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Brownian motion and the formation of singularities in the heat flow for harmonic maps
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  • Published: September 1996

Brownian motion and the formation of singularities in the heat flow for harmonic maps

  • Anton Thalmaier1 

Probability Theory and Related Fields volume 105, pages 335–367 (1996)Cite this article

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Summary

We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of takin expectations is replaced by the concept of “martingale means”, namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat flow for harmonic maps with small initial energy.

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Authors and Affiliations

  1. NWF I-Mathematik, Universität Regensburg, D-93040, Regensburg, Germany

    Anton Thalmaier

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  1. Anton Thalmaