Heat kernel estimates for Δ + Δα/2 under gradient perturbation

For alpha is an element of (0, 2) and M > 0, we consider a family of nonlocal operators {Delta + a(alpha) Delta(alpha/2), a is an element of (0, M]} on le under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a -> 0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Levy process with singular drift. (C) 2015 Elsevier B.V. All rights reserved.