Exponential decay of positivity preserving semigroups on L^p

Abstract

In recent studies on invariant measures of diffusions on infinite-dimensional spaces, several approaches have been considered. In [6], existence, uniqueness and regularity of invariant measures and asymptotic properties of solutions of stochastic evolution equations were investigated. In purely analytic approaches, a study of invariant measures for Markovian semigroups on L associated with quadratic forms was made in [5] as a generalization of [19, 20, 21, 10], while the regularity of the measures μ solving some elliptic equations L∗μ = 0 was proved in [4, 3], where L is an operator of type Lu = tr(Au′′) +B · ∇u. In this paper, for given Markovian (or more generally, positivity preserving) semigroups {Pt}, we discuss conditions for the existence of invariant measures, and for the exponential decay of {Pt} to a projection operator. We treat this problem in an analytic way, so we always impose the condition that {Pt} is also a strongly continuous semigroup on L for some p ∈ (1,∞). As for the existence of invariant measures, the situation becomes quite simple if {Pt} is eventually compact. But we do not assume this since such compactness seems hard to be expected in the case that the underlying space is infinite dimensional. In Gross’ paper [9] concerning physical ground states for Hermitian operators, similar situations were dealt with and the so-called hyperboundedness of semigroups was used as a replacement of compactness. We apply his idea to our problem; under the condition (I) regarding integrability (see Definition 2.1) for semigroups or resolvents, we prove the existence of invariant measures by approximating the underlying space by a sequence of finite number of sets. The result improves the corresponding ones in [5, 10]. In order to discuss the exponential decay of {Pt} in the L sense, we introduce a kind of ergodicity condition, (E) (see Definition 3.1). We may say that it is a substitution for strict positivity of transition densities; we do not expect the existence of such densities in our concerning infinite-dimensional cases. This type of condition appeared in Kusuoka’s