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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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