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Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds ; ; Thalmaier, Anton in Stochastic Partial Differential Equations: Analysis and Computations (in press) 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 ▲] Detailed reference viewed: 72 (16 UL)Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 115 (9 UL)Functional inequalities on path space of sub-Riemannian manifolds and applications Cheng, Li Juan ; ; Thalmaier, Anton in Nonlinear Analysis: Theory, Methods and Applications (2021), 210(112387), 1-30 We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians ... [more ▼] We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in L^2. With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein-Uhlenbeck operator. [less ▲] Detailed reference viewed: 164 (37 UL)Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds ; ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 124 (14 UL)Some inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance ; Thalmaier, Anton ; 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 ▲] Detailed reference viewed: 62 (8 UL)Exponential contraction in Wasserstein distance on static and evolving manifolds Cheng, Li Juan ; Thalmaier, Anton ; in Revue Roumaine de Mathématiques Pures et Appliquées (2021), 66(1), 107-129 In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not ... [more ▼] In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures. [less ▲] Detailed reference viewed: 174 (35 UL)Gradient Estimates on Dirichlet and Neumann Eigenfunctions ; Thalmaier, Anton ; 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 ▲] Detailed reference viewed: 613 (112 UL)Radial processes for sub-Riemannian Brownian motions and applications ; ; et al in Electronic Journal of Probability (2020), 25(paper no. 97), 1-17 We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating ... [more ▼] We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng's type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group. [less ▲] Detailed reference viewed: 195 (22 UL)Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 240 (46 UL)Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces Thompson, James ; Thalmaier, Anton in Bernoulli (2020), 26(3), 2202-2225 In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian ... [more ▼] In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of RCD*(K,N) spaces. In each case the necessary ingredients are an Ito formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrodinger operators. [less ▲] Detailed reference viewed: 215 (45 UL)Stochastic completeness and gradient representations for sub-Riemannian manifolds ; Thalmaier, Anton 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 ▲] Detailed reference viewed: 296 (54 UL)Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations ; ; et al in Calculus of Variations and Partial Differential Equations (2019), 58:130(4), 1-38 We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical ... [more ▼] We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations. [less ▲] Detailed reference viewed: 289 (54 UL)Derivative and divergence formulae for diffusion semigroups Thalmaier, Anton ; Thompson, James in Annals of Probability (2019), 47(2), 743-773 Detailed reference viewed: 522 (110 UL)Uniform gradient estimates on manifolds with a boundary and applications Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Analysis and Mathematical Physics (2018), 8(4), 571-588 We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using ... [more ▼] We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform quantitative gradient estimates, firstly for bounded C^2 functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians. [less ▲] Detailed reference viewed: 255 (65 UL)Functional inequalities on manifolds with non-convex boundary Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James 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 ▲] Detailed reference viewed: 277 (43 UL)Quantitative C1-estimates by Bismut formulae Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Journal of Mathematical Analysis and Applications (2018), 465(2), 803-813 For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a ... [more ▼] For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δu. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings. [less ▲] Detailed reference viewed: 359 (65 UL)Evolution systems of measures and semigroup properties on evolving manifolds Cheng, Li Juan ; Thalmaier, Anton in Electronic Journal of Probability (2018), 23(20), 1-27 An evolving Riemannian manifold (M,g_t)_{t\in I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-\infty,T). Given an ... [more ▼] An evolving Riemannian manifold (M,g_t)_{t\in I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-\infty,T). Given an additional C^{1,1} family of vector fields (Z_t)_{t\in I} on M. We study the family of operators L_t=\Delta_t +Z_t where \Delta_t denotes the Laplacian with respect to the metric g_t. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by L_t, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established. [less ▲] Detailed reference viewed: 355 (69 UL)Spectral gap on Riemannian path space over static and evolving manifolds Cheng, Li Juan ; Thalmaier, Anton in Journal of Functional Analysis (2018), 274(4), 959-984 In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along ... [more ▼] In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber’s recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space. [less ▲] Detailed reference viewed: 339 (43 UL)Characterization of pinched Ricci curvature by functional inequalities Cheng, Li Juan ; Thalmaier, Anton in Journal of Geometric Analysis (2018), 28(3), 2312-2345 In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient ... [more ▼] In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, L^p-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities. [less ▲] Detailed reference viewed: 378 (54 UL)Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part II Grong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 131-164 Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semi-group P_t ... [more ▼] Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semi-group P_t corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of P_t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold. [less ▲] Detailed reference viewed: 328 (33 UL) |
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