A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

In this paper, we consider the following type of non-local (pseudo-differential) operators L on Rd: Lu(x) = 1 2 d ∑ i,j=1 ∂ ∂xi ( aij(x) ∂ ∂xj ) + lim ε↓0 ∫ {y∈Rd: |y−x|>ε} (u(y) − u(x))J(x, y)dy, where A(x) = (aij(x))1≤i,j≤d is a measurable d × d matrix-valued function on Rd that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial nonnegative kernel on Rd × Rd satisfying certain conditions. Corresponding to L is a symmetric strong Markov process X on Rd that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of L and parabolic Harnack principle for positive parabolic functions of L. Moreover, two-sided sharp heat kernel estimates are derived for such operator L and jump-diffusion X. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on Rd. To establish these results, we employ methods from both probability theory and analysis. AMS 2000 Mathematics Subject Classification: Primary 60J35, 47G30, 60J45; Secondary 31C05, 31C25, 60J75.