\bigr)
, respectively
\bigr)
.
Let \omega be such that Y\varepsilon (\tau \varepsilon (\delta )) = \delta , i.e., the process Y\varepsilon hits \delta earlier than - \delta . Then for this \omega we
have
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigl(
| X\varepsilon (t) - X+(t)| + | Y\varepsilon (t) - Y +(t)|
\bigr)
\leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\Bigl( \bigm| \bigm| X\varepsilon (\tau \varepsilon (\delta ) + t) - X+,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| +
+
\bigm| \bigm| Y\varepsilon (\tau \varepsilon (\delta ) + t) - Y +,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| \Bigr) +
+ \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\Bigl( \bigm| \bigm| X+,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t) - X+(\tau \varepsilon (\delta ) + t)
\bigm| \bigm| +
+
\bigm| \bigm| Y +,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t) - Y +(\tau \varepsilon (\delta ) + t)
\bigm| \bigm| \Bigr) +
+ \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\Bigl( \bigm| \bigm| X+(\tau \varepsilon (\delta ) + t) - X+(t)
\bigm| \bigm| + \bigm| \bigm| Y +(\tau \varepsilon (\delta ) + t) - Y +(t)
\bigm| \bigm| \Bigr) +
+ \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,\tau \varepsilon (\delta )]
\bigl(
| X\varepsilon (t) - x0| + | Y\varepsilon (t)|
\bigr)
=
= I1 + . . .+ I4.
Select small \delta > 0 and after that select \varepsilon 0 > 0 from (3.2). It follows from (3.3), (3.4), and
construction of (X+, Y +) in Subsection 2.1 that I2, I3, I4 are small with high probability.
To estimate I1 we need the following statement on integral equations. Let f(t) =
\bigl(
fX(t), fY (t)
\bigr)
be a nonrandom continuous function, and functions X\pm ,x,y
(f) , Y \pm ,x,y
(f) satisfy the integral equation
X\pm ,x,y
(f) (t) = x+
t\int
0
\psi \pm \bigl( X\pm ,x,y
(f) (s), Y \pm ,x,y
(f) (s)
\bigr)
ds+ fX(t),
Y \pm ,x,y
(f) (t) =
t\int
0
\varphi \pm \bigl( X\pm ,x,y
(f) (s), Y \pm ,x,y
(f) (s)
\bigr) \bigl(
Y \pm ,x,y
(f)
\bigr) \gamma
(s) ds+ fY (t), t \in [0, T ],
X\pm ,x,y
(f) (0) = x, Y \pm ,x,y
(f) (0) = y.
Remark 3.2. We do not assume that a pair X\pm ,x,y
(f) , Y \pm ,x,y
(f) is a unique solution. Recall also that
the domains of \psi \pm , \varphi \pm is the whole space.
Lemma 3.2.
\forall \delta > 0 \forall R \geq 1 \exists \alpha > 0 \forall x \in [ - R,R] \forall t \in [0, T ] \forall f : \| f\| \infty < \alpha ,
\forall y \in
\biggl[
1
R
,R
\biggr]
:
\bigm| \bigm| X+,x,y
(f) (t) - X+,x,y(t)
\bigm| \bigm| + \bigm| \bigm| Y +,x,y
(f) (t) - Y +,x,y(t)
\bigm| \bigm| \leq \delta ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1268 A. KULIK, A. PILIPENKO
\forall y \in
\biggl[
- R, - 1
R
\biggr]
:
\bigm| \bigm| X - ,x,y
(f) (t) - X - ,x,y(t)
\bigm| \bigm| + \bigm| \bigm| Y - ,x,y
(f) (t) - Y - ,x,y(t)
\bigm| \bigm| \leq \delta .
The proof of the lemma is standard. Notice that if \alpha is small enough, then Y \pm ,x,y
(f) (t) \not = 0,
t \in [0, T ], and coefficients of the integral equations are locally Lipschitz continuous if y \not = 0.
Let \omega be such that Y\varepsilon (\tau \varepsilon (\delta )) = \delta . Then\bigm| \bigm| X\varepsilon (\tau \varepsilon (\delta ) + t) - X+,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| + \bigm| \bigm| Y\varepsilon (\tau \varepsilon (\delta ) + t) - Y +,X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| =
=
\Bigl( \bigm| \bigm| X+,x,\delta
(f) (t) - X+,x,\delta (t)
\bigm| \bigm| + \bigm| \bigm| Y +,x,\delta
(f) (t) - Y +,x,\delta (t)
\bigm| \bigm| \Bigr) \bigm| \bigm| \bigm|
x=X\varepsilon (\tau \varepsilon (\delta ))
,
where
f(t) :=
\left( \varepsilon \tau \varepsilon (\delta )+t\int
\tau \varepsilon (\delta )
b
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dB(s), \varepsilon
t\int
0
\beta
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW (s)
\right) .
Since b and \beta are bounded we have the uniform convergence in probability:
\varepsilon \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\left(
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\tau \varepsilon (\delta )+t\int
\tau \varepsilon (\delta )
b
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dB(s)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\beta
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW (s)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \bfP \rightarrow 0, \varepsilon \rightarrow 0,
for any \delta > 0.
This, (3.3), (3.5), and Lemma 3.2 give us convergence
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\Bigl( \bigm| \bigm| X\varepsilon (\tau \varepsilon (\delta ) + t) - XX\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| +
+
\bigm| \bigm| Y\varepsilon (\tau \varepsilon (\delta ) + t) - Y X\varepsilon (\tau \varepsilon (\delta )),Y\varepsilon (\tau \varepsilon (\delta ))(t)
\bigm| \bigm| \Bigr) \bfP \rightarrow 0, \varepsilon \rightarrow 0,
for any \delta > 0.
Step 5. The proof of the theorem follows from Step 4 and the next estimate of probabilities
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \pm \delta
\bigr)
.
Lemma 3.3. For all \mu > 0, \delta 0 > 0 exists to \delta \in (0, \delta 0) such that
p+ - \mu \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
\varepsilon \rightarrow 0
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
\leq p+ + \mu ,
p - - \mu \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
\varepsilon \rightarrow 0
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = - \delta
\bigr)
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = - \delta
\bigr)
\leq p - + \mu ,
where p\pm are defined in (2.2).
Proof. Let \nu > 0 be arbitrary. Select \delta 1 > 0 such that\bigm| \bigm| \varphi \pm (x, y) - \varphi \pm (x0, 0)
\bigm| \bigm| \leq \nu , 0 <
\bigl(
\beta \pm (x0, 0)
\bigr) 2 - \nu <
\bigl(
\beta \pm (x, y)
\bigr) 2
<
\bigl(
\beta \pm (x0, 0)
\bigr) 2
+ \nu (3.6)
as | x - x0| < \delta 1, | y| \in [0, \delta 1].
Set \sigma \varepsilon (\delta ) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : | X\varepsilon (t) - x0| \geq \delta
\bigr\}
.
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1269
It follows from (3.3) that \bfP
\Bigl(
\sigma \varepsilon
\bigl(
\delta
1 - \gamma
6
\bigr)
< \tau \varepsilon (\delta )
\Bigr)
< C\delta
1 - \gamma
6 for small \varepsilon . Hence, if \delta
1 - \gamma
6 < \delta 1,
then with probability greater than 1 - C\delta
1 - \gamma
6 the process Y\varepsilon exits [ - \delta , \delta ] before X\varepsilon exits [ - \delta 1, \delta 1].
Therefore, without loss of generality we will assume that (3.6) is satisfied for all (x, y).
Set
s\varepsilon (y) :=
\left\{
\int y
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2(\varphi +(x0, 0) + \nu )z\gamma +1
\varepsilon 2(\gamma + 1)(\beta +(x0, 0)2 - \nu )
\biggr\}
dz, y \geq 0,
\int y
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2(\varphi - (x0, 0) - \nu )| z| \gamma +1
\varepsilon 2(\gamma + 1)(\beta - (x0, 0)2 + \nu )
\biggr\}
dz, y \leq 0.
Then
\varphi (x, y)y\gamma s\prime \varepsilon (y) +
\varepsilon 2
2
\beta 2(x, y)s\prime \prime \varepsilon (y) \leq 0
for all x, y (recall that we assume that (3.6) is satisfied for all (x, y)).
So,
0 \geq \bfE s\varepsilon
\bigl(
Y\varepsilon (\tau \varepsilon (\delta ))
\bigr)
= s\varepsilon (\delta )\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
+ s\varepsilon ( - \delta )\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = - \delta
\bigr)
=
= s\varepsilon (\delta )\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
+ s\varepsilon ( - \delta )
\bigl(
1 - \bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr) \bigr)
=
=
\bigl(
s\varepsilon (\delta ) - s\varepsilon ( - \delta )
\bigr)
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
+ s\varepsilon ( - \delta ).
