Global heat kernel estimates for \(\Delta+\Delta^{\alpha/2}\) in half-space-like domains. (English) Zbl 1247.60115

Summary: Suppose that \(d\geq 1\) and \(\alpha\in (0, 2)\). In this paper, we establish by using probabilistic methods sharp two-sided pointwise estimates for the Dirichlet heat kernels of \(\{\Delta+ a^\alpha \Delta^{\alpha/2}; \;a\in (0, 1]\}\) on half-space-like \(C^{1, 1}\) 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 \(a \in (0, 1]\) in the sense that the constants in the estimates are independent of \(a\in (0, 1]\). Thus they yield the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking \(a\to 0\). Integrating the heat kernel estimates with respect to the time variable \(t\), we obtain uniform sharp two-sided estimates for the Green functions of \(\{\Delta+ a^\alpha \Delta^{\alpha/2}; \;a\in (0, 1]\}\) in half-space-like \(C^{1, 1}\) domains in \(\mathbb{R}^d\).


60J35 Transition functions, generators and resolvents
47G20 Integro-differential operators
60J75 Jump processes (MSC2010)
47D07 Markov semigroups and applications to diffusion processes
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