References of "Cheng, Li-Juan"
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See detailBismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds
Chen, Qin-Qian; Cheng, Li-Juan; Thalmaier, Anton UL

E-print/Working paper (2021)

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock ... [more ▼]

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds. [less ▲]

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See detailHessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds
Cao, Jun; Cheng, Li-Juan; Thalmaier, Anton UL

E-print/Working paper (2021)

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the Lp-Calderón-Zygmund inequalities extend to a reasonable class of non-compact ... [more ▼]

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the Lp-Calderón-Zygmund inequalities extend to a reasonable class of non-compact Riemannian manifolds without the assumption of a positive injectivity radius. In the present paper, we give a positive answer for 1 < p < 2 under the natural assumption of a lower bound on the Ricci curvature. For p > 2, we complement the study in Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality by adding Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role throughout the proofs. [less ▲]

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See detailSome inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance
Cheng, Li-Juan; Thalmaier, Anton UL; Wang, Feng-Yu

E-print/Working paper (2021)

For a complete connected Riemannian manifold M let V∊ C^2(M) be such that µ(dx)=exp(-V(x))vol(dx) is a probability measure on M. Taking µ as reference measure, we derive inequalities for probability ... [more ▼]

For a complete connected Riemannian manifold M let V∊ C^2(M) be such that µ(dx)=exp(-V(x))vol(dx) is a probability measure on M. Taking µ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds. [less ▲]

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See detailDimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows
Cheng, Li-Juan; Thalmaier, Anton UL

E-print/Working paper (2021)

Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. In this article, we give a probabilistic ... [more ▼]

Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on M which is generated by a Schrödinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows. [less ▲]

Detailed reference viewed: 77 (2 UL)