Rate of Growth of Local Times of Strongly Symmetric Markov Processes

Abstract

Let S be a locally compact metric space with a countable base and let X = (Ω, f _t , X _t , P ^x), t ∈ R ⁺,be a strongly symmetric standard Markov process with state space S. Let m be a σ-finite measure on S. What is actually meant by “strongly symmetric” is explained in [MR] but for our purposes it is enough to note that it is equivalent to X being a standard Markov process for which there exists a symmetric transition density function p _t (x, y), (with respect to m). This implies that X has a symmetric 1-potential density

$${u^1}\left( {x,y} \right) = \int_0^\infty {{e^{ - t}}{p_t}\left( {x,y} \right)dt} $$

(1)

We assume that

$${u^1}\left( {x,y} \right) > \infty \forall x,y \in S$$

(2)

which implies that there exists a local time $L = \left\{ {L_t^y,\left( {t,y} \right) \in {R^ + } \times S} \right\}$ for X which we normalize by setting

$${E^x}\left( {\int_0^\infty {{e^{ - t}}dL_t^y} } \right) = {u^1}\left( {x,y} \right)$$

(3)

It is easy to see, as is shown in [MR], that u ¹(x, y) is positive definite on S × S. Therefore, we can define a mean zero Gaussian process G = {G(y), y ∈ S}, with covariance

$$E\left( {G\left( x \right)G\left( y \right)} \right) = {u^1}\left( {x,y} \right)\forall x,y \in S$$

The processes X and G, which we take to be independent, are related through the 1-potential density u ¹(x, y) and are referred to as associated processes.