Heat Kernel for Fractional Diffusion Operators with Perturbations

Details Heat Kernel for Fractional Diffusion Operators with Perturbations Heat Kernel for Fractional Diffusion Operators with Perturbations

2012-04-23

arxiv.org/abs/1204.4956v1

Physics

Mathematical Physics

19 pages

Scientific paper

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 $L^{(\aa)}$ be the $\aa$-stable subordination of $L$ for $\aa\in (1,2).$ We found some classes $\mathbb K_\aa^{\gg,\bb} (\bb,\gg\in [0,\aa))$ of time-space functions containing the Kato class, such that for any measurable $b: [0,\infty)\times M\to TM$ and $c: [0,\infty)\times M\to M$ with $|b|, c\in \mathbb K_\aa^{1,1},$ the operator $$L_{b,c}^{(\aa)}(t,x):= L^{(\aa)}(x)+ +c(t,x),\ \ (t,x)\in [0,\infty)\times M$$ has a unique heat kernel $p_{b,c}^{(\aa)}(t,x;s,y), 0\le s1$, where $\rr$ is the Riemannian distance. The estimate of $\nabla_yp^{(\aa)}_{b,c}$ and the H\"older continuity of $\nn_x p_{b,c}^{(\aa)}$ are also considered. The resulting estimates of the gradient and its H\"older continuity are new even in the standard case where $L=\DD$ on $\R^d$ and $b,c$ are time-independent.

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