Constructing stochastic flows of kernels

TL;DR

This paper introduces a new method for constructing stochastic flows of kernels in locally compact metric spaces, ensuring their consistency with given finite-dimensional distributions and enabling a measurable, invariant presentation.

Contribution

It provides a novel construction of stochastic flows of kernels from consistent Feller transition functions, with a measurable idempotent presentation and invariance properties.

Findings

Existence of stochastic flows of kernels consistent with given finite-dimensional distributions.

Construction of a measurable, invariant presentation of the flow.

Flow invariance under a specific idempotent measurable presentation.

Abstract

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 $n$-point motions of $K$ are determined by $\mathsf{P}^{(n)}.$ Presented construction allows to find a single idempotent measurable presentation $\mathfrak{p}$ of distributions of all kernels $K_{s,t}$ from the flow, and to construct a flow that is invariant under $\mathfrak{p}$ and is jointly measurable in all arguments.