Distribution dependent SDEs for Landau type equations

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution...

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Bibliographic Details
Published inStochastic processes and their applications Vol. 128; no. 2; pp. 595 - 621
Main Author Wang, Feng-Yu
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2018
Subjects
Distribution dependent SDEs
Exponential convergence
Harnack inequality
Homogeneous Landau equation
Wasserstein distance
Harnack inequality
60J75
Exponential convergence
47G20
60G52
Distribution dependent SDEs
Wasserstein distance
Homogeneous Landau equation
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Summary:The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup Pt∗ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup Pt∗ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2017.05.006