Heat kernel for non-local operators with variable order

Let α(x) be a measurable function taking values in [α1,α2] for 0<α1⩽α2<2, and κ(x,z) be a positive measurable function that is symmetric in z and bounded between two positive constants. Under uniform Hölder continuous assumptions on α(x) and x↦κ(x,z), we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order Lf(x)=∫Rd(f(x+z)−f(x)−〈∇f(x),z〉1{|z|⩽1})κ(x,z)|z|d+α(x)dz. In particular, we show that the operator L generates a conservative Feller process on Rd having strong Feller property, which is usually assumed a priori in the literature to study analytic properties of L via probabilistic approaches. Our near-diagonal estimates and lower bound estimates of the heat kernel depend on the local behavior of index function α(x). When α(x)≡α∈(0,2), our results recover some results by Chen and Kumagai (2003) and Chen and Zhang (2016).