Polylinear Additive Functionals of Superprocesses

Abstract

LetXbe a superdiffusion in a domainEof Rd. A polylinear additive functional ofXcorresponding to a positive Borel functionρis given by the formulaA(B)=∫Bdt1, …, dtk∫Ekρ(t1, z1; …; tk, zk)Xt1(dz1)…Xtk(dzk).By a passage to the limit, we extend this definition to a certain class of generalized functionsρ. More precisely, we associate an additive functionalAνof (Xt, Pμ) with every measureνsubject to the condition∫ν(dt1, dz1; …; dtk, dzk)ν(dt′1, dz′1; …; dt′k, dz′k) emsp;×qμ(t1, z1; …; tk, dzk; t′1, z′1; …; t′k, dz′k)<∞for a certain kernelqμ. We prove that, ifν{t:ti=s}=0 fori=1, …, kand for alls, then measureAνhas, a.s., the same property. This result is applicable, in particular, to self-intersection local times ofX.