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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: 1682 (77 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: 360 (57 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: 446 (47 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: 376 (47 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: 475 (41 UL)Heat equation in vector bundles with time-dependent metric Philipowski, Robert ; Thalmaier, Anton in Journal of the Mathematical Society of Japan (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: 351 (36 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: 293 (29 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: 338 (32 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. Série I. Mathématique (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: 300 (14 UL)Entropy and lowest eigenvalue on evolving manifolds Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Pacific Journal of Mathematics (2013), 264(1), 61-81 Detailed reference viewed: 291 (22 UL)The differentiation of hypoelliptic diffusion semigroups ; Thalmaier, Anton in Don Burkholder: A Collection of Articles in His Honor (2012) Detailed reference viewed: 423 (40 UL)A stochastic approach to a priori estimates and Liouville theorems for harmonic maps Thalmaier, Anton ; in Bulletin des Sciences Mathématiques (2011), 135(6-7), 816-843 Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type ... [more ▼] Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour. [less ▲] Detailed reference viewed: 270 (17 UL)Brownian motion and negative curvature ; Thalmaier, Anton in Random walks, boundaries and spectra (2011) Detailed reference viewed: 289 (6 UL)Horizontal diffusion in C¹ path space ; ; Thalmaier, Anton in Séminaire de Probabilités XLIII (2011) Detailed reference viewed: 299 (13 UL)Paul Malliavin (10 September 1925 - 3 June 2010) Thalmaier, Anton in European Mathematical Society. Newsletter (2011), 81 Detailed reference viewed: 185 (11 UL)Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis ; Thalmaier, Anton in Probabilistic approach to geometry (2010) Detailed reference viewed: 314 (20 UL)Stochastic differential equations with coefficients in Sobolev spaces ; ; Thalmaier, Anton in Journal of Functional Analysis (2010), 259(5), 1129-1168 Detailed reference viewed: 274 (5 UL)The differentiation of hypoelliptic diffusion semigroups ; Thalmaier, Anton in Illinois Journal of Mathematics (2010), 54(4), 1285-1311 Detailed reference viewed: 157 (5 UL)Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric ; ; Thalmaier, Anton in Journal of Functional Analysis (2010), 259(12), 3037-3079 Detailed reference viewed: 246 (9 UL)Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds ; Thalmaier, Anton ; in Stochastic Processes & Their Applications (2009), 119(10), 3653-3670 Detailed reference viewed: 267 (21 UL) |
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