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Dirichlet heat kernel in the exterior of a compact set. (English) Zbl 1037.58018
Let \(M\) be a complete, noncompact Riemannian manifold. For a compact set \(K\subset M\) denote \(\Omega=M\setminus K\) and consider the Dirichlet heat kernel \(p_\Omega(t,x,y)\) in \(\Omega\) which by definition, as a function of \(t\) and \(x\), is a minimal positive solution of the problem \[ \partial_tu=\Delta u,\qquad u| _{\partial\Omega}=0,\qquad u| _{t=0}=\delta_y \] with \(\Delta=\) Laplace-Beltrami operator on \(M\) and Dirac’s delta \(\delta_y.\)
The general goal of the very interesting paper under review is to estimate \(p_\Omega\) away from the boundary \(\partial\Omega.\) Surprisingly, the answer is nontrivial even if \(M={\mathbb R}^n,\) \(n>1,\) and \(K\) the unit ball.
To be more precise, denote by \(p(t,x,y)\) the global heat kernel on \(M\) which is defined as the minimal positive fundamental solution to the heat equation in \(M.\) It follows \(p_\Omega\leq p\) by the comparison principle and the natural question to ask is whether \(p_\Omega\) is smaller than \(p\) or comparable to \(p\) when staying away from \(\partial\Omega.\) The answer depends on whether the manifold \(M\) is parabolic. The Riemannian manifold \(M\) is called parabolic, if the Brownian motion \((X_t)_{t\geq0}\) on \(M\) is recurrent, and nonparabolic if the Brownian motion is transient (e.g., \({\mathbb R}^2\) is parabolic but \({\mathbb R}^3\) is not). When \(M\) is nonparabolic then there is a positive probability that \((X_t)_{t\geq0}\) will never hit \(\partial\Omega\) started in \(\Omega,\) which suggests that \(p\) and \(p_\Omega\) should be comparable. If \(M\) is parabolic, then \((X_t)_{t\geq0}\) hits \(\partial\Omega\) with probability 1. Thus one expects the probability of getting from \(x\) to \(y\) without touching \(\partial\Omega\) may be significantly smaller than in the absence of \(\partial\Omega,\) i.e., \(p_\Omega\ll p.\)
The authors prove and quantify the validity of the above heuristic when \(M\) is a Riemannian manifold with nonnegative Ricci curvature. Set \(d(x,y)\) for the geodesic distance between \(x,\;y\in M\) and let \(B(x,r)\) be the geodesic ball of radius \(r\) centered at \(x,\) having volume \(V(x,r).\) The main result in the nonparabolic case is as follows:
Theorem 1.1. Let \(M\) be a complete, noncompact Riemannian manifold of nonnegative Ricci curvature. Let \(K\) be a compact set with nonempty interior and \(\Omega=M\setminus K.\) Assume \(M\) is nonparabolic. Then \[ p_\Omega(t,x,y)\asymp {1\over{V(x,\sqrt t)}} e^{-d^2(x,t)/t} \] for all \(t>0\) and all \(x\) and \(y\) far enough from \(K\) (\(\asymp\) substitutes “comparable to”).
To state the result in the parabolic case, for a given compact \(K\) fix a point \(o\in K\) and denote \[ | x| :=d(x,K)=\inf\{d(x,y)\colon\;y\in K\}, \] \[ H(r):= 1+ \int_0^r {{s e^{-1/s}}\over {V(o,s)}}\,ds, \] \[ D(t,x,y):= { {H(| x| )H(| y| )}\over {(H(| x| )+H(\sqrt t))(H(| y| )+H(\sqrt t))}} \] with \(t>0\) and \(x,\;y\in M.\) It is possible to prove that if \(M\) is nonparabolic then \(H(r)\) stays between two positive constants, and hence so does \(D(t,x,y).\) On the contrary, if \(M\) has nonnegative Ricci curvature and is parabolic, then \[ \int^\infty {{dt}\over {V(x,\sqrt t)}}=+\infty \] and therefore \(H(r)\) is unbounded which implies \(D(t,x,y)=0.\) Further on, given a point \(o\in M,\) the pointed manifold \((M,o)\) satisfies the condition of relatively connected annuli (RCA) if there is \(A>1\) such that for any \(r>A^2\) and all \(x,\;y\in \partial B(o,r)\) there exists a continuous path in \(B(o,Ar)\setminus B(o,A^{-1}r)\) connecting \(x\) to \(y.\) This is the main result in the parabolic case:
Theorem 1.2. Let \(M\) be a complete, noncompact Riemannian manifold of nonnegative Ricci curvature. Let \(K\) be a compact set with nonempty interior and \(\Omega=M\setminus K.\) Assume \(M\) is parabolic and satisfies (RCA). Then \[ p_\Omega(t,x,y)\asymp {{D(t,x,y)}\over{V(x,\sqrt t)}} e^{-d^2(x,t)/t} \] for all \(t>0\) and all \(x\) and \(y\) with large enough \(| x| \) and \(| y| ,\) respectively.
Examples of application of the above results to various manifolds are also proposed.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K05 Heat equation
58J65 Diffusion processes and stochastic analysis on manifolds
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