9,925 research outputs found
Diffusion semigroup on manifolds with time-dependent metrics
Let , on a differential manifold equipped
with time-depending complete Riemannian metric , where
is the Laplacian induced by and is a
family of -vector fields. We first present some explicit criteria for
the non-explosion of the diffusion processes generated by ; then establish
the derivative formula for the associated semigroup; and finally, present a
number of equivalent semigroup inequalities for the curvature lower bound
condition, which include the gradient inequalities, transportation-cost
inequalities, Harnack inequalities and functional inequalities for the
diffusion semigroup
Weak Poincar\'e Inequality for Convolution Probability Measures
By using Lyapunov conditions, weak Poincar\'e inequalities are established
for some probability measures on a manifold . These results are further
applied to the convolution of two probability measures on . Along with
explicit results we study concrete examples
A probabilistic method for gradient estimates of some geometric flows
In general, gradient estimates are very important and necessary for deriving
convergence results in different geometric flows, and most of them are obtained
by analytic methods. In this paper, we will apply a stochastic approach to
systematically give gradient estimates for some important geometric quantities
under the Ricci flow, the mean curvature flow, the forced mean curvature flow
and the Yamabe flow respectively. Our conclusion gives another example that
probabilistic tools can be used to simplify proofs for some problems in
geometric analysis.Comment: 22 pages. Minor revision to v1. Accepted for publication in
Stochastic Processes and their Application
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