Therefore,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfP
\bigl(
Y\varepsilon (\tau \varepsilon (\delta )) = \delta
\bigr)
\leq \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
- s\varepsilon ( - \delta )
s\varepsilon (\delta ) - s\varepsilon ( - \delta )
=
= \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
\int 0
- \delta
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
- 2(\varphi - (x0, 0) - \nu )| z| \gamma +1
\varepsilon 2(\gamma + 1)((\beta - (x0, 0))2 + \nu )
\Biggr\}
dz
\int 0
- \delta
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
- 2(\varphi - (x0, 0) - \nu )| z| \gamma +1
\varepsilon 2(\gamma + 1)((\beta - (x0, 0))2 + \nu )
\Biggr\}
dz +
\int \delta
0
\mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{
- 2(\varphi +(x0, 0) + \nu )z\gamma +1
\varepsilon 2(\gamma + 1)(
\bigl(
\beta +(x0, 0)
\bigr) 2 - \nu )
\Biggr\}
dz
=
=
\Biggl(
\varphi +(x0, 0) + \nu \bigl(
\beta +(x0, 0)
\bigr) 2 - \nu
\Biggr) 1
\gamma +1
\biggl(
\varphi - (x0, 0) - \nu
(\beta - (x0, 0))2 + \nu
\biggr) 1
\gamma +1
+
\Biggl(
\varphi +(x0, 0) + \nu \bigl(
\beta +(x0, 0)
\bigr) 2 - \nu
\Biggr) 1
\gamma +1
.
Here we used the following:
\delta \int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- Az\gamma +1
\varepsilon 2
\biggr\}
dz =
1
1 + \gamma
\biggl(
\varepsilon 2
A
\biggr) 1
1+\gamma
A\delta \gamma +1
\varepsilon 2\int
0
e - tt
1
1+\gamma
- 1
dt \sim
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1270 A. KULIK, A. PILIPENKO
\sim 1
1 + \gamma
\biggl(
\varepsilon 2
A
\biggr) 1
1+\gamma
\Gamma
\biggl(
1
1 + \gamma
\biggr)
, \varepsilon \rightarrow 0,
for any A > 0, \delta > 0.
Since \nu was arbitrary, this completes the proof of Lemma 3.3 and Theorem 2.1.
4. Proof of Theorem 2.2. At the beginning notice that
Y\varepsilon \Rightarrow 0, \varepsilon \rightarrow 0. (4.1)
Indeed, by Itô’s formula we have
Y 2
\varepsilon (t) \leq C\varepsilon 2 + 2\varepsilon
t\int
0
Y\varepsilon (s)\beta
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW (s),
where C is independent of \varepsilon . Hence we get an estimate
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bfE Y 2
\varepsilon (t) \leq C\varepsilon 2 \forall \varepsilon > 0.
It follows from the Doob inequality that
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 2\varepsilon
t\int
0
Y\varepsilon (s)\beta
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW (s)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bfP \rightarrow 0, \varepsilon \rightarrow 0.
This completes the proof of (4.1).
Let \delta > 0 be a fixed number. Notice that
\varepsilon - \delta Y\varepsilon (t) = \varepsilon \delta (\gamma - 1)
t\int
0
\varphi
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr) \bigl(
\varepsilon - \delta Y\varepsilon (t)
\bigr) \gamma
dt+
+\varepsilon 1 - \delta - \delta (\gamma - 1)
2 \varepsilon
\delta (\gamma - 1)
2
t\int
0
\beta
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dW (s) =
=
t\int
0
\varphi
\bigl(
X\varepsilon (s), \varepsilon
\delta \varepsilon - \delta Y\varepsilon (s)
\bigr) \bigl(
\varepsilon - \delta Y\varepsilon (t)
\bigr) \gamma
d
\bigl(
\varepsilon \delta (\gamma - 1)t
\bigr)
+
+\varepsilon 1 -
\delta (\gamma +1)
2
t\int
0
\beta
\bigl(
X\varepsilon (s), \varepsilon
\delta \varepsilon - \delta Y\varepsilon (s)
\bigr)
dW\varepsilon
\bigl(
\varepsilon \delta (\gamma - 1)s
\bigr)
,
where W\varepsilon (t) = \varepsilon
\delta (\gamma - 1)
2 W
\bigl(
\varepsilon - \delta (\gamma - 1)t
\bigr)
is a Wiener process.
If 1 - \delta (\gamma + 1)
2
= 0, i.e., \delta =
2
\gamma + 1
, then the process \~Y\varepsilon (t) := \varepsilon - \delta Y\varepsilon (t) = \varepsilon
- 2
\gamma +1Y\varepsilon (t) satisfies
the SDE
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1271
\~Y\varepsilon (t) =
t\int
0
\varphi
\bigl(
X\varepsilon (s), \varepsilon
2
\gamma +1 \~Y\varepsilon (s)
\bigr)
\~Y \gamma
\varepsilon (s) d
\Bigl(
\varepsilon
2(\gamma - 1)
\gamma +1 s
\Bigr)
+
t\int
0
\beta
\bigl(
X\varepsilon (s), \varepsilon
2
\gamma +1 \~Y\varepsilon (s)
\bigr)
dW\varepsilon
\Bigl(
\varepsilon
2(\gamma - 1)
\gamma +1 s
\Bigr)
.
Set \~\varepsilon = \varepsilon
2(1 - \gamma )
\gamma +1 . Therefore,
dX\varepsilon (t) = a\~\varepsilon
\bigl(
X\varepsilon (t), \~Y\varepsilon (t)
\bigr)
dt+ b\~\varepsilon
\bigl(
X\varepsilon (t), \~Y\varepsilon (t)
\bigr)
dB(t),
d \~Y\varepsilon (t) = \alpha \~\varepsilon (X\varepsilon (t), \~Y\varepsilon (t)) d\~\varepsilon
- 1t+ \beta \~\varepsilon
\bigl(
X\varepsilon (t), \~Y\varepsilon (t)
\bigr)
dW\~\varepsilon
\bigl(
\~\varepsilon - 1t
\bigr)
,
(4.2)
where
a\~\varepsilon (x, y) = \psi
\bigl(
x, \varepsilon
2
\gamma +1 y
\bigr)
= \psi
\bigl(
x, \~\varepsilon
1
1 - \gamma y
\bigr)
,
b\~\varepsilon (x, y) = \varepsilon b
\bigl(
x, \varepsilon
2
\gamma +1 y
\bigr)
= \~\varepsilon (\gamma +1)/2(1 - \gamma )b
\bigl(
x, \~\varepsilon
1
1 - \gamma y
\bigr)
,
\alpha \~\varepsilon (x, y) = \varphi
\bigl(
x, \varepsilon
2
\gamma +1 y
\bigr)
y\gamma = \varphi
\bigl(
x, \~\varepsilon
1
1 - \gamma y
\bigr)
y\gamma ,
\beta \~\varepsilon (x, y) = \beta
\bigl(
x, \varepsilon
2
\gamma +1 y
\bigr)
= \beta
\bigl(
x, \~\varepsilon
1
1 - \gamma y
\bigr)
.
We see that the system (4.2) has the form (2.4). Let us apply Theorem 2.3, where k = 1,
a\varepsilon (x, y) := a\~\varepsilon (x, y), \sigma \varepsilon (x, y) = c\varepsilon (x, y, u) = C\varepsilon (x, y, z) = 0,
A\varepsilon (x, y) := \alpha \~\varepsilon (x, y), \Sigma \varepsilon (x, y) := \beta \~\varepsilon (x, y),
\xi \varepsilon (t) := \~\varepsilon (\gamma +1)/2(1 - \gamma )
t\int
0
b(X\varepsilon (s), \~\varepsilon
1
1 - \gamma \~Y\~\varepsilon (s))dB(s).
Conditions H0 , H1 , and H2 are obviously true.
Functions from condition \bfH 3 are
A(x, y) =
\bigl(
\varphi +(x, 0)1y>0 + \varphi - (x, 0)1y<0
\bigr)
y\gamma ,
\Sigma (x, y) = \beta +(x, 0)1y\geq 0 + \beta - (x, 0)1y<0, C(x, y, z) = 0.
