Fundamental solutions of nonlocal Hörmander’s operators II

Abstract

Consider the following nonlocal integro-differential operator: for α∈(0,2)α∈(0,2): L(α)σ,bf(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x), Lσ,b(α)f(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x), where σ:Rd→Rd⊗Rdσ:Rd→Rd⊗Rd and b:Rd→Rdb:Rd→Rd are smooth functions and have bounded partial derivatives of all orders greater than 11, δδ is a small positive number, p.v. stands for the Cauchy principal value and LL is a bounded linear operator in Sobolev spaces. Let B1(x):=σ(x)B1(x):=σ(x) and Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x)Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x) for j∈Nj∈N. Suppose Bj∈C∞b(Rd;Rd⊗Rd)Bj∈Cb∞(Rd;Rd⊗Rd) for each j∈Nj∈N. Under the following uniform Hormander’s type condition: for some j0∈Nj0∈N, infx∈Rdinf|u|=1∑j=1j0|uBj(x)|2>0, 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(α)σ,bLσ,b(α). In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].