Superprocesses and their linear additive functionals

Abstract

Let X = ( X t , P ) X = ({X_t},P) be a measure-valued stochastic process. Linear functionals of X X are the elements of the minimal closed subspace L L of L 2 ( P ) {L^2}(P) which contains all X t ( B ) {X_t}(B) with ∫ X t ( B ) 2 d P > ∞ \smallint {{X_t}{{(B)}^2}\;dP\; > \infty } . Various classes of L L -valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case—"weighted occupation times"—such equations have been established earlier by Iscoe. We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.