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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

##### MSC:
 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.)