Note that A and \Sigma are discontinuous at y = 0 and formally \bfH 3 is not satisfied. However, the only
place in Theorem 2.3, where we used the continuity of A and \Sigma , is the identification of limit points
for the sequence \{ y\varepsilon n\} in the proof of Proposition 5.2. Since diffusion coefficients \beta \pm are separated
from zero functions, it can be seen that processes \{ y\varepsilon n\} spend small time in small neighborhoods of
0 uniformly on \{ \varepsilon n\} . This yields that any limit point of \{ y\varepsilon n\} solves (2.8) and Proposition 5.2 holds
true. This is all we need for Theorem 2.3 application.
Condition \bfH 4 is satisfied with \nu = \mu = 0.
Without loss of generality we will assume that
\varphi \pm (x, 0) \leq c < 0 for all x \in \BbbR d, (4.3)
where c is a constant. The general case can be considered using a localization. Hence, condition H5
is satisfied with \kappa = \gamma .
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1272 A. KULIK, A. PILIPENKO
Consider equation with frozen coefficients
dy(x)(t) =
\bigl(
\varphi +(x, 0)1y(x)(t)>0 + \varphi - (x, 0)1y(x)(t)<0
\bigr)
(y(x)(t))\gamma dt+
+
\Bigl(
\beta +(x, 0)1y(x)(t)\geq 0 + \beta - (x, 0)1y(x)(t)<0
\Bigr)
dW (t). (4.4)
Existence and uniqueness of a weak solution to equation with frozen coefficients, and the strong
Markov property follows from [10]. Hence, condition F0 holds true.
To verify condition F1 , we modify the argument from [16] (Section 3.3.2). Because the diffusion
coefficient in (4.4) is discontinuous, we do not have a good reference to state that the transition
probability density p(x)(t, y, y\prime ) is continuous in x, y, y\prime . In order to overcome this minor difficulty
we use the following localization argument. Consider the SDE
dy(x,+)(t) = \varphi +(x, 0)
\bigl(
| y(x,+)(t)| \wedge 2
\bigr) \gamma
\mathrm{s}\mathrm{g}\mathrm{n}
\bigl(
y(x,+)(t)
\bigr)
dt+ \beta +(x, 0) dW (t). (4.5)
This is an SDE with a constant diffusion coefficient and bounded and Hölder continuous drift coeffi-
cient, hence the standard analytic theory (see, e.g., [11]) yields that its transition probability density
p(x,+)(t, y, y\prime ) is continuous in x, y, y\prime . Then for y0 = 1 and every t0 > 0 it holds that
\mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
\bigm\| \bigm\| P (x,+)
t0
(y, dy\prime ) - P
(x,+)
t0
(y0, dy
\prime )
\bigm\| \bigm\|
TV
=
= \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
\int \bigm| \bigm| p(x,+)
t0
(y, y\prime ) - p
(x,+)
t0
(y0, y
\prime )
\bigm| \bigm| dy\prime \rightarrow 0, y \rightarrow y0.
The coefficients of the equations (4.4), (4.5) coincide on [0, 2], and thus the laws of the solutions to
these equations, stopped at the moment of exit from [0, 2], coincide. Taking t0 small enough, we
can guarantee that each of these solutions stay in [0, 2] up to the time t0 with probability \geq 5
6
if the
initial value y stays in
\biggl[
1
2
,
3
2
\biggr]
. By the coupling characterization of the TV distance (the “Coupling
lemma”, e.g., [16], Theorem 2.2.2), this yields that, for such t0,
\mathrm{s}\mathrm{u}\mathrm{p}
y1,y2\in [ 12 ,
3
2
],| x| \leq R
\bigm\| \bigm\| \bigm\| P (x)
t0
(y1, dy
\prime ) - P
(x,+)
t0
(y2, dy
\prime )
\bigm\| \bigm\| \bigm\|
TV
\leq
\biggl(
1 - 5
6
\biggr)
+
\biggl(
1 - 5
6
\biggr)
=
1
3
.
Combining these two estimates we see that there exist t0 > 0 and r > 0 small enough, so that
\mathrm{s}\mathrm{u}\mathrm{p}
y1,y2\in [1 - r,1+r],| x| \leq R
\bigm\| \bigm\| \bigm\| P (x)
t0
(y1, dy
\prime ) - P
(x)
t0
(y2, dy
\prime )
\bigm\| \bigm\| \bigm\|
TV
<
3
4
;
in the right-hand side we could actually take any number >
1
3
+
1
3
=
2
3
. This proves the local
Dobrushin condition in a small ball centered at y0 = 1. To extend this condition to a large ball
| y| \leq R, we use another standard argument, based on the support theorem. Namely, y(x) can be
represented as an image of a Brownian motion under the time change and the change of measure (see
[13]). Since the Wiener measure in C0(0,\infty ) has a full topological support, it is easy to show using
this representation that, for any t1 > 0, there exists \delta > 0 such that
P
(x)
t1
\bigl(
y, [1 - r, 1 + r]
\bigr)
\geq \delta , | x| \leq R, | y| \leq R.
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1273
Take h = t0 + t1 and for x, y1, y2 with | x| \leq R, | y1| \leq R, | y2| \leq R consider two processes Y 1
t ,
Y 2
t which start at y1, y2, respectively, solve (4.4) independently up to the time t1, and then provide
the maximal coupling probability on the time interval [t1, t1 + t0], conditioned on their values at
the time t1 (we can construct such a process using the Coupling lemma for probability kernels [16],
Theorem 2.2.4). Then \bigm\| \bigm\| \bigm\| P (x)
h (y1, dy
\prime ) - P
(x)
h (y2, dy
\prime )
\bigm\| \bigm\| \bigm\|
TV
\leq \bfP
\bigl(
Y 1
h \not = Y 2
h
\bigr)
=
=
\int
\bfR 2
4
\bigm\| \bigm\| \bigm\| P (x)
t0
(z1, dy
\prime ) - P
(x)
t0
(z2, dy
\prime )
\bigm\| \bigm\| \bigm\|
TV
P
(x)
h (y1, dz1)P
(x)
h (y2, dz2) \leq
\leq 1 - P
(x)
t1
\bigl(
y1, [1 - r, 1 + r]
\bigr)
P
(x)
t1
\bigl(
y2, [1 - r, 1 + r]
\bigr)
+
+
\int
[1 - r,1+r]2
\bigm\| \bigm\| \bigm\| P (x)
t0
(z1, dy
\prime ) - P
(x)
t0
(z2, dy
\prime )
\bigm\| \bigm\| \bigm\|
TV
P
(x)
h (y1, dz1)P
(x)
h (y2, dz2) \leq
\leq 1 +
\biggl(
- 1 +
3
4
\biggr)
P
(x)
t1
\bigl(
y1, [1 - r, 1 + r]
\bigr)
P
(x)
t1
\bigl(
y2, [1 - r, 1 + r]
\bigr)
\leq
\leq 1 - \delta 2
4
for any | x| \leq R, | y1| \leq R, | y2| \leq R, which completes the proof of F1 .
The IPM \pi (x)(dy) equals (see [14], Exercise 5.40):
\pi (x)(dy) = c(x)
\biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
\varphi +(x, 0)
(\beta +(x, 0))2
y\gamma +1
(\gamma + 1)
\biggr\}
1y\geq 0 + \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
\varphi - (x, 0)
(\beta - (x, 0))2
y\gamma +1
(\gamma + 1)
\biggr\}
1y<0
\biggr)
dy,
where
c(x) - 1 =
\int
\BbbR
\biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
\varphi +(x, 0)
(\beta +(x, 0))2
y\gamma +1
(\gamma + 1)
\biggr\}
1y\geq 0 + \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
\varphi - (x, 0)
(\beta - (x, 0))2
y\gamma +1
(\gamma + 1)
\biggr\}
1y<0
\biggr)
dy =
=
\Gamma ( 1
\gamma +1)
(\gamma + 1)
\Biggl( \biggl(
(\gamma + 1)(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
+
\biggl(
(\gamma + 1)(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
\Biggr)
.
Condition H6 is satisfied with a(x, y) = \psi +(x, 0)1y>0 + \psi - (x, 0)1y<0, \sigma (x, y) = c(x, y, z) = 0,
and B = \BbbR d \times \{ 0\} .
The averaged coefficient
a(x) = \psi +(x, 0)\pi (x)([0,\infty )) + \psi - (x, 0)\pi (x)(( - \infty , 0)) =
= \psi +(x, 0)
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
+
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
+
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1274 A. KULIK, A. PILIPENKO
+\psi - (x, 0)
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
\biggl(
(\beta - (x, 0))2
\varphi - (x, 0)
\biggr) 1
\gamma +1
+
\biggl(
(\beta +(x, 0))2
\varphi +(x, 0)
\biggr) 1
\gamma +1
is Lipschitz continuous, \=b(x) = 0, K(\rho )(x, dv) = K
(\rho )
(x, dv) = 0. So, condition A0 holds true and
the corresponding martingale problem has a unique solution.
