Long-time behaviour for distribution dependent SDEs with local Lipschitz coefficients

TL;DR

This paper establishes global existence, uniqueness, and long-term stability results for distribution dependent stochastic differential equations with local Lipschitz coefficients, using measure-dependent techniques and Lyapunov functions.

Contribution

It introduces a novel approach using local Wasserstein distance and integral conditions, advancing the understanding of distribution dependent SDEs with local Lipschitz coefficients.

Findings

Proved global existence and uniqueness of solutions.

Established $r$-th moment exponential stability.

Studied invariant measures under strong monotonicity.

Abstract

By using a classical truncated argument and introducing the local Wasserstein distance, the global existence and uniqueness are proved for the distribution dependent SDEs with local Lipschitz coefficients. Due to the measure dependence, the conditions in the sense of pointwise for classical cases can be simplified to the conditions in the sense of integral. On the basis of the well-posedness, we prove the $r$-th moment exponential stability using the measure dependent Lyapunov functions and the existence and uniqueness of invariant probability measure is studied under the integrated strong monotonicity condition. Finally, some examples are given to illustrate the results in this paper.