Constructing Strong Markov Processes

TL;DR

This paper presents a method to construct strong Markov processes from finite-dimensional Markov transition functions by embedding the state space into Euclidean space using resolvent-induced structures.

Contribution

It introduces a novel approach to construct strong Markov processes on intrinsic state spaces derived from finite-dimensional Markov transition functions.

Findings

Constructs strong Markov processes on intrinsic state spaces.

Provides examples including non-strong Markov processes.

Shows the embedding of state spaces into Euclidean space.

Abstract

The construction presented in this paper can be briefly described as follows: starting from any "finite-dimensional" Markov transition function p_t, on a measurable state space (E,B), we construct a strong Markov process on a certain "intrinsic" state space that is, in fact, a closed subset of a finite dimensional Euclidean space R^d. Of course we must explain the meaning of finite-dimensionality and intrinsity. Starting with p_t, we consider the range of the nonnegative bounded measurable functions under the action of the resolvent. This class of functions induces a uniform structure on E. We say that E is finite-dimensional if this uniformity is finitely generated. In such cases we then map E into R^d. The intrinsic state space is the closure of the range of this mapping. On this enlarged state space we construct a strong Markov process, which corresponds quite naturally to p_t. We give several examples including the usual examples of nonstrong Markov process.