This with (4.1) concludes the proof.
5. Proof of Theorem 2.3. The weak compactness of the family \{ X\varepsilon , \varepsilon > 0\} in \BbbD ([0,\infty ),\BbbR d)
follows, in a standard way, from the negligibility assumption H0 and the boundedness assumptions
\bfH 1, \bfH 2. Under the assumptions of the theorem, for any C\infty
0 -function \varphi the function L\varphi is conti-
nuous and bounded. Hence, in order to prove that any weak limit point of the family \{ X\varepsilon , \varepsilon > 0\} as
\varepsilon \rightarrow 0 solves the martingale problem (2.9), it is enough to show that, for any C\infty
0 -function \varphi , any
s1, . . . , sq < s < t, and any continuous and bounded function \Phi : \BbbR d\times q \rightarrow \BbbR
\bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \left[ \varphi \bigl( X\varepsilon (t)
\bigr)
- \varphi (X\varepsilon (s)) -
t\int
s
L\varphi (X\varepsilon (r)) dr
\right] \rightarrow 0, \varepsilon \rightarrow 0, (5.1)
we denote by \bfE \varepsilon the expectation w.r.t. \bfP \varepsilon . Denote\widetilde X\varepsilon (t) = X\varepsilon (t) - \xi \varepsilon (t) =
= X\varepsilon (0) +
t\int
0
a\varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (t)
\bigr)
ds+
t\int
0
\sigma \varepsilon
\bigl(
X\varepsilon (s), Y\varepsilon (s)
\bigr)
dB\varepsilon
s+
+
t\int
0
\int
\BbbR m
c\varepsilon
\bigl(
X\varepsilon (s - ), Y\varepsilon (s - ), u
\bigr) \Bigl[
N \varepsilon (du, ds) - 1| u| \leq \rho \nu
\varepsilon (du)ds
\Bigr]
. (5.2)
Observe that L\varphi is a bounded and continuous function. So, by H0 relation (5.1) is equivalent to
\bfE \varepsilon \Phi
\bigl( \widetilde X\varepsilon (s1), . . . , \widetilde X\varepsilon (sq)
\bigr) \left[ \varphi \bigl( \widetilde X\varepsilon (t)
\bigr)
- \varphi ( \widetilde X\varepsilon (s)) -
t\int
s
L\varphi ( \widetilde X\varepsilon (r)) dr
\right] \rightarrow 0, \varepsilon \rightarrow 0. (5.3)
Denote
b\varepsilon (x, y) = \sigma \varepsilon (x, y)
\bigl(
\sigma \varepsilon (x, y)
\bigr) \ast
, K\varepsilon
(\rho )(x, y,A) = \nu \varepsilon
\bigl(
\{ u : | u| \leq \rho , c\varepsilon (x, y, u) \in A\}
\bigr)
,
K(\rho ),\varepsilon (x, y,A) = \nu \varepsilon
\bigl(
\{ u : | u| > \rho , c\varepsilon (x, y, u) \in A\}
\bigr)
,
and
\scrL \varepsilon \varphi (x, y) = \nabla \varphi (x) \cdot a\varepsilon (x, y) + 1
2
\nabla 2\varphi (x) \cdot b\varepsilon (x, y)+
+
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x) - \nabla \varphi (x) \cdot v
\Bigr)
K\varepsilon
(\rho )(x, dv)+
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1275
+
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x)
\Bigr)
K(\rho ),\varepsilon (x, y, dv).
Then, by the Itô formula, we have
\varphi
\bigl( \widetilde X\varepsilon (t)
\bigr)
- \varphi ( \widetilde X\varepsilon (s)) =
t\int
s
\scrL \varepsilon \varphi (X\varepsilon (r), Y\varepsilon (r)) dr + (martingale part). (5.4)
Applying H0 once again, we get that, to prove (5.1) and (5.3), it is enough to prove, for any
s1, . . . , sq < t,
\bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr)
\rightarrow 0, \varepsilon \rightarrow 0. (5.5)
Before proving (5.5), we formulate and prove two auxiliary statements.
5.1. Auxiliaries, I: uniform ergodic rate for the frozen microscopic dynamics.
Proposition 5.1. Let conditions H1 – H5 , F0 , F1 hold. If \kappa \in (0, 1) and p > 0 are from these
conditions, then for every R > 0 there exists C such that for any x, y with | x| \leq R, | y| \leq R\bigm\| \bigm\| P (x)
t (y, dy\prime ) - \pi (x)(dy\prime )
\bigm\| \bigm\|
TV
\leq Ct -
p+\kappa - 1
1 - \kappa . (5.6)
If \kappa \geq 1, then there exists a > 0 such that, for every R > 0 and any x, y with | x| \leq R, | y| \leq R,\bigm\| \bigm\| P (x)
t (y, dy\prime ) - \pi (x)(dy\prime )
\bigm\| \bigm\|
TV
\leq Ce - at
with a constant C depending on R.
Proof. The required statement is actually obtained, though not in this precise form, in [16]
(Section 3). The difference between the current situation and the one studied in [16] is that the
ergodic rates were obtained there for individual processes (while here we have a family indexed
by x) and separately for diffusions and Lévy driven SDEs (while here we have both types of the
noise involved simultaneously). This difference is not crucial, and we just give a short outline of the
argument, referring to [16] for details.
The convergence conditions H3 , H4 yield that the bounds from the conditions H1 , H2 and the
drift condition H5 remain true for the limiting coefficients A(x, y), \Sigma (x, y), C(x, y, z) and Lévy
measure \mu (du). Then we have the following: if V \in C2 is a function such that V (y) \geq 1 and
V (y) = | y| p, | y| \geq 2, then, for any x \in \BbbR d, the semimartingale decomposition holds
V (y(x)(t)) = V
\bigl(
y(x)(0)
\bigr)
+
t\int
0
\scrA V
\bigl(
x, y(x)(s)
\bigr)
ds+ (martingale part), (5.7)
where the function \scrA V (x, y) satisfies
\scrA V (x, y) \leq
\left\{ CV - aV V (y)
p+\kappa - 1
p , \kappa \in (0, 1),
CV - aV V (y), \kappa \geq 1,
(5.8)
with some constants CV , aV > 0. For the proof of this statement, see [17] (Proposition 2.5).
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1276 A. KULIK, A. PILIPENKO
Given (5.7), (5.8) we can proceed analogously to [16] (Sections 3.3, 3.4). Namely, for \kappa \in (0, 1)
we use [16] (Theorem 3.2.3) and [16] (Example 3.2.6) to show that
\bfE y
\widetilde V \bigl( y(x)(h)\bigr) - \widetilde V (y) \leq \widetilde CV - \widetilde cV \widetilde V (y)
p+\kappa - 1
p , (5.9)
where h is the same as in the assumption F1 , \widetilde CV , \widetilde cV > 0 are some new constants, and \widetilde V is a new
function which is equivalent to V in the sense that, for some positive constants c1, c2
c1V \leq \widetilde V \leq c2V.
Following the proof of [16] (Theorem 3.2.3) and calculations of [16] (Example 3.2.6) line by line,
we easily see that, because the constants CV , cV in (5.8) do not depend on x, the constants \widetilde CV , \widetilde cV ,
c1, c2 and the function \widetilde V can be chosen uniformly for x \in \BbbR d.
Inequality (5.9) is actually the Lyapunov condition for the skeleton chain y
(x,h)
k = y(x)(kh),
k \geq 0, for the process y(x), see [16] (Section 2.8). Combined with the local Dobrushin condition
assumed in F1 , we get by [16] (Corollary 2.8.10) the inequality\bigm\| \bigm\| \bigm\| P (x)
kh (y, dy\prime ) - \pi (x)(dy\prime )
\bigm\| \bigm\| \bigm\|
TV
\leq C(1 + k) -
p+\kappa - 1
1 - \kappa \widetilde V (y), \kappa \in (0, 1),
where we have used the identity
p+ \kappa - 1
p
\biggl(
1 - p+ \kappa - 1
p
\biggr) - 1
=
p+ \kappa - 1
1 - \kappa
.
