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      Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincar\'e Inequalities

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          Abstract

          We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the \(L^2\)-convergence rate by using weak Poincar\'e inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.

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          Author and article information

          Journal
          11 October 2021
          Article
          2110.05536
          35e5bcba-58e5-4948-9346-c544c1108b52

          http://creativecommons.org/licenses/by/4.0/

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          Custom metadata
          37A25, 47D07, 35B40, 37J25, 60H10
          math.PR math.FA

          Functional analysis,Probability
          Functional analysis, Probability

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