Comparisons for measure valued processes with interactions

Abstract

This paper considers some measure-valued processes $\{X_t\dvtx t \in [0,T]\}$ based on an underlying critical branching particle structure with random branching rates. In the case of constant branching these processes are Dawson--Watanabe processes. Sufficient conditions on functionals $\Phi$ of the process are given that imply that the expectations $E(\Phi(X_T))$ are comparable to the constant branching case. Applications to hitting estimates and regularity of solutions are discussed. The result is established via the martingale optimality principle of stochastic control theory. Key steps, which are of independent interest, are the proof of a version of Itô's lemma for $\Phi(X_t)$, suitable for a large class of functions of measures (Theorem 3) and the proof of various smoothing properties of the Dawson--Watanabe transition semigroup (Section 3).