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

For α∈(0,2)α∈(0,2) and M>0M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]}{Δ+aαΔα/2,a∈(0,M]} on RdRd 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 aa and recover the sharp estimates for Brownian motion with drift as a→0a→0. Each fundamental solution determines a conservative Feller process XX. We characterize XX as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.