Hitting probabilities for Brownian motion on Riemannian manifolds.
Let \(M\) be a complete non-compact Riemannian manifold, with heat kernel \(p(t,x,y)\) and Wiener measure \(\mathbb{P}_x\), and \(K\) be a compact subset of \(M\). The aim of this article is to estimate the hitting probability function: \(\psi(t,x):= \mathbb{P}_x((\exists s\in [0,t]) X_s\in K)\), when \(t> 0\) and \(d(x,K)\geq \delta> 0\), and also \(\partial_t\psi(t,x)\). As a main tool, the authors first establish the following potential-theoretic formula: for any precompact neighborhood of \(K\), say open \(L\), and for any \(x\in M\setminus K\), \(t> 0\): \[ \partial_t \psi(t,x)= 2\int_{L\setminus K} p_K(t,x,\cdot)_x |\nabla\varphi|^2 \,d\mu- \int_{L\setminus K}\partial_t p_K(t,x,\cdot)_x \varphi^2 \,d\mu, \] where \(p_K(t,x,y)\) is the Dirichlet heat kernel of \(L\), relative to some fixed measure \(\mu\) on \(M\), and where \(\varphi\) is the equilibrium potential of the capacitor \((K,L)\). Many fine estimates are then deduced, in various cases and ranges, for \(\psi\) and \(\partial_t\psi\), from both sides. In particular it is proved that if \(\text{cap}(K,L)> 0\) and \(x\not\in L\): \[ \psi(t,x)\leq 2\text{\,cap}(K,L)\times \int^t_0 \sup_{y\in L\setminus K}\, p_K(s,x,y)\,ds \] and \[ \psi(\infty,x)- \psi(t,x)\leq 2\text{\,cap}(K, L)\times \int^\infty_t \sup_{y\in L\setminus K}\, p_K(s,x,y)\,ds+ \mu(L\setminus K)\sup_{y\in L\setminus K}\, p_K(t,x,y). \] In the transient case, it is shown that \[ \psi(t,x)\geq \text{cap}(K)\times \int^t_0 \inf_{y\in\partial K} p(s,x,y)\,ds \] and \[ \psi(\infty, x)- \psi(t,x)\leq \text{cap}(K)\times \int^\infty_t \sup_{y\in\partial K}\, p(s,x,y)\,ds. \] More specific bounds are obtained under geometric conditions on \(M\), in both recurrent and transient cases. In the example \(M= \mathbb{R}^2\), \(K= \langle| x|\leq 1\}\), the following bounds are obtained: \(c_0\exp(- C_1| x|^2/t)\leq \log| x|\times \psi(t, x)\leq C_1\exp(- C_0| x|^2/t)\) if \(0< t< 2| x|^2\); and \(\psi(t, x)\approx (\log\sqrt{t}- \log| x|)/\log\sqrt{t}\), \(\partial_t \psi(t,x)\approx {\log| x|\over t\log^2t}\) if \(t\geq 2| x|^2\).
scientific article; zbMATH DE number 1985988
scientific article