Fundamental solutions of nonlocal Hörmander's operators II

Consider the following nonlocal integro-differential operator: for α ∈ (0, 2): where σ: R^d →R^d ⊗ R^d and b: R^d → R^d are smooth functions and have bounded partial derivatives of all orders greater than 1, δ 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) := σ(x) and B_j+1(x) := b(x) · ∇B_j (x)-∇b(x) · B_j (x) for j ∈ N. Suppose B_j ∈ C ^∞ _b (R^d R^d ⊗ R^d ) for each j ∈ N. Under the following uniform Hörmander's type condition: for some j₀ ∈ N, by using Bismut's approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator L^(α)_σ,b. In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46-49] and Varadhan [Sankhyā A 73 (2011) 50-51].