Heat kernel bounds for a large class of Markov process with singular jump

Abstract

Let Z=(Z1,…,Zd) be the d-dimensional Lévy processes where Zi’s are independent 1-dimensional Lévy processes with jump kernel Jϕ,1(u,w)=|u−w|−1ϕ(|u−w|)−1 for u,w∈R. Here ϕ is an increasing function with weak scaling condition of order α̲,α¯∈(0,2). Let J(x,y)≍Jϕ(x,y) be the symmetric measurable function where Jϕ(x,y)≔Jϕ,1(xi,yi)if xi≠yi for some i and xj=yj for all j≠i0if xi≠yi for more than one index iCorresponding to the jump kernel J, we show the existence of non-isotropic Markov processes X≔(X1,…,Xd) and obtain sharp two-sided heat kernel estimates for the transition density functions.