Since Lyapunov condition and the local Dobrushin condition are uniform in x, the constant C here
can be chosen uniformly for x \in \BbbR d; one can easily check this following line by line the proofs
of [16] (Corollary 2.8.10) and the theorems it is based on: [16] (Theorems 2.7.5 and 2.8.6). Since
the total variation distance
\bigm\| \bigm\| P (x)
t (y, dy\prime ) - \pi (x)(dy\prime )
\bigm\| \bigm\|
TV
is nonincreasing in t and V (y) is locally
bounded, this completes the proof of the required statement in the case \kappa \in (0, 1).
For \kappa \geq 1, we can argue in a completely analogous way, using [16] (Corollary 2.8.3).
5.2. Auxiliaries, II: weak convergence of the microscopic dynamics to the frozen one. Consider
the following microscopic analogue of (2.4). Assume that (x\varepsilon , y\varepsilon ) is a solution (maybe nonunique)
to the equations
x\varepsilon (t) = x\varepsilon (0) + \varepsilon
t\int
0
a\varepsilon
\bigl(
x\varepsilon (s), y\varepsilon (t)
\bigr)
ds+ \varepsilon 1/2
t\int
0
\sigma \varepsilon
\bigl(
x\varepsilon (s), y\varepsilon (s)
\bigr)
db\varepsilon s+
+
t\int
0
\int
\BbbR m
C\varepsilon
\bigl(
x\varepsilon (s - ), y\varepsilon (s - ), u
\bigr) \Bigl[
n\varepsilon (du, ds) - 1| u| \leq \rho \varepsilon \nu
\varepsilon (du)ds
\Bigr]
+ \zeta \varepsilon (t),
y\varepsilon (t) = y\varepsilon (0) +
t\int
0
A\varepsilon
\bigl(
x\varepsilon (s), y\varepsilon (s)
\bigr)
ds+
t\int
0
\Sigma \varepsilon
\bigl(
x\varepsilon (s), y\varepsilon (s)
\bigr)
dw\varepsilon
s+
+
t\int
0
\int
\BbbR l
c\varepsilon
\bigl(
x\varepsilon (s - ), y\varepsilon (s - ), z
\bigr) \Bigl[
q\varepsilon (dz, ds) - 1| z| \leq \rho \mu
\varepsilon (dz)ds
\Bigr]
,
(5.10)
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1277
where b\varepsilon t , w
\varepsilon
t are Brownian motions and n\varepsilon (du, dt), q\varepsilon (dz, dt) are Poisson point measures on a
common filtered probability space
\bigl( \widetilde \Omega \varepsilon , \widetilde \scrF \varepsilon , \widetilde \bfP \varepsilon
\bigr)
, and the random measures n\varepsilon (du, dt), q\varepsilon (dz, dt)
have the intensity measures \varepsilon \nu \varepsilon (du)dt and \mu \varepsilon (dz)dt, respectively, \zeta \varepsilon (t) is an adapted càdlàg process.
System (5.10) naturally appears, e.g., if we consider the original system (2.4) at the “microscopic
time scale” \varepsilon t with an initial time shift by t0 :
x\varepsilon (t) = X\varepsilon (t0 + \varepsilon t), y\varepsilon (t) = Y\varepsilon (t0 + \varepsilon t), and \zeta \varepsilon (t) = \xi \varepsilon (t0 + \varepsilon t) - \xi \varepsilon (t0). (5.11)
For a fixed pair of functions (\rho (\varepsilon ), \varrho (\varepsilon )) such that \rho (\varepsilon ) \rightarrow 0 and \varrho (\varepsilon ) \rightarrow 0 as \varepsilon \rightarrow 0, and
constants R > 0, T > 0 denote by \scrK (\rho , \varrho ,R, T ) the class of all families
\bigl\{
(x\varepsilon , y\varepsilon ), \varepsilon > 0
\bigr\}
which
satisfy (5.10) on some probability space with nonrandom initial values x\varepsilon (0), y\varepsilon (0), | x\varepsilon (0)| \leq R,
| y\varepsilon (0)| \leq R and
\widetilde \bfP \varepsilon
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\leq T
| \zeta \varepsilon (s)| > \rho (\varepsilon )
\biggr)
\leq \varrho (\varepsilon ).
Proposition 5.2. Let conditions H1 – H5 , F0 hold. Then, for any 0 < t < T and any bounded
continuous function f : \BbbR d \times \BbbR k \rightarrow \BbbR and R > 0,
\mathrm{s}\mathrm{u}\mathrm{p}
\{ (x\varepsilon ,y\varepsilon )\} \in \scrK (\rho ,\varrho ,R,T )
\bigm| \bigm| \bigm| \widetilde \bfE \varepsilon f(x\varepsilon (t), y\varepsilon (t)) - P frozen
t f(x, y)
\bigm| \bigm| \bigm|
x=x\varepsilon (0),y=y\varepsilon (0)
\bigm| \bigm| \bigm| \rightarrow 0, \varepsilon \rightarrow 0, (5.12)
where
P frozen
t f(x, y) =
\int
\BbbR k
f(y\prime )P
(x)
t (y, dy\prime ), t \geq 0.
Proof. Assuming the contrary, we will have that there exists a sequence x\varepsilon n(\cdot ), y\varepsilon n(\cdot ) of
solutions to (5.10) with | x\varepsilon n(0)| \leq R, | y\varepsilon n(0)| \leq R such that\Bigl( \widetilde \bfE \varepsilon nf(x\varepsilon n(t), y\varepsilon n(t)) - P
(x)
t f(x, y)
\bigm| \bigm| \bigm|
x=x\varepsilon n (0),y=y\varepsilon n (0)
\Bigr)
\not \rightarrow 0, n\rightarrow \infty . (5.13)
Without loss of generality, after passing to a subsequence, we can assume that x\varepsilon n(0) \rightarrow x\ast and
y\varepsilon n(0) \rightarrow y\ast as n\rightarrow \infty . Then it is easy to show that, for any c > 0,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\widetilde \bfP \varepsilon n
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
| x\varepsilon n(s) - x\ast | > c
\Biggr)
= 0. (5.14)
Next, denote by \bfP \ast the law in \bfD
\bigl(
[0, T ],\BbbR k
\bigr)
of y(x\ast ) with y(x\ast )(0) = y\ast . Since the \bfP \ast -probability
for y(\cdot ) to have a jump at the point t is 0, the function F (y(\cdot )) = f(y(t)) is a.s. continuous on
\bfD
\bigl(
[0, T ],\BbbR k
\bigr)
. Thus, in order to prove that (5.13) fails, it is enough to show that the laws of y\varepsilon n ,
n \geq 1, weakly converge in \bfD
\bigl(
[0, T ],\BbbR k
\bigr)
to \bfP \ast . Such a statement is quite standard, and we just
outline its proof here.
By (5.14), the continuity assumption H3 , and convergence of the noise H4 it is easy to prove that
any weak limit point to \{ y\varepsilon n\} solves (2.8). By the weak uniqueness assumption F0 , this yields that
any weak limit point to \{ y\varepsilon n\} has the law \bfP \ast .
That is, to prove the required weak convergence it is enough to prove that \{ y\varepsilon n\} is weakly
compact in \bfD
\bigl(
[0, T ],\BbbR k
\bigr)
.
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1278 A. KULIK, A. PILIPENKO
To prove the weak compactness, we use L2-moment bounds for the increments of the process
y\varepsilon n combined with a truncation of the large jumps. Namely, by H2 for any fixed \delta > 0 there exists
Q\delta such that
\widetilde \bfP \varepsilon
\Bigl(
N
\bigl(
[0, T ]\times \{ | z| > Q\delta \}
\bigr)
> 0
\Bigr)
< \delta , \varepsilon > 0.
Thus it is enough to prove weak compactness for every “truncated” family \{ y\varepsilon n,Q\} , Q > 0, where
y\varepsilon ,Q satisfies an analogue of (5.10) with the integral for q\varepsilon taken over \{ | z| \leq Q\} instead of \BbbR l. For
such a “truncated” family, applying [17] (Proposition 2.5) we get
\bigm| \bigm| y\varepsilon n,Q(s)\bigm| \bigm| 2 = \bigm| \bigm| y\varepsilon n,Q(0)\bigm| \bigm| 2 + s\int
0
H(r) dr + (martingale part), (5.15)
where H is bounded. Combining this with the maximal martingale inequality, we get that
\widetilde \bfE \varepsilon n \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
\bigm| \bigm| y\varepsilon n,Q(s)\bigm| \bigm| 2
is bounded. Since the coefficient A\varepsilon (x, y) is bounded locally in y, the above bound and the (uniform)
bounds for C\varepsilon , \mu \varepsilon from H1 , H2 yield the required weak compactness of \{ y\varepsilon n\} . Summarizing all the
above, we have that \{ y\varepsilon n\} weakly converges to \bfP \ast . Combined with (5.14), this contradicts to (5.13)
and proves the required statement.
