Global Dirichlet Heat Kernel Estimates for Symmetric Levy Processes in Half-Space

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Levy processes on half spaces for all . These L,vy processes may or may not have Gaussian component. When Levy density is comparable to a decreasing function with damping exponent, our estimate is explicit in terms of the distance to the boundary, the Levy exponent and the damping exponent of Levy density.