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Register a new userExponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs
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The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(mu_t,mu_infty)^2 +{rm Ent}(mu_t|mu_infty)le c {rm e}^{-lambda t} minbig{W_2(mu_0, mu_infty)^2,{rm Ent}(mu_0|mu_infty)big}, tge 1,$$ where $c,lambda>0$ are constants, $mu_t$ is the distribution of the solution at time $t$, $mu_infty$ is the unique invariant probability measure, ${rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.
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By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present a general result deducing the uniform ergodicity of McKean-Vlasov SDEs from that of classical SDEs. As an application, the $L^1$-exponential convergence is derived for a class of non-symmetric singular granular media equations.
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Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It^{o} representations of their limiting forms as the approximation parameter $delta rightarrow 0$. All filters require the solution of a Poisson equation defined on $mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
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