5.3. End of the proof of Theorem 2.3. In this subsection we complete the proof of (5.5). This
will conclude proof of the theorem. Denote
\scrL \varphi (x, y) = \nabla \varphi (x) \cdot a(x, y) + 1
2
\nabla 2\varphi (x) \cdot b(x, y)+
+
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x) - \nabla \varphi (x) \cdot v
\Bigr)
K(\rho )(x, y, dv)+
+
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x)
\Bigr)
K(\rho )(x, y, dv) =
= \nabla \varphi (x) \cdot a(x, y) + 1
2
\nabla 2\varphi (x) \cdot b(x, y)+
+
\int
\BbbR m
\Bigl(
\varphi (x+ c(x, y, u)) - \varphi (x) - \nabla \varphi (x) \cdot c(x, y, u)1| u| \leq \rho
\Bigr)
\nu (du). (5.16)
Next, since the set B from the condition H6 is open, there exists a sequence of continuous functions
\chi j(x, y), j \geq 1, such that
(i) 0 \leq \chi j(x, y) \leq 1, j \geq 1;
(ii) each \chi j has a support compactly embedded to B;
(iii) for each x, y, \chi j(x, y) \nearrow \chi \infty (x, y) = 1B(x, y), j \rightarrow \infty .
Recall the notation f(x) =
\int
\BbbR k
f(x, y)\pi (x)(dy).
The following lemma collects several simple statements used in the proof.
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1279
Lemma 5.1. The following properties hold:
(a) there exists C > 0 such that | \scrL \varepsilon \varphi (x, y)| \leq C, \varepsilon > 0, for all x, y;
(b) \scrL \varepsilon \varphi \rightarrow \scrL \varphi , \varepsilon \rightarrow 0, uniformly on each compactum K \subset B;
(c) there exists C > 0 such that | \scrL \varphi (x, y)| \leq C for all (x, y) \in B;
(d) \chi j(x) \rightarrow 1, j \rightarrow \infty , uniformly on \{ | x| \leq R\} for each R > 0;
(e) \chi j\scrL \varphi (x) \rightarrow L\varphi (x), j \rightarrow \infty , uniformly on \{ | x| \leq R\} for each R > 0, where L is defined
in (2.9);
(f) for any T > 0,
\bfE \varepsilon
\bigm| \bigm| \widetilde X\varepsilon (t) - \widetilde X\varepsilon (s)
\bigm| \bigm| 2 \wedge 1 \leq C| t - s| , s, t \in [0, T ],
where \widetilde X is defined in (5.2), and
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ],\varepsilon >0
\bfP \varepsilon
\bigl(
| X\varepsilon (t)| > R
\bigr)
\rightarrow 0, R\rightarrow \infty ;
(g) for any T > 0,
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ],\varepsilon >0
\bfP \varepsilon
\bigl(
| Y\varepsilon (t)| > R
\bigr)
\rightarrow 0, R\rightarrow \infty .
Proof. Statement (a) follows directly from the assumptions H1 , H2 . Statement (b) can be
derived, in a standard way, using the convergence assumptions H4 , H6 and the bounds from the
assumptions H1 , H2 . Statement (c) follows from (a) and (b).
To prove statement (d), we first mention that each function \chi j is continuous by the assumption
F0 . These functions converge monotonously, at each x \in \BbbR d, to the function
\chi \infty (x) =
\int
\BbbR k
1B(x, y)\pi
(x)(dy) \equiv 1,
where the last identity holds by the assumption H6 . Then the required uniform convergence follow
by the Dini theorem.
To prove statement (e), we first use statements (c) and (d) to get\bigm| \bigm| \chi j\scrL \varphi (x) - \scrL \varphi (x)
\bigm| \bigm| = \bigm| \bigm| \chi j\scrL \varphi (x) - \chi \infty \scrL \varphi (x)
\bigm| \bigm| \leq C(1 - \chi j(x)) \rightarrow 0, j \rightarrow \infty ,
uniformly for x with | x| \leq R. Then the required statement follows by the identity
\scrL \varphi (x) =
\int
\BbbR k
\scrL \varphi (x, y)\pi (x)(dy) =
=
\int
\BbbR k
\biggl(
\nabla \varphi (x) \cdot a(x, y) + 1
2
\nabla 2\varphi (x) \cdot b(x, y)
\biggr)
\pi (x)(dy)+
+
\int
\BbbR k
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x) - \nabla \varphi (x) \cdot v
\Bigr)
K(\rho )(x, dv)\pi
(x)(dy)+
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1280 A. KULIK, A. PILIPENKO
+
\int
\BbbR k
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x)
\Bigr)
K(\rho )(x, y, dv)\pi (x)(dy) =
= \nabla \varphi (x) \cdot a(x) + 1
2
\nabla 2\varphi (x) \cdot b(x) +
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x) - \nabla \varphi (x) \cdot v
\Bigr)
K(\rho )(x, dv)+
+
\int
\BbbR m
\Bigl(
\varphi (x+ v) - \varphi (x)
\Bigr)
K
(\rho )
(x, dv) =
= L\varphi (x).
Statement (f) can be obtained using the same “truncation of large jumps” argument as in the proof of
Proposition 5.2 and the bounds from the assumptions H1 , \bfH 2; we omit the details.
To prove statement (g), we treat Y\varepsilon (t) as the value of the process y\varepsilon from (5.10) taken at the
(large) time instant \tau = \varepsilon - 1t with \zeta \varepsilon (\tau ) = \xi \varepsilon (\varepsilon \tau ), i.e., y\varepsilon (\tau ) = Y\varepsilon (\varepsilon \tau ). Without loss of generality
we can assume that the constant \kappa in the assumption H5 satisfies \kappa \leq 1. Then by [17] (Theorem 2.8),
for every pY < p+ \kappa - 1,
\mathrm{s}\mathrm{u}\mathrm{p}
\tau \geq 0,\varepsilon >0
\bfE \varepsilon
\bigm| \bigm| y\varepsilon (\tau )\bigm| \bigm| pY <\infty ,
here we have used that the initial values y\varepsilon (0) = Y\varepsilon (0) are bounded. This immediately yields (g).
Lemma 5.1 is proved.
Now we are ready to prove (5.5). Fix N > 0, and write denote by \bfP \varepsilon
t - \varepsilon N ,\bfE
\varepsilon
t - \varepsilon N the conditional
probability and conditional expectation w.r.t. \scrF \varepsilon
t - \varepsilon N . For \varepsilon small enough, we have sq < t - \varepsilon N and
thus
\bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr)
=
= \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr)
\bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr)
.
By the assumption \bfH 0, there exist functions \rho (\varepsilon ) \rightarrow 0, \varrho (\varepsilon ) \rightarrow 0 such that
\bfP \varepsilon
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
| \xi \varepsilon (s)| > \rho (\varepsilon )
\Biggr)
\leq \varrho (\varepsilon ),
where T > t is a fixed number. For a given R > 0, consider the \scrF \varepsilon
t - \varepsilon N -measurable set
\Omega \varepsilon
t,N,R =
\Biggl\{
\omega : \bfP \varepsilon
t - \varepsilon N
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
| \xi \varepsilon (s)| > \rho (\varepsilon )
\Biggr)
\leq R\varrho (\varepsilon )
\Biggr\}
,
then by the Markov inequality
\bfP \varepsilon (\Omega \varepsilon \setminus \Omega \varepsilon
t,N,R) \leq
1
R\varrho (\varepsilon )
\bfE \varepsilon
\Biggl[
\bfP \varepsilon
t - \varepsilon N
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
| \xi \varepsilon (s)| > \rho (\varepsilon )
\Biggr) \Biggr]
=
=
1
R\varrho (\varepsilon )
\bfP \varepsilon
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,T ]
| \xi \varepsilon (s)| > \rho (\varepsilon )
\Biggr)
\leq
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1281
\leq \varrho (\varepsilon )
R\varrho (\varepsilon )
=
1
R
.
We have seen in the proof of Lemma 5.1 that L\varphi = \chi \infty \scrL \varphi , thus by statement (c) of this lemma the
function L\varphi is bounded. The functions \Phi , \scrL \varepsilon are bounded, as well,\bigm| \bigm| \bigm| \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| \leq
\leq C\bfP \varepsilon (| X\varepsilon (t - \varepsilon N)| > R) + C\bfP \varepsilon (| Y\varepsilon (t - \varepsilon N)| > R) +
C
R
+
+C\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
\bigm| \bigm| \bigm| \bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| , (5.17)
where we denote
\widetilde \Omega \varepsilon
t,N,R = \Omega \varepsilon
t,N,R \cap
\bigl\{
| X\varepsilon (t - \varepsilon N)| \leq R, | Y\varepsilon (t - \varepsilon N)| \leq R
\bigr\}
.
