Construction of $${L^p}$$ -Strong Feller Processes

Abstract

In this chapter we provide a general construction scheme for \({L^p}\)-strong Feller processes on locally compact separable metric spaces. The construction result yields that starting from certain regularity conditions on the semigroup associated with a symmetric Dirichlet form, one obtains a diffusion process which solves the corresponding martingale problem for every starting point from an explicitly known set. In Theorem 2.3.10 we mention further useful properties of the process, formulated also as pointwise statements. In Section 2.4 we provide concrete examples. Our results and their proofs are based on [AKR03] and [Doh05]. We got also many ideas from [FG07], [FG08] and [Sti10]. For the construction of classical Feller processes from strongly continuous contraction semigroups on spaces of continuous functions vanishing at infinity, see e.g. [BG68, Ch. I, Theo 9.4]. There are also results on the construction of Hunt processes from resolvents of kernels, see [Sto83] and Remark 2.3.1 below.