An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions

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 ▲]

A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

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 ▲]

A note on Chow's entropy functional for the Gauss curvature flow

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 ▲]

The differentiation of hypoelliptic diffusion semigroups

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

Paul Malliavin (10 September 1925 - 3 June 2010)

in European Mathematical Society. Newsletter (2011), 81

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

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 ▲]

Stochastic differential equations with coefficients in Sobolev spaces

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

The differentiation of hypoelliptic diffusion semigroups

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

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

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

Bild von Möglichem und Unmöglichem

Article for general public (2007)