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Non-divergence form operators and variations on Yau’s explosion criterion. (English) Zbl 1101.58304
Summary: Let $$\Delta$$ denote the standard (i.e., Levi-Civita) Laplacian for some non-compact, connected, complete, separable Riemannian manifold $$M$$. S.-T. Yau [J. Math. Pures Appl. (9) 57, 191–201 (1978; Zbl 0405.35025)] proved that when the Ricci curvature is bounded uniformly below, then the only bounded solution to the heat equation $$\partial_t \mu=\Delta\mu$$ on $$[0, \infty) \times M$$ which vanishes at $$t=0$$ is the one which vanishes everywhere. Equivalently, no matter where it starts, Brownian motion on $$M$$ never explodes. Yau’s original statement was improved or extended in various directions by a long list of authors. With this paper, the present author joins the list.

##### MSC:
 58J65 Diffusion processes and stochastic analysis on manifolds 35J60 Nonlinear elliptic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 60H30 Applications of stochastic analysis (to PDEs, etc.)
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##### References:
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