Markov Processes with Equal Capacities

Abstract

Let X and $\hat X$ be transient standard Markov processes in weak duality with respect to a σ-finite measure m. Let (Y, Ŷ, μ) be a second “dual pair” with the same state space E as (X, $\hat X$, m). Let Cap^X and Cap^Y be the 0-order capacities associated with (X, $\hat X$, m) and (Y, Ŷ, μ), and let V and $\hat V$ denote the potential kernels for Y and Ŷ. Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap^X(B)=Cap^Y(B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t≥0 of (X, $\hat X$, m) such that

$$U_D (x,dy) + \hat U_D (x,dy) = V(x,dy) + \hat V(x,dy)$$

with the exception of a capacity-zero set of x's. Here U _D (resp. Û _D) is the potential kernel of the time-changed process $X_{D^{ - 1} (t)}$ (resp. $\hat X_{D^{ - 1} (t)} )$, t≥0. In particular, if both X and Y are symmetric processes, then the equality of the capacities Cap^X and Cap^Y implies that X and Y are time changes of one another. This derivation rests on a generalization of a formula of Choquet concerning the “differentiation” of capacities. In the symmetric case, our main result extends a theorem of Glover et al.⁽²³⁾