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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:
[1] Grigor’yan, A.A. (1987). On stochastically complete manifolds.Soviet Math. Dokl. (English translation),34(2), 310–313. · Zbl 0632.58041
[2] Grigor’yan, A.A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. To appear.
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[5] Yau, S.-T. (1978). On the heat kernel of a complete Riemannian manifold.J. Math. Pure Appl., ser. 9,57, 191–201. · Zbl 0405.35025
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