# Pilipenko A. Yu.

### Limit theorem for coiuntable systems of stochastic differential equations

Pilipenko A. Yu., Tantsiura M. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1380-1402

We consider infinite systems of stochastic differential equations used to describe the motion of interacting particles in random media. It is assumed that mass of each particle tends to zero and the density of particles infinitely increases in a proper way. It is proved that the sequence of the corresponding measure-valued processes converges in distribution. We also prove existence and uniqueness of a strong solution for the limit equation.

### On the Limit Behavior of a Sequence of Markov Processes Perturbed in a Neighborhood of the Singular Point

Pilipenko A. Yu., Prikhod’ko Yu. E.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 499-516

We study the limit behavior of a sequence of Markov processes whose distributions outside any neighborhood of a “singular” point are attracted to a certain probability law. In any neighborhood of this point, the limit behavior can be irregular. As an example of application of the general result, we consider a symmetric random walk with unit jumps perturbed in the neighborhood of the origin. The invariance principle is established for the standard time and space scaling. The limit process is a skew Brownian motion.

### On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1241-1256

For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.

### Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1663–1673

Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from $x$. We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singular component of the stochastic measure-valued process $µ_t = µ ○ ϕ_t^{−1}$, which is an image of some absolutely continuous measure $\mu$ for random mapping $\varphi_t(\cdot)$. We prove that the contraction of the Hausdorff measure $H^{d-1}$ onto a support of the singular component is $\sigma$-finite. We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to $H^{d-1}$.

### Measure-Valued Diffusions and Continual Systems of Interacting Particles in a Random Medium

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1289–1301

We consider continual systems of stochastic equations describing the motion of a family of interacting particles whose mass can vary in time in a random medium. It is assumed that the motion of every particle depends not only on its location at given time but also on the distribution of the total mass of particles. We prove a theorem on unique existence, continuous dependence on the distribution of the initial mass, and the Markov property. Moreover, under certain technical conditions, one can obtain the measure-valued diffusions introduced by Skorokhod as the distributions of the mass of such systems of particles.

### Properties of the Flows Generated by Stochastic Equations with Reflection

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1069 – 1078

We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in $\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set $\varphi_t(\mathbb{R}_+^d)$. We prove that the Hausdorff dimension of the boundary $\partial \varphi_t(\mathbb{R}_+^d)$ is not greater than $d - 1$.

### Stroock–Varadhan Theorem for Flows Generated by Stochastic Differential Equations with Interaction

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 227-236

We prove a theorem that characterizes the support of a flow generated by a system of stochastic differential equations with interaction.

### Nonlinear Transformations of Smooth Measures on Infinite-Dimensional Spaces

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1226-1250

We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.

### On the existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 863–871

We study the problem of existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process. For the conditional mathematical expectation of a solution, we obtain a partial differential equation.

### On the properties of an operator of stochastic differentiation constructed on a group

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 563-568

We construct a differential operator by an admissible group in the space *L* _{2} (ℝ^{m},*P*)and study its properties.