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