Global Dirichlet Heat Kernel Estimates for Symmetric L\'evy 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) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may or may not have Gaussian component. When L\'evy density is comparable to a decreasing function with damping exponent $\beta$,our estimate is explicit in terms of the distance to the boundary, the L\'evy exponent and the damping exponent $\beta$ of L\'evy density.