References of "Philipowski, Robert 50002850"
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See detailOn gradient solitons of the Ricci-Harmonic flow
Guo, Hongxin; Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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See detailMartingales on manifolds with time-dependent connection
Guo, Hongxin UL; Philipowski, Robert UL; Thalmaier, Anton UL

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 ▲]

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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 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 detailStochastic particle approximations for the Ricci flow on surfaces and the Yamabe flow
Philipowski, Robert UL

in Potential Analysis (2011), 35

We present stochastic particle approximations for the normalized Ricci flow on surfaces and for the non-normalized Yamabe flow on manifolds of arbitrary dimension.

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See detailCoupling of Brownian motions and Perelman's L-functional
Kuwada, Kazumasa; Philipowski, Robert UL

in Journal of Functional Analysis (2011), 260

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a ... [more ▼]

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non-decreasing function and the normalized L-distance. In particular, it provides a new proof of a recent result of Topping [P. Topping, L-optimal transportation for Ricci flow, J. Reine Angew. Math. 636 (2009) 93–122]. [less ▲]

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See detailNon-explosion of diffusion processes on manifolds with time-dependent metric
Kuwada, Kazumasa; Philipowski, Robert UL

in Mathematische Zeitschrift (2011), 268

We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric ... [more ▼]

We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon et al. (arXiv:0904.2762, to appear in Sém. Prob.). As an important tool which is of independent interest we derive an Itô formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15:1491–1500, 1987). [less ▲]

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