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Pré-Publication, Document De Travail Année : 2022

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

Résumé

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a onedimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.
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Dates et versions

hal-03536620 , version 1 (20-01-2022)

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  • HAL Id : hal-03536620 , version 1

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Arnaud Guillin, Alain Durmus, Andreas Eberle, Katharina Schuh. Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement. 2022. ⟨hal-03536620⟩
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