Fundamental solutions of nonlocal Hörmander’s operators

Consider the following nonlocal integro-differential operator: for α ∈ (0, 2), (formula presented) where σ:ℝ^d → ℝ^d ⊗ ℝ^d and b:ℝ^d → ℝ^d are smooth and have bounded firstorder derivatives, and p.v. stands for the Cauchy principal value. Let B₁(x):= σ(x) and B_j+1(x):= b(x). ∇Bj (x)-∇b(x). Bj (x) for j ∈ ℕ. Under the following Hörmander’s type condition: for any x ∈ ℝ^d and some n = n(x) ∈ ℕ, (formula presented) by using the Malliavin calculus, we prove the existence of the heat kernel ρ_t (x, y) to the operator L^(α) _σ,b as well as the continuity of x (formula presented) ρ_t (x,.) in L¹(ℝ^d) as a density function for each t > 0.Moreover, when σ(x) = σ is constant and B_j ∈ C_b^∞ b for each j ∈ ℕ, under the following uniform Hörmander’s type condition: for some j₀ ∈ ℕ, (formula presented), we also show the smoothness (t, x, y) (formula presented) ρ_t(x, y) with ρ_t (formula presented) (ℝ^d × ℝ^d).