Fix j \geq 1, decompose
\bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr)
=
= \bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- \scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
+
+\bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- \scrL \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
+
+
\Bigl(
\bfE \varepsilon
t - \varepsilon N\scrL \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- P frozen
N (\chi j\scrL \varphi )
\bigl(
X\varepsilon (t - \varepsilon N), Y\varepsilon (t - \varepsilon N)
\bigr) \Bigr)
+
+
\Bigl(
P frozen
N (\chi j\scrL \varphi )
\bigl(
X\varepsilon (t - \varepsilon N), Y\varepsilon (t - \varepsilon N)
\bigr)
- \chi j\scrL \varphi (X\varepsilon (t - \varepsilon N))
\Bigr)
+
+
\Bigl(
\chi j\scrL \varphi (X\varepsilon (t - \varepsilon N)) - L\varphi (X\varepsilon (t - \varepsilon N))
\Bigr)
+
+
\Bigl(
L\varphi (X\varepsilon (t - \varepsilon N)) - L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr)
. (5.18)
Let us estimate each term in the decomposition (5.18). For the first term, we simply write using
Lemma 5.1(a) \bigm| \bigm| \bigm| \bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- \scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| \leq
\leq C\bfE \varepsilon
t - \varepsilon N
\Bigl(
1 - \chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
. (5.19)
For the second term, we recall that the support of \chi j is compactly embedded to B, thus, by
Lemma 5.1(b),\bigm| \bigm| \bigm| \bfE \varepsilon
t - \varepsilon N
\Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- \scrL \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \bigm| \bigm| \bigm| \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x,y
\bigm| \bigm| \scrL \varepsilon \varphi (x, y) - \scrL \varphi (x, y))
\bigm| \bigm| \chi j(x, y) \rightarrow 0, \varepsilon \rightarrow 0. (5.20)
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1282 A. KULIK, A. PILIPENKO
To estimate the third term in (5.18), observe first that the function \chi j\scrL \varphi is continuous, which folows
from H4 , H6 similarly to Lemma 5.1(b). Next, define the pair x\varepsilon , y\varepsilon by (5.11) with t0 = t - \varepsilon N and
take \widetilde \bfP \varepsilon = \bfP \varepsilon
t - \varepsilon N,\omega , the regular version of the conditional probability. Then, for a.a. \omega \in \widetilde \Omega \varepsilon
t,N,R,
the pair x\varepsilon , y\varepsilon w.r.t. the probability \bfP \varepsilon
t - \varepsilon N,\omega belongs to the class \scrK (\rho , 2R\varrho ,R, 2N) in the notation
introduced before Proposition 5.2. Applying this proposition, we get
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
\bigm| \bigm| \bigm| \bfE \varepsilon
t - \varepsilon N\scrL \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
\chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
-
- P frozen
N (\chi j\scrL \varphi )
\bigl(
X\varepsilon (t - \varepsilon N), Y\varepsilon (t - \varepsilon N)
\bigr) \bigm| \bigm| \bigm| \rightarrow 0, \varepsilon \rightarrow 0. (5.21)
To estimate the fourth term, we use Proposition 5.1; without loss of generality we assume that \kappa < 1.
Since the function \chi j\scrL \varphi is bounded, Proposition 5.1 yields
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
\bigm| \bigm| \bigm| P frozen
N (\chi j\scrL \varphi )
\bigl(
X\varepsilon (t - \varepsilon N), Y\varepsilon (t - \varepsilon N)
\bigr)
- \chi j\scrL \varphi (X\varepsilon (t - \varepsilon N))
\bigm| \bigm| \bigm| \leq CN - p+\kappa - 1
1 - \kappa . (5.22)
For the fifth term, we have
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
\bigm| \bigm| \bigm| \bigl( \chi j\scrL \varphi (X\varepsilon (t - \varepsilon N)) - L\varphi (X\varepsilon (t - \varepsilon N))
\bigr) \bigm| \bigm| \bigm| \leq \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
\bigm| \bigm| \chi j\scrL \varphi (x) - L\varphi (x)
\bigm| \bigm| . (5.23)
For the sixth term, we get
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
\bigm| \bigm| \bigm| L\varphi (X\varepsilon (t - \varepsilon N)) - L\varphi
\bigl(
X\varepsilon (t)
\bigr) \bigm| \bigm| \bigm| \rightarrow 0, \varepsilon \rightarrow 0, (5.24)
by Lemma 5.1(f) and uniform continuity of L\varphi on compacts. Summarizing the estimates (5.17) and
(5.19) – (5.24), we get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bigm| \bigm| \bigm| \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon (| X\varepsilon (s)| > R) + C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon
\bigl(
| Y\varepsilon (s)| > R
\bigr)
+
C
R
+
+CN - p+\kappa - 1
1 - \kappa + \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
\bigm| \bigm| \scrL \varphi \chi j(x) - L\varphi (x)
\bigm| \bigm| +
+C \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
E\varepsilon
t - \varepsilon N
\Bigl(
1 - \chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
. (5.25)
Similarly to (5.21) – (5.23), we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfE \varepsilon 1\widetilde \Omega \varepsilon
t,N,R
E\varepsilon
t - \varepsilon N
\Bigl(
1 - \chi j
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr) \Bigr)
\leq CN - p+\kappa - 1
1 - \kappa + \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
(1 - \chi j(x)),
thus,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bigm| \bigm| \bigm| \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon (| X\varepsilon (s)| > R) + C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon
\bigl(
| Y\varepsilon (s)| > R
\bigr)
+
C
R
+
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1283
+CN - p+\kappa - 1
1 - \kappa + \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
\bigm| \bigm| \scrL \varphi \chi j(x) - L\varphi (x)
\bigm| \bigm| + C \mathrm{s}\mathrm{u}\mathrm{p}
| x| \leq R
(1 - \chi j(x)). (5.26)
The constants R, N, j in the above inequality are arbitrary. Taking first j \rightarrow \infty , N \rightarrow \infty for a fixed
R, we get, by Lemma 5.1(d), (e), that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bigm| \bigm| \bigm| \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon (| X\varepsilon (s)| > R) + C \mathrm{s}\mathrm{u}\mathrm{p}
s\leq t,\varepsilon >0
\bfP \varepsilon
\bigl(
| Y\varepsilon (s)| > R
\bigr)
+
C
R
. (5.27)
Then by Lemma 5.1(f), (g) we can pass to the limit R\rightarrow \infty and finally get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bigm| \bigm| \bigm| \bfE \varepsilon \Phi
\bigl(
X\varepsilon (s1), . . . , X\varepsilon (sq)
\bigr) \Bigl(
\scrL \varepsilon \varphi
\bigl(
X\varepsilon (t), Y\varepsilon (t)
\bigr)
- L\varphi
\bigl(
X\varepsilon (t)
\bigr) \Bigr) \bigm| \bigm| \bigm| = 0.
This proves (5.5) and completes the entire proof.
6. Appendix. Proof of Lemma 3.1. Set
\zeta \varepsilon (t) = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ s \geq 0 :
s\int
0
b2\varepsilon (z)dz \geq t
\right\} .
Making the change of time \~\eta \varepsilon (t) := \eta \varepsilon (\zeta \varepsilon (t)), we see that \~\eta \varepsilon (t) satisfies assumptions of this lemma
with another constant \~A > 0 and a new Wiener process \~W (t) =W (\zeta \varepsilon (t)) but with \~b\varepsilon (t) \equiv 1. Since
(C2)
- 2t \leq \zeta \varepsilon (t) \leq (C1)
- 2t without loss of generality we will assume that b\varepsilon (t) \equiv 1.
Set L\varepsilon := Ax\gamma
d
dx
+
\varepsilon 2
2
d2
dx2
. Denote
v\varepsilon (x) :=
| x| \int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2Ay\gamma +1
(\gamma + 1)\varepsilon 2
\biggr\} \left( y\int
0
2
\varepsilon 2
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2Az\gamma +1
(\gamma + 1)\varepsilon 2
\biggr\}
dz
\right) dy.
We have L\varepsilon v\varepsilon (x) \geq 1, \mathrm{s}\mathrm{g}\mathrm{n}(x)v\prime \varepsilon (x) \geq 0, and v\varepsilon (0) = 0.
