Hypercontractivity for Functional Stochastic Differential Equations

Abstract

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated to a class of functional stochastic differential equations. Consequently, the semigroup $P_t$ converges exponentially to its unique invariant probability measure $\mu$ in entropy, $L^2(\mu )$ and the totally variational norm, and it is compact in $L^2(\mu )$ for 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.

20 pages

References in corpus (1)

• Harnack inequality and applications for stochastic generalized porous media equations

Cited by in corpus (1)

• Hypercontractivity and Its Applications for Functional SDEs of Neutral Type