Global Properties

Abstract

The sample path properties in Chap. 5 describe the behaviour of the stochastic process locally. We will now turn our attention to global properties of the process and its semigroup; following Fukushima–Oshima–Takeda [118] we mean by global behaviour of a process questions like transience, recurrence, irreducibility, functional inequalities and so on. When studying such properties, one often needs not only Feller but also L ²-semigroups. We refer, in particular, to Sect. 1.5 for situations when a Feller semigroup is also an L ^p-semigroup. Unless otherwise stated, we work again with $E = {\mathbb{R}}^{d}$.