Inverse Subordination of Excessive Functions

Abstract

Let X be a right process, and let M be a multiplicative functional (MF) of X. We use the standard terminology and notation for right processes. See [BG], [S] or [DM87]. In particular let S := inf{t : M _t = 0} and E _M := {x : P^x(S > 0) = 1} = {x : P ^x(M ₀ = 1) = 1}. The operator P ¹ _M is defined for q ≥ 0 by

$$P_M^qf(x): = \left\{ {\begin{array}{*{20}{c}} {{P^x}\smallint _0^\infty {e^{ - qt}}f({X_t})( - d{M_t}),x \in {E_M}} \\ {f(x),x{ \notin _M}.} \end{array}} \right.$$

(1.1)

Here and elsewhere in this paper, f, g, u, v; with or without affixes denote positive universally measurable functions (i.e., elements of pɛ*) unless stated otherwise.