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Hypercontractivity for functional stochastic differential equations

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  • Bao, Jianhai
  • Wang, Feng-Yu
  • Yuan, Chenggui

Abstract

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated with a class of functional stochastic differential equations. Consequently, the semigroup Pt converges exponentially to its unique invariant probability measure μ in both L2(μ) and the totally variational norm ‖⋅‖var, and it is compact in L2(μ) for sufficiently large t>0. This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated. As the associated Dirichlet form does not satisfy the log-Sobolev inequality, the standard argument using functional inequalities does not work.

Suggested Citation

  • Bao, Jianhai & Wang, Feng-Yu & Yuan, Chenggui, 2015. "Hypercontractivity for functional stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3636-3656.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:9:p:3636-3656
    DOI: 10.1016/j.spa.2015.04.001
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    References listed on IDEAS

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    1. Reiß, M. & Riedle, M. & van Gaans, O., 2006. "Delay differential equations driven by Lévy processes: Stationarity and Feller properties," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1409-1432, October.
    2. Wang, Feng-Yu & Yuan, Chenggui, 2011. "Harnack inequalities for functional SDEs with multiplicative noise and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2692-2710, November.
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    Cited by:

    1. Wujun Lv & Xing Huang, 2021. "Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 34(2), pages 827-851, June.

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