Heat kernel for fractional diffusion operators with perturbations

Feng-Yu Wang; Xi-Cheng Zhang

doi:10.1515/forum-2012-0074

Artikel

Heat kernel for fractional diffusion operators with perturbations

Feng-Yu Wang und Xi-Cheng Zhang

Veröffentlicht/Copyright: 7. März 2013

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Aus der Zeitschrift Forum Mathematicum Band 27 Heft 2

Abstract

Let L be an elliptic differential operator on a complete connected Riemannian manifold M such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let L(α) be the α-stable subordination of L for α∈(1,2). We found some classes 𝕂αγ,β(β,γ∈[0,α)) of time-space functions containing the Kato class, such that for any measurable functions b:[0,∞)×M→TM and c:[0,∞)×M→M with |b|,c∈𝕂α1,1, the operator

Lb,c(α)(t,x):=L(α)(x)+〈b(t,x),∇·〉+c(t,x),(t,x)∈[0,∞)×M,

has a unique heat kernel pb,c(α)(t,x;s,y), 0≤s<t, x,y∈M, which is jointly continuous and satisfies

t-sC{ρ(x,y)∨(t-s)1α}d+α≤pb,c(α)(t,x;s,y)≤C(t-s){ρ(x,y)∨(t-s)1α}d+α

and

|∇xpb,c(α)(t,x;s,y)|≤C(t-s)α-1α{ρ(x,y)∨(t-s)1α}d+α,0≤s<t,x,y∈M,

for some constant C > 1, where ρ is the Riemannian distance. The estimate of ∇ypb,c(α) and the Hölder continuity of ∇xpb,c(α) are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where L=Δ on ℝ^d and b, c are time-independent.