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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: 98 (20 UL)Gradient Estimates on Dirichlet and Neumann Eigenfunctions ; Thalmaier, Anton ; E-print/Working paper (2018) 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: 51 (11 UL)Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow ; Thalmaier, Anton E-print/Working paper (2017) We prove a completely 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 ... [more ▼] We prove a completely 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: 46 (5 UL)Evolution systems of measures and semigroup properties on evolving manifolds Cheng, Li Juan ; Thalmaier, Anton E-print/Working paper (2017) 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: 86 (27 UL)Characterization of pinched Ricci curvature by functional inequalities Cheng, Li Juan ; Thalmaier, Anton in Journal of Geometric Analysis (The) (2017) 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: 133 (29 UL)Functional inequalities on manifolds with non-convex boundary Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James E-print/Working paper (2017) 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: 39 (12 UL)Derivative and divergence formulae for diffusion semigroups Thalmaier, Anton ; Thompson, James E-print/Working paper (2017) Detailed reference viewed: 152 (35 UL)Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations ; ; et al E-print/Working paper (2017) 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: 58 (8 UL)Quantitative C1-estimates by Bismut formulae Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James E-print/Working paper (2017) 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: 124 (32 UL)Geometry of subelliptic diffusions Thalmaier, 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: 329 (67 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: 198 (28 UL)Stochastic completeness and gradient representations for sub-Riemannian manifolds ; Thalmaier, Anton E-print/Working paper (2016) 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: 63 (10 UL)Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part I Grong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 99-130 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 ... [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: 240 (51 UL)On gradient solitons of the Ricci-Harmonic flow ; Philipowski, Robert ; Thalmaier, Anton in Acta Mathematica Sinica (2015), 31(11), 1798-1804 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 ... [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: 279 (45 UL)Martingales on manifolds with time-dependent connection Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Journal of Theoretical Probability (2015), 28(3), 1038-1062 We define martingales on manifolds with time-dependent connection, extending in this way the theory of stochastic processes on manifolds with time-changing geometry initiated by Arnaudon, Coulibaly and ... [more ▼] We define martingales on manifolds with time-dependent connection, extending in this way the theory of stochastic processes on manifolds with time-changing geometry initiated by Arnaudon, Coulibaly and Thalmaier (2008). We show that some, but not all properties of martingales on manifolds with a fixed connection extend to this more general setting. [less ▲] Detailed reference viewed: 193 (44 UL)An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Potential Analysis (2015), 42(2), 483-497 We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is ... [more ▼] We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained. [less ▲] Detailed reference viewed: 268 (35 UL)Heat equation in vector bundles with time-dependent metric Philipowski, Robert ; Thalmaier, Anton in Journal of the Mathematical Society of Japan [=JMSJ] (2015), 67(4), 1759-1769 We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well ... [more ▼] We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well. As a corollary we obtain a vanishing theorem for bounded ancient solutions under a curvature condition. Our results apply in particular to the case of differential forms. [less ▲] Detailed reference viewed: 207 (27 UL)A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Stochastic Processes and their Applications (2014), 124(11), 3535-3552 We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems ... [more ▼] We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations. [less ▲] Detailed reference viewed: 156 (28 UL)Equivalent Harnack and gradient inequalities for pointwise curvature lower bound ; Thalmaier, Anton ; in Bulletin des Sciences Mathématiques (2014), 138(5), 643-655 Detailed reference viewed: 148 (24 UL)A note on Chow's entropy functional for the Gauss curvature flow Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Comptes Rendus de l'Académie des Sciences (2013), 351(21-22), 833-835 Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional which is monotone along the unnormalized flow and whose critical point is a shrinking ... [more ▼] Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional which is monotone along the unnormalized flow and whose critical point is a shrinking self-similar solution. [less ▲] Detailed reference viewed: 88 (9 UL) |
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