Georgii V. Riabov

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• PhD
• Senior Researcher at National Academy of Sciences of Ukraine

In the paper we suggest a new construction of stochastic flows of kernels in a locally compact separable metric space $M$. Starting from a consistent sequence of Feller transtition function $(\mathsf{P}^{(n)}: n\geq 1)$ on $M$ we prove existence of a stochastic flow of kernels $K=(K_{s,t}: -\infty<s\leq t<\infty)$ in $M,$ such that distributions of...

UDC 519.21 We consider stochastic flows of measurable mappings in a locally compact separable metric space ( M , ρ ) and propose a new construction that produces strong measurable continuous modifications for certain stochastic flows of measurable mappings in metric graphs.

The paper develops a general methodology for analyzing policies with path-dependency (hysteresis) in stochastic models with forward looking optimizing agents. Our main application is a macro-climate model with a path-dependent climate externality. We derive in closed form the dynamics of the optimal Pigouvian tax, that is, its drift and diffusion c...

We study the distribution of a Brownian motion conditioned to start from the boundary of an open set G and to stay in G for a finite period of time. The characterizations of distributions of this kind in terms of certain singular stochastic differential equations are obtained. The accumulated results are applied to the study of boundaries of the cl...

Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing stoc...

UDC 519.21 Distribution of a Brownian motion conditioned to start from the boundary of an open set and to stay in for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing s...

This work is devoted to long-time properties of the Arratia flow with drift – a stochastic flow on R whose one-point motions are weak solutions to a stochastic differential equation dX(t)=a(X(t))dt+dw(t) that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow that gives rise to a...

For a class of coalescing stochastic flows on the real line the existence of dual flows is proved. A stochastic flow and its dual are constructed as a forward and backward perfect cocycles over the same metric dynamical system. The metric dynamical system itself is defined on a new state space for coalescing flows. General results are applied to Ar...

This work is devoted to long-time properties of the Arratia flow with drift -- a stochastic flow on $\mathbb{R}$ whose one-point motions are weak solutions to a stochastic differential equation $dX(t)=a(X(t))dt+dw(t)$ that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow (constr...

Existence of random dynamical systems for a class of coalescing stochastic flows on R is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

Existence of random dynamical systems for a class of coalescing stochastic flows on $\mathbb{R}$ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

A representation for the Kantorovich--Rubinstein distance between probability measures on an abstract Wiener space in terms of the extended stochastic integral (or, divergence) operator is obtained.

In the article we present chaotic decomposition and analog of the Clark formula for the local time of Gaussian integrators. Since the integral with respect to Gaussian integrator is understood in Skorokhod sense, then there exist more than one Clark representation for the local time. We present different representations and discuss the representati...

In the article we present chaotic decomposition and analog of the Clark formula for the local time of Gaussian integrators. Since the integral with respect to Gaussian integrator is understood in Skorokhod sense, then there exist more than one Clark representation for the local time. We present different representations and discuss the representati...

A representation for the Kantorovich-Rubinstein distance between probability measures on a separable Banach space X in the case when this distance is defined by the Cameron-Martin norm of a centered Gaussian measure μ on X is obtained in terms of the extended stochastic integral (or divergence) operator.

We consider a class of measures absolutely continuous with respect to the distribution of the stopped Wiener process $w(\cdot\wedge\tau)$. Multiple stochastic integrals, that lead to the analogue of the It\^o-Wiener expansions for such measures, are described. An analogue of the Krylov-Veretennikov formula for functionals $f=\varphi(w(\tau))$ is ob...

The structure of square integrable functionals measurable with respect to the $n-$point motion of the Arratia flow is studied. Relying on the change of measure technique, a new construction of multiple stochastic integrals along trajectories of the flow is presented. The analogue of the It\^o-Wiener expansion for square integrable functionals from...

In this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of $L^2$ space generated by the process $\eta(\cdot)=w(\min(\tau,\cdot)),$ where $w$ is a Brownian motion and $\tau$ is the first moment when $w$ hits the given continuous function $g$ is considered. We present a ne...

The finite absolute continuity of probability measures on an abstract Wiener space (X,H,μ) with respect to a Gaussian measure μ is studied. The limit theorem for the tails of such measures is proved.

In this paper we study finite absolute continuity with respect to Wiener measure. A general condition of finite absolute continuity in terms of disintegration of measures is given. Finite absolute continuity of Itô processes and conditional distributions of the Wiener process is studied.

We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite absolute continuity of Gaussian measures is equivalent to the condition of the...