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Potential theory in conical domains. (English) Zbl 0918.31008
The main subject of this lengthy paper are estimates on the heat kernel of the Laplacian with Dirichlet boundary conditions in a conical domain in \({\mathbb R}^d\), as well as estimates for the corresponding object in case of a discrete potential theory in a conical domain.
More precisely, let \(\Omega \subset {\mathbb R}^d\), \(\Omega \neq {\mathbb R}^d\), be a conical domain with vertex \(0\) having a Lipschitz boundary. Let \(u\) be the réduite of \(\Omega\), that is, \(u\) is the unique (up to a multiplicative constant) positive harmonic function in \(\Omega\) vanishing on the boundary \(\partial \Omega\), let \((B_t, t\geq 0)\) be a standard Brownian motion in \({\mathbb R}^d\), and let \(\tau=\inf \{t>0: B_t\notin \Omega\}\) be the exit time from \(\Omega\). The heat kernel and the “probability of life” are defined by \(p_t(x,y)={\mathbb P}_x[t<\tau, B_t\in dy]\), and \(P(t,x)= \int p_t(x,y) dy ={\mathbb P}_x[t<\tau]\), \(t>0\), \(x,y,\in \Omega\). By using the boundary parabolic Harnack principle, the author first proves the following estimates: \[ u(x)\leq u(x+y),\quad x,y \in \Omega, \text{ and } P(1,x)\leq C u(x),\quad x\in \Omega. \] It follows by scaling that \(P(t,x) \leq C u(x)/ t^{\alpha /2}\), and then by standard methods the following estimate for the heat kernel: for all \(0<\epsilon <1\) there exists \(C=C_{\epsilon}\) such that \[ p_t(x,y)\leq C\frac{u(x)u(y)}{t^{\alpha +d/2}} \exp \left(- \frac{| x-y| ^2}{(4+\epsilon)t}\right) . \] Here \(\alpha=\text{deg}(\Omega) >0\) is the homogeneity degree of \(\Omega\).
Most of the paper is devoted to proving similar estimates in case of a random walk instead of the Brownian motion. Let \(\mu \) be a centered probability measure on \({\mathbb R}^d\) with isotropic covariance and bounded support, let \((S_n, n\geq 0)\) be a random walk in \({\mathbb R}^d\) based on \(\mu\), and let \(\tau=\inf\{j: S_j\notin \Omega\}\) be the exit time from \(\Omega\). The \(n\)-step transition probability for the killed random walk is defined by \(P(n;x,y)={\mathbb P}_x[S_n=y, \tau >n]\) (in case of a lattice walk), i.e., \(P(n;x,y)dy ={\mathbb P}_x[S_n\in dy , \tau>n]\) (in case of absolutely continuous \(\mu\)), and the “probability of life” by \(P(n;x)={\mathbb P}_x[\tau>n]\). Under certain technical conditions (\(\Omega\) should be in “general position”) the main result of the paper states the following: Let \(x_0\in \Omega\) be fixed, and let \(h(x)=u(x+x_0)\). Then \[ P(n;x,y)\leq C\frac{h(x)h(y)}{n^{\alpha+d/2}}, \;\;x,y\in \Omega, \;\;n\geq 1 , \] \[ P(n;x)\leq C\frac{(| x| ^{\alpha-1/2}+1)(d(x,\partial \Omega)^{1/2}+1)}{n^{\alpha/2}},\;\;x\in \Omega, \;\;n\geq 1 , \] \[ P(n;x)\geq C \min\{1, n^{-\alpha/2}(u(x)-C| x| ^{\alpha -1} -C)\}, \;\;x \in \Omega, \;d(x,\partial \Omega)\geq C, \;n\geq C| x| ^2+C . \] The proof is quite involved and uses continuous time estimates, Dirichlet forms techniques and abstract Hardy-Littlewood theory, as well as a lengthy and complicated construction of a sub(super)-harmonic function for the random walk.

MSC:
31C45 Other generalizations (nonlinear potential theory, etc.)
60J45 Probabilistic potential theory
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