arXiv subjects
Publications and source records attributed to Georgii V. Riabov.
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 stochastic flows on $\mathbb{R}.$
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 Arratia flows with drift.
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 (constructed in \cite{Riabov}) that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed.
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 $η(\cdot)=w(\min(τ,\cdot)),$ where $w$ is a Brownian motion and $τ$ is the first moment when $w$ hits the given continuous function $g$ is considered. We present a new construction of multiple stochastic integrals with respect to the process $η.$ Our approach is based on the change of measure technique. The analogue of the Itô-Wiener expansion for the space $L^2(η)$ is constructed.