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Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds. (English) Zbl 1178.58013
Let $$M$$ be a non-compact complete Riemannian manifold. Let $$X_s(x)$$ be an $$L$$-diffusion process with starting point $$x$$ and explosion time $$\xi(x)$$, where $$L= \Delta+\nabla V$$ for some $$C^2$$ function $$V$$. Let $$P_t$$ be the Dirichlet diffusion semigroup generated by $$L$$ defined by $P_t f(x)= \mathbb E[f(X_t(x))1_{\{t<\xi(x)\}}].$
The authors prove the following gradient-entropy inequality: There exists a continuous positive function $$F$$ on $$]0,1]\times M$$ such that $$\forall \, \delta>0$$, $$x\in M$$, $$t>0$$, $$f\in {\mathcal B}_b^+$$, we have
$|\nabla P_t f(x)|\leq \delta (P_t f\log f- P_t f\log P_t f)(x)+\left(F(\delta\wedge 1,x) \left(\frac{1} {\delta (t\wedge 1)} +1\right)+\frac{2\delta}{\text{e}}\right)P_t f(x).$

##### MSC:
 58J65 Diffusion processes and stochastic analysis on manifolds 58J35 Heat and other parabolic equation methods for PDEs on manifolds 60H30 Applications of stochastic analysis (to PDEs, etc.)
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##### References:
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