Then, by Ito’s formula, we have
\bfE v\varepsilon (\eta \varepsilon (\tau \varepsilon (\delta ) \wedge n)) = \bfE
\tau \varepsilon (\delta )\wedge n\int
0
\bigl(
a\varepsilon (s)\eta
\gamma
\varepsilon (s)v
\prime
\varepsilon (\eta \varepsilon (s)) +
\varepsilon 2
2
v\prime \prime \varepsilon (\eta \varepsilon (s))
\bigr)
ds \geq
\geq \bfE
\tau \varepsilon (\delta )\wedge n\int
0
\bigl(
A\eta \gamma \varepsilon (s)v
\prime
\varepsilon (\eta \varepsilon (s)) +
\varepsilon 2
2
v\prime \prime \varepsilon (\eta \varepsilon (s))
\bigr)
ds = \bfE
\tau \varepsilon (\delta )\wedge n\int
0
L\varepsilon v\varepsilon (\eta \varepsilon (s))ds \geq
\geq \bfE
\tau \varepsilon (\delta )\wedge n\int
0
1ds = \bfE \tau \varepsilon (\delta ) \wedge n.
Passing n\rightarrow \infty and applying the Fatou lemma we get a.s. finiteness of \tau \varepsilon (\delta ). Since v\varepsilon (\eta \varepsilon (\tau \varepsilon (\delta ))) =
= v\varepsilon (\delta ) = v\varepsilon ( - \delta ), we get the estimate
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1284 A. KULIK, A. PILIPENKO
\bfE \tau \varepsilon (\delta ) \leq v\varepsilon (\delta ).
Let x > 0 be arbitrary. Changing the variables s :=
z\gamma +1
\varepsilon 2
and t :=
y\gamma +1
\varepsilon 2
we obtain
v\varepsilon (x) =
2\varepsilon
2
\gamma +1
(\gamma + 1)\varepsilon 2
| x| \gamma +1
\varepsilon 2\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2At
\gamma + 1
\biggr\} \left(
t
1
\gamma +1 \varepsilon
2
\gamma +1\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2Az\gamma +1
(\gamma + 1)\varepsilon 2
\biggr\}
dz
\right) t - \gamma
\gamma +1dt =
=
2\varepsilon
4
\gamma +1
(\gamma + 1)2\varepsilon 2
| x| \gamma +1
\varepsilon 2\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2At
\gamma + 1
\biggr\} \left( t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2As
\gamma + 1
\biggr\}
s
- \gamma
\gamma +1ds
\right) t - \gamma
\gamma +1dt =
=
2
(\gamma + 1)2
\varepsilon
2(1 - \gamma )
\gamma +1
| x| \gamma +1
\varepsilon 2\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- 2At
\gamma + 1
\biggr\} \left( t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2As
\gamma + 1
\biggr\}
s
- \gamma
\gamma +1ds
\right) t - \gamma
\gamma +1dt. (6.1)
It follows from L’Hôpital’s rule that for any \alpha > 0 and \beta > - 1:
t\int
0
e\alpha ss\beta ds \sim \alpha - 1e\alpha tt\beta , t\rightarrow +\infty .
So,
t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2As
\gamma + 1
\biggr\}
s
- \gamma
\gamma +1ds \sim \gamma + 1
2A
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2At
\gamma + 1
\biggr\}
t
- \gamma
\gamma +1 , t\rightarrow +\infty .
Applying this and L’Hôpital’s rule, we get
u\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- - 2At
\gamma + 1
\biggr\} \left( t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2As
\gamma + 1
\biggr\}
s
- \gamma
\gamma +1ds
\right) t - \gamma
\gamma +1dt \sim
\sim \gamma + 1
2A
u\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
- - 2At
\gamma + 1
\biggr\} \biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
2At
\gamma + 1
\biggr\}
t
- \gamma
\gamma +1
\biggr)
t
- \gamma
\gamma +1dt =
=
\gamma + 1
2A
u\int
0
t
- 2\gamma
\gamma +1 dt =
(\gamma + 1)2
2A(1 - \gamma )
u
1 - \gamma
\gamma +1 , u\rightarrow +\infty .
Therefore, we get from (6.1) the following equivalence for any fixed x \not = 0 as \varepsilon \rightarrow 0 :
v\varepsilon (x) \sim
\varepsilon \rightarrow 0
K\varepsilon
2(1 - \gamma )
\gamma +1
\biggl(
| x| \gamma +1
\varepsilon 2
\biggr) - 2\gamma
\gamma +1
+1
=
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ON REGULARIZATION BY A SMALL NOISE OF MULTIDIMENSIONAL ODES . . . 1285
= K2\varepsilon
2(1 - \gamma )
\gamma +1
\biggl(
| x| \gamma +1
\varepsilon 2
\biggr) 1 - \gamma
\gamma +1
=
= K2\varepsilon
2(1 - \gamma )
\gamma +1 | x| 1 - \gamma \varepsilon
2(\gamma - 1)
\gamma +1 = K2| x| 1 - \gamma ,
where K is a constant independent of \delta .
This yields that for any fixed \delta \geq 0:
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfE \tau \varepsilon (\delta ) \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\varepsilon \rightarrow 0
\bfE v\varepsilon (\delta ) = K2\delta
1 - \gamma .
Lemma 3.1 is proved.
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Received 27.08.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
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| id | umjimathkievua-article-6292 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:27:00Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/93/721b8735d8eaa7c6f6744222b6903793.pdf |
| spelling | umjimathkievua-article-62922022-03-26T11:02:09Z On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. UDC 519.21 In this paper we solve a selection problem for multidimensional SDE $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$It is assumed that $X^{\epsilon}(0)=x^0\in H,$ the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H,$ and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution.We show that if the drift pushes the solution away from $H,$ then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to $H,$ then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new. Работа посвящена нахождению предела решений многомерных стохастических дифференциальных уравнений $d X^\e(t)=a(X^\e(t))\, d t+\eps \sigma(X^\e(t))\, d W(t)$ при $\ve\to 0,$ где коэффициенты переноса и диффузии являются локально липшицевскими функциями вне фиксированной гиперплоскости $H$. Предполагается, что $X^\e(0)=x^0\in H$, коэффициент переноса $a(x)$ имеет гельдеровскую асимптотику, когда $x$ приближается к $H$, и предельное обыкновенное дифференциальное уравнение $d X(t)=a(X(t))\, d t$ может не иметь единственное решения. Доказано, что если перенос отталкивает решение от гиперплоскости $H$, то предельный процесс с определенными вероятностями выбирает некоторые экстремальные решения предельного дифференциального уравнения. Если перенос является притягивающим к $H$, то предельный процесс удовлетворяет некоторому обыкновенному дифференциальному уравнению с усредненными коэффициентами. Для доказательства последнего результата сформулирован новый достаточно общий принцип усреднения. УДК 519.21 Про регуляризацiю малим шумом багатовимiрних звичайних диференцiальних рiвнянь з нелiпшицевими коефiцiєнтами Статтю присвячено знаходженню границі розв'язків багатовимірних стохастичних диференціальних рівнянь $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t+{\epsilon}\sigma(X^{\epsilon}(t))\, d W(t)$ при ${\epsilon} \to 0,$ де коефіцієнти переносу та дифузії є локально ліпшицевими функціями ззовні фіксованої гіперплощини $H.$Припускається, що $X^{\epsilon}(0)=x^0\in H,$ коефіцієнт переносу $a(x)$ має гельдерову асимптотику, коли $x$ наближається до $H,$ і граничне звичайне диференціальне рівняння $d X(t)=a(X(t))\, d t$ може не мати єдиного розв'язку.Доведено, що якщо перенос відштовхує розв'язок від гіперплощини $H,$ то граничний процес із певними ймовірностями вибирає деякі екстремальні розв'язки граничного диференціального рівняння. Якщо перенос притягує розв'язокдо $H,$ то граничний процес задовольняє звичайне диференціальне рівняння з усередненими коефіцієнтами. Для доведення останнього результату сформульовано новий достатньо загальний принцип усереднення. Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6292 10.37863/umzh.v72i9.6292 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1254-1285 Український математичний журнал; Том 72 № 9 (2020); 1254-1285 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6292/8754 Copyright (c) 2020 Андрій Пилипенко, Олексій Кулик |
| spellingShingle | Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. Pilipenko, A. Kulik, A. On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_alt | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_full | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_fullStr | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_full_unstemmed | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_short | On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients |
| title_sort | on regularization by a small noise of multidimensional odes with non-lipschitz coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6292 |
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