Abstract
We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker–Planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures, endowed with the Wasserstein distance.
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Ambrosio, L., Savaré, G. & Zambotti, L. Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145, 517–564 (2009). https://doi.org/10.1007/s00440-008-0177-3
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DOI: https://doi.org/10.1007/s00440-008-0177-3
Keywords
- Reversible Markov processes
- Log-concave probability measures
- Gradient flows
- Optimal transportation
- Relative entropy