Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Li-Juan Cheng () and Kun Zhang ()
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Li-Juan Cheng: Zhejiang University of Technology
Kun Zhang: City University of Hong Kong

Journal of Theoretical Probability, 2017, vol. 30, issue 4, 1334-1368

Abstract: Abstract Let $$L_t:=\Delta _t+Z_t$$ L t : = Δ t + Z t for a $$C^{\infty }$$ C ∞ -vector field Z on a differentiable manifold M with boundary $$\partial M$$ ∂ M , where $$\Delta _t$$ Δ t is the Laplacian operator, induced by a time dependent metric $$g_t$$ g t differentiable in $$t\in [0,T_\mathrm {c})$$ t ∈ [ 0 , T c ) . We first establish the derivative formula for the associated reflecting diffusion semigroup generated by $$L_t$$ L t . Then, by using parallel displacement and reflection, we construct the couplings for the reflecting $$L_t$$ L t -diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.

Keywords: Geometric flow; Ricci flow; Curvature; Second fundamental form; Transportation-cost inequality; Harnack inequality; Coupling; Primary 58J65; Secondary 60J60; 53C44 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-016-0678-4

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