Expansion and attraction of RDS: long time behavior of the solution to singular SDE

TL;DR

This paper develops a framework to analyze the long-term behavior of solutions to singular stochastic differential equations, focusing on the expansion and attraction properties of the flow in Euclidean space.

Contribution

It introduces a new method for studying the long-time dynamics of SDEs with singular coefficients, including conditions for the existence of pullback attractors.

Findings

Established conditions for the existence of pullback attractors in singular SDEs.

Provided a framework for analyzing the expansion rate of flows under singular dynamics.

Applied the theory to specific classes of SDEs with singular drift terms.

Abstract

We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of the SDE can be split into two terms, one of which is singular and the radial component of the other term has a radial component of sufficient strength in the direction of the origin, then the random dynamical system generated by the SDE admits a pullback attractor.