In this talk, 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 $t>0$. These Levy 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.
This is a joint work with Zhen-Qing Chen.