Global heat kernel estimates for ∆ + ∆ α / 2 in half - space - like domains ∗

Suppose that d ≥ 1 and α ∈ (0, 2). In this paper, we establish by using probabilistic methods sharp two-sided pointwise estimates for the Dirichlet heat kernels of {∆ + a∆; a ∈ (0, 1]} on half-space-like C domains for all time t > 0. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in a ∈ (0, 1] in the sense that the constants in the estimates are independent of a ∈ (0, 1]. Thus they yield the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking a → 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain uniform sharp two-sided estimates for the Green functions of {∆+a∆; a ∈ (0, 1]} in half-space-like C domains in R.