Exponential Ergodicity for Singular McKean–Vlasov Stochastic Differential Equations in Weighted Variation Metric

Shanshan Hu () and Yue Wang ()
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Shanshan Hu: Tianjin University
Yue Wang: Shenzhen Polytechnic University

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-34

Abstract: Abstract In this paper, we prove exponential ergodicity for McKean–Vlasov stochastic differential equations (SDEs) with singular drift under a weighted variation metric. The McKean–Vlasov SDE is perturbed by a singular potential and does not completely satisfy the typical dissipative condition in the x-variable. Our conclusion extends some ergodicity results in total variation norm with or without dependence on distribution and indicates ergodicity under Wasserstein distance if the weight function is chosen in a particular way. Furthermore, we apply the main result to nonlinear Fokker–Planck equations, in particular, to non-symmetric singular granular media equations, and observe the long-time behavior of the SDEs.

Keywords: Ergodicity; McKean–Vlasov SDEs; Invariant probability measures; Weighted variation metric; 60H10; 60G65 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01398-2

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