L^p estimates for Feynman-Kac propagators with time-dependent reference measures

TL;DR

This paper develops L^p estimates for Feynman-Kac propagators with time-dependent measures, extending symmetric Markov semigroup theory to non-Markovian, time-inhomogeneous operators, with applications to sequential MCMC methods.

Contribution

It introduces a new class of time-inhomogeneous Feynman-Kac operators and derives L^p estimates using weighted inequalities, generalizing existing semigroup results.

Findings

Derived L^p-L^p and L^p-L^q estimates for the operators

Estimates depend on the value of p due to non-Markovian nature

Applicable to sequential Markov Chain Monte Carlo methods

Abstract

We introduce a class of time-inhomogeneous transition operators of Feynman-Kac type that can be considered as a generalization of symmetric Markov semigroups to the case of a time-dependent reference measure. Applying weighted Poincar\'e and logarithmic Sobolev inequalities, we derive L^p-L^p and L^p-L^q estimates for the transition operators. Since the operators are not Markovian, the estimates depend crucially on the value of p. Our studies are motivated by applications to sequential Markov Chain Monte Carlo methods.