Abstract
Let † denote the standard (i.e., Levi-Civita) Laplacian for some non-compact, connected, complete, separable Riemannian manifild M. In a much cited article, Yau [5] proved that when the Ricci curvature is bounded uniformly below, then the only bounded solution to the heat equation ∂ t μ=Δμ on [0, ∞) × M which vanishes at t=0 is the one which vanishes evarywhere. 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.
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Stroock, D.W. Non-divergence form operators and variations on Yau's explosion criterion. The Journal of Fourier Analysis and Applications 4, 565–574 (1998). https://doi.org/10.1007/BF02498225
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DOI: https://doi.org/10.1007/BF02498225