References of "Thalmaier, Anton 50003189"
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See detailAn entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
Guo, Hongxin UL; Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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See detailHeat equation in vector bundles with time-dependent metric
Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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See detailA stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric
Guo, Hongxin UL; Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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See detailEquivalent Harnack and gradient inequalities for pointwise curvature lower bound
Arnaudon, Marc; Thalmaier, Anton UL; Wang, Feng-Yu

in Bulletin des Sciences Mathématiques (2014), 138(5), 643-655

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See detailA note on Chow's entropy functional for the Gauss curvature flow
Guo, Hongxin UL; Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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See detailEntropy and lowest eigenvalue on evolving manifolds
Guo, Hongxin UL; Philipowski, Robert UL; Thalmaier, Anton UL

in Pacific J. Math. (2013), 264(1), 61-81

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See detailThe differentiation of hypoelliptic diffusion semigroups
Arnaudon, Marc; Thalmaier, Anton UL

in Don Burkholder: A Collection of Articles in His Honor (2012)

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See detailHorizontal diffusion in C¹ path space
Arnaudon, Marc; Coulibaly, Koléhè Abdoulaye; Thalmaier, Anton UL

in Séminaire de Probabilités XLIII (2011)

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See detailPaul Malliavin (10 September 1925 - 3 June 2010)
Thalmaier, Anton UL

in European Mathematical Society. Newsletter (2011), 81

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See detailBrownian motion and negative curvature
Arnaudon, Marc; Thalmaier, Anton UL

in Random walks, boundaries and spectra (2011)

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See detailA stochastic approach to a priori estimates and Liouville theorems for harmonic maps
Thalmaier, Anton UL; Wang, Feng-Yu

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 ▲]

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See detailBrownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric
Airault, Hélène; Malliavin, Paul; Thalmaier, Anton UL

in Journal of Functional Analysis (2010), 259(12), 3037-3079

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See detailStochastic differential equations with coefficients in Sobolev spaces
Fang, Shizan; Luo, Dejun; Thalmaier, Anton UL

in Journal of Functional Analysis (2010), 259(5), 1129-1168

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See detailLi-Yau type gradient estimates and Harnack inequalities by stochastic analysis
Arnaudon, Marc; Thalmaier, Anton UL

in Probabilistic approach to geometry (2010)

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See detailThe differentiation of hypoelliptic diffusion semigroups
Arnaudon, Marc; Thalmaier, Anton UL

in Illinois Journal of Mathematics (2010), 54(4), 1285-1311

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See detailExistence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature
Arnaudon, Marc; Thalmaier, Anton UL; Ulsamer, Stefanie

in Mathematische Zeitschrift (2009), 263(2), 369-409

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See detailGradient estimates and Harnack inequalities on non-compact Riemannian manifolds
Arnaudon, Marc; Thalmaier, Anton UL; Wang, Feng-Yu

in Stochastic Processes & Their Applications (2009), 119(10), 3653-3670

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See detailBrownian motion with respect to a metric depending on time: definition, existence and applications to Ricci flow
Arnaudon, Marc; Coulibaly, Kolehe Abdoulaye; Thalmaier, Anton UL

in Comptes Rendus. Mathématique (2008), 346(13-14), 773-778

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See detailBild von Möglichem und Unmöglichem
Mortini, Raymond; Thalmaier, Anton UL

Article for general public (2007)

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See detailGradient estimates for positive harmonic functions by stochastic analysis
Arnaudon, Marc; Driver, Bruce K.; Thalmaier, Anton UL

in Stochastic Processes & Their Applications (2007), 117(2), 202-220

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