Ergodic Properties of Non-Linear Density-Dependent Perturbations of the Ornstein-Uhlenbeck Process

Denis Belomestny; Ekaterina Morozova

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arXiv · 2606.18877

Ergodic Properties of Non-Linear Density-Dependent Perturbations of the Ornstein-Uhlenbeck Process

Denis Belomestny·Ekaterina Morozova

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Abstract

The present paper considers McKean-Vlasov SDEs with density-dependent spatially unbounded drift, which may be viewed as a non-linear density-dependent perturbation of the Ornstein-Uhlenbeck process. We develop a comprehensive theoretical framework for this class of equations. First, we establish strong well-posedness and derive optimal Gaussian pointwise bounds for both the solution density and its gradient. Then we derive an explicit expression for the stationary density and show that it satisfies logarithmic Sobolev and Poincar\'e inequalities. Finally, we prove exponential convergence to equilibrium in the $\chi^2$-metric.

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Denis Belomestny, Ekaterina Morozova. 2026-06-17. Ergodic Properties of Non-Linear Density-Dependent Perturbations of the Ornstein-Uhlenbeck Process. https://arxiv.org/abs/2606.18877

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