Reference : Reflecting diffusion semigroup on manifolds carrying geometric flow
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Reflecting diffusion semigroup on manifolds carrying geometric flow
Cheng, Li Juan mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Zhang, Kun []
Journal of Theoretical Probability
Yes (verified by ORBilu)
New York
[en] Geometric flow ; Ricci flow ; curvature ; second fundamental form ; coupling ; Harnack inequality ; transportation-cost inequality
[en] Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differentiable manifold $M$ with boundary $\partial M$, where $\Delta_t$ is the Laplacian operator, induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first establish the derivative formula for the associated reflecting diffusion semigroup generated by $L_t$; then construct the couplings for the reflecting $L_t$-diffusion processes by parallel displacement and reflection, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup; and finally, by using the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary, including the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.
Fonds National de la Recherche Luxembourg
Researchers ; Professionals

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