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On stochastically complete manifolds. (English. Russian original) Zbl 0632.58041
Sov. Math., Dokl. 34, 310-313 (1987); translation from Dokl. Akad. Nauk SSSR 290, 534-537 (1986).
The author sketches the proofs of the following two theorems:
Theorem 1. Let M be a complete Riemannian manifold, and let V(s) be the volume of a geodesic ball \(B_ r\) of radius r with fixed center \(0\in M\). If \(\int^{\infty}(r/V(r))dr=\infty\), then M is stochastically complete.
Theorem 2. Suppose that M is a complete Riemannian manifold, and n(x,t) is a solution of the initial value problem \(\partial u/\partial t=\Delta n\), \(u|_{t=0}=0\) defined in the strip \(M_ T=M\times [0,T]\). Suppose that for any \(R>0\) \[ \int^{T}_{0}\int_{B_ R}u^ 2(t,x)dxdt<e^{f(R)}, \] where f is a monotonically increasing function such that \(\int^{\infty}(r/f(r))dr=\infty\). Then \(u\equiv 0\) in \(M_ T\).
Reviewer: N.Jacob

58J65 Diffusion processes and stochastic analysis on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)