Hypercontractivity and Its Applications for Functional SDEs of Neutral Type

TL;DR

This paper investigates hypercontractivity properties of Markov semigroups generated by functional SDEs of neutral type, demonstrating exponential convergence to invariant measures in various metrics.

Contribution

It establishes hypercontractivity for these semigroups and proves exponential convergence in entropy, L^2, and total variation norms.

Findings

Proves hypercontractivity for semigroups of neutral type SDEs

Shows exponential convergence to invariant measure in entropy

Demonstrates convergence in L^2 and total variation norms

Abstract

In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$ converges exponentially to its unique invariant probability measure $\mu$ in entropy, $L^2 (\mu)$ and $\|\cdot\|_{\mbox{var}}$, respectively.