Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

in Analysis and PDE (in press)

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 ▲]

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

in Stochastic Partial Differential Equations: Analysis and Computations (2023), 11(2), 685-713

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 ▲]

Variations of the sub-Riemannian distance on Sasakian manifolds with applications to coupling

E-print/Working paper (2022)

On Sasakian manifolds with their naturally occurring sub-Riemannian structure, we consider parallel and mirror maps along geodesics of a taming Riemannian metric. We show that these transport maps have ... [more ▼]

On Sasakian manifolds with their naturally occurring sub-Riemannian structure, we consider parallel and mirror maps along geodesics of a taming Riemannian metric. We show that these transport maps have well-defined limits outside the sub-Riemannian cut-locus. Such maps are not related to parallel transport with respect to any connection. We use this map to obtain bounds on the second derivative of the sub-Riemannian distance. As an application, we get some preliminary result on couplings of sub-Riemannian Brownian motions. [less ▲]

Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

E-print/Working paper (2022)

Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds

E-print/Working paper (2021)

We address some fundamental questions about geometric analysis on Riemannian manifolds. The L^p-Calderón-Zygmund inequality is one of the cornerstones in the regularity theory of elliptic equations, and ... [more ▼]

We address some fundamental questions about geometric analysis on Riemannian manifolds. The L^p-Calderón-Zygmund inequality is one of the cornerstones in the regularity theory of elliptic equations, and it has been asked under which geometric conditions it holds for a reasonable class of non-compact Riemannian manifolds, and to what extent assumptions on the derivative of curvature and on the injectivity radius of the manifold are necessary. In the present paper, for 1<p<2, we give a positive answer for the validity of the L^p-Calderón-Zygmund inequality on a Riemannian manifold assuming only a lower bound on the Ricci curvature. It is well known that this alone is not sufficient for p>2. In this case we complement the study of Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality under additional Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Bounds in the Kato class are integral conditions and much weaker than pointwise bounds. Throughout the proofs, probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role. [less ▲]

Gradient Estimates on Dirichlet and Neumann Eigenfunctions

in International Mathematics Research Notices (2020), 2020(20), 7279-7305

By methods of stochastic analysis on Riemannian manifolds, we derive explicit two-sided gradient estimates for Dirichlet eigenfunctions on a d-dimensional compact Riemannian manifold D with boundary ... [more ▼]

By methods of stochastic analysis on Riemannian manifolds, we derive explicit two-sided gradient estimates for Dirichlet eigenfunctions on a d-dimensional compact Riemannian manifold D with boundary. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper. [less ▲]

Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow

in Annales de l'Institut Fourier (2020), 70(1), 437-456

We prove a new integral criterion for the existence and completeness of the wave operators W_{\pm}(-\Delta_h,-\Delta_g, I_{g,h}) corresponding to the (unique self-adjoint realizations of) the Laplace ... [more ▼]

We prove a new integral criterion for the existence and completeness of the wave operators W_{\pm}(-\Delta_h,-\Delta_g, I_{g,h}) corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators -\Delta_j, j=g,h, that are induced by two quasi-isometric complete Riemannian metrics g and h on an open manifold M. In particular, this result provides a criterion for the absolutely continuous spectra of -\Delta_g and -\Delta_h to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow. [less ▲]

Stochastic completeness and gradient representations for sub-Riemannian manifolds

in Potential Analysis (2019), 51(2), 219-254

Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a ... [more ▼]

Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of L/2, while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry. [less ▲]

Derivative and divergence formulae for diffusion semigroups

in Annals of Probability (2019), 47(2), 743-773

Functional inequalities on manifolds with non-convex boundary

in Science China Mathematics (2018), 61(8), 1421-1436

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non ... [more ▼]

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary. [less ▲]