References of "Thalmaier, Anton 50003189"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 1 to 20 of 65 1 2 3 4     Exponential contraction in Wasserstein distance on static and evolving manifoldsCheng, Li Juan ; Thalmaier, Anton ; Zhang, Shao-Qinin Revue Roumaine de Mathématiques Pures et Appliquées (2021)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: 116 (19 UL) Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flowsCheng, Li-Juan; Thalmaier, Anton E-print/Working paper (2020)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: 52 (1 UL) Gradient Estimates on Dirichlet and Neumann EigenfunctionsArnaudon, Marc; Thalmaier, Anton ; Wang, Feng-Yuin International Mathematics Research Notices (2020), 2020(20), 7279-7305By 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: 552 (112 UL) Radial processes for sub-Riemannian Brownian motions and applicationsBaudoin, Fabrice; Grong, Erlend; Kuwada, Kazumasa et alin Electronic Journal of Probability (2020), 25(paper no. 97), 1-17We 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: 172 (22 UL) Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flowGüneysu, Batu; Thalmaier, Anton in Annales de l'Institut Fourier (2020), 70(1), 437-456We 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: 218 (46 UL) Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spacesThompson, James ; Thalmaier, Anton in Bernoulli (2020), 26(3), 2202-2225In 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: 184 (44 UL) Functional inequalities on path space of sub-Riemannian manifolds and applicationsCheng, Li Juan ; Grong, Erlend; Thalmaier, Anton E-print/Working paper (2019)For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a well-defined gradient operator. By using ... [more ▼]For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a well-defined gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincaré inequalities. [less ▲]Detailed reference viewed: 53 (9 UL) Stochastic completeness and gradient representations for sub-Riemannian manifoldsGrong, Erlend; Thalmaier, Anton in Potential Analysis (2019), 51(2), 219-254Given 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: 272 (53 UL) Sub-Laplacian comparison theorems on totally geodesic Riemannian foliationsBaudoin, Fabrice; Grong, Erlend; Kuwada, Kazumasa et alin Calculus of Variations and Partial Differential Equations (2019), 58:130(4), 1-38We 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: 251 (52 UL) Derivative and divergence formulae for diffusion semigroupsThalmaier, Anton ; Thompson, James in Annals of Probability (2019), 47(2), 743-773Detailed reference viewed: 460 (107 UL) Uniform gradient estimates on manifolds with a boundary and applicationsCheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Analysis and Mathematical Physics (2018), 8(4), 571-588We 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: 228 (65 UL) Functional inequalities on manifolds with non-convex boundaryCheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Science China Mathematics (2018), 61(8), 1421-1436In 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: 236 (41 UL) Quantitative C1-estimates by Bismut formulaeCheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Journal of Mathematical Analysis and Applications (2018), 465(2), 803-813For 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: 320 (65 UL) Evolution systems of measures and semigroup properties on evolving manifoldsCheng, Li Juan ; Thalmaier, Anton in Electronic Journal of Probability (2018), 23(20), 1-27An 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: 323 (66 UL) Spectral gap on Riemannian path space over static and evolving manifoldsCheng, Li Juan ; Thalmaier, Anton in Journal of Functional Analysis (2018), 274(4), 959-984In 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: 311 (42 UL) Characterization of pinched Ricci curvature by functional inequalitiesCheng, Li Juan ; Thalmaier, Anton in Journal of Geometric Analysis (The) (2018), 28(3), 2312-2345In 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: 343 (53 UL) Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part IGrong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 99-130We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub ... [more ▼]We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part II. [less ▲]Detailed reference viewed: 335 (57 UL) Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part IIGrong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 131-164Using 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: 304 (33 UL) Geometry of subelliptic diffusionsThalmaier, Anton in Barilari, Davide; Boscain, Ugo; Sigalotti, Mario (Eds.) Geometry, Analysis and Dynamics on sub-Riemannian Manifolds. Volume II (2016)The lectures focus on some probabilistic aspects related to sub-Riemannian geometry. The main intention is to give an introduction to hypoelliptic and subelliptic diffusions. The notes are written from a ... [more ▼]The lectures focus on some probabilistic aspects related to sub-Riemannian geometry. The main intention is to give an introduction to hypoelliptic and subelliptic diffusions. The notes are written from a geometric point of view trying to minimize the weight of probabilistic baggage'' necessary to follow the arguments. We discuss in particular the following topics: stochastic flows to second order differential operators; smoothness of transition probabilities under Hörmander's brackets condition; control theory and Stroock-Varadhan's support theorems; Malliavin calculus; Hörmander's theorem. The notes start from well-known facts in Geometric Stochastic Analysis and guide to recent on-going research topics, like hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semigroups; notions of curvature related to sub-Riemannian diffusions. [less ▲]Detailed reference viewed: 1134 (73 UL) On gradient solitons of the Ricci-Harmonic flowGuo, Hongxin; Philipowski, Robert ; Thalmaier, Anton in Acta Mathematica Sinica (2015), 31(11), 1798-1804In this paper we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the ... [more ▼]In this paper we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds. [less ▲]Detailed reference viewed: 405 (47 UL) 1 2 3 4