Hypercontractivity for functional stochastic differential equations

Jianhai Bao, Feng-Yu Wang and Chenggui Yuan

Stochastic Processes and their Applications, 2015, vol. 125, issue 9, 3636-3656

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.

Keywords: Hypercontractivity; Compactness; Exponential ergodicity; Functional stochastic differential equation; Harnack inequality (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2015.04.001

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