Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S-b=(Delta) over bar (alpha/2) + b . where Delta over bar alpha/2 b.del where (Delta) over bar (alpha/2) is the truncated fractional Laplacian, alpha is an element of (1, 2) and b is an element of K-d(alpha-1). In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance divide vertical bar x - y vertical bar divide in short time.