Hypercontractivity for functional stochastic differential equations

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 PtPt converges exponentially to its unique invariant probability measure μμ in both L2(μ)L2(μ) and the totally variational norm ‖⋅‖var, and it is compact in L2(μ)L2(μ) for sufficiently large t>0t>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.