Heat kernels and analyticity of non-symmetric jump diffusion semigroups

الأصل · EN

Let d≥ 1 and α∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ᵈ: καf(x):=p.v.∫ᵈ(f(x+z)-f(x))κ(x,z)|z|ᵈ⁺α z, where 0<κ₀≤ κ(x,z)≤ κ₁, κ(x,z)=κ(x,-z), and |κ(x,z)-κ(y,z)|≤κ₂|x-y|β for some β∈(0,1). Using Levi's method, we construct the fundamental solution (also called heat kernel) pκα(t, x, y) of κα, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates of the heat kernel. We also show that pκα(t, x, y) is jointly Hölder continuous in (t, x). The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of κα gives rise a Feller process {X, ₓ, x∈ ᵈ} on ᵈ. We determine the Lévy system of X and show that ₓ solves the martingale problem for (κα, C²b(ᵈ)). Furthermore, we obtain the analyticity of the non-symmetric semigroup associated with κα in Lᵖ-spaces for every p∈[1,∞). A maximum principle for solutions of the parabolic equation ∂ₜ u =καu is also established.