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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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