Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

...they map M -valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157 (ii)]), it is natural to study them using stochastic methods, and this has been done in a number of papers, e.g. =-=[2, 8, 14, 15, 30, 34]-=-. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in...