Fundamental solutions of nonlocal Hörmander.s operators II.

Consider the following nonlocal integro-differential operator: for ..(0,2): L(.).,bf(x):=p.v..|z|<.f(x+.(x)z).f(x)|z|d+.dz+b(x)..f(x)+Lf(x), where .:Rd.Rd.Rd and b:Rd.Rd are smooth functions and have bounded partial derivatives of all orders greater than 1, . is a small positive number, p.v. stands for the Cauchy principal value and L is a bounded linear operator in Sobolev spaces. Let B1(x):=.(x) and Bj+1(x):=b(x)..Bj(x)..b(x).Bj(x) for j.N. Suppose Bj.C.b(Rd;Rd.Rd) for each j.N. Under the following uniform Hörmander.s type condition: for some j0.N, infx.Rdinf|u|=1.j=1j0|uBj(x)|2>0, by using Bismut.s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator L(.).,b. In particular, we answer a question proposed by Nualart [Sankhy. A 73 (2011) 46.49] and Varadhan [Sankhy. A 73 (2011) 50.51].