Renormalized Self-Intersection Local Times and Wick Power Chaos Processes

Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric Levy processes in $Rm$, $m=1,2$. In $R2$ these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In $R1$ these include stable processes of index $3/4<betale 1$ and many processes in their domains of attraction.Let $(Omega, mathcal F(t),X(t), Px)$ be one of these radially symmetric Levy processes with 1-potential density $u1(x,y)$. Let $mathcal G2n$ denote the class of positive finite measures $mu$ on $Rm$ for which $int!!int (u1(x,y))2n,dmu(x),dmu(y)<infty$. For $muinmathcal G2n$, let $alphan,epsilon(mu,lambda) oversettextdefto=int!!int0leq t1leq cdots leq tnleq lambda fepsilon(X(t1)-x)prodj=2n fepsilon(X(tj)- X(tj-1)),dt1cdots,dtn,dmu(x)$ where $fepsilon$ is an approximate $delta-$function at zero and $lambda$ is an random exponential time, with mean one, independent of $X$, with probability measure $Plambda$.The renormalized self-intersection local time of $X$ with respect to the measure $mu$ is defined as $gamman(mu)=limepsilonto 0,sumk=0n-1(-1)k n-1 choose k(u1epsilon(0))k alphan-k,epsilon(mu,lambda)$ where $u1epsilon(x)oversettextdefto= int fepsilon(x-y)u1(y),dy$, with $u1(x)oversettextdef to= u1(x+z,z)$ for all $zin Rm$. Conditions are obtained under which this limit exists in $L2(Omegatimes R+,Pylambda)$ for all $yin Rm$, where $Pylambdaoversettextdefto= Pytimes Plambda$. Let $mux,xin Rm$ denote the set of translates of the measure $mu$.The main result in this paper is a sufficient condition for the continuity of $gamman(mux),,xin Rm$ namely that this process is continuous $Pylambda$ almost surely for all $yin Rm$, if the corresponding 2$n$-th Wick power chaos process, $:G2nmux:,,xin Rm$ is continuous almost surely. This chaos process is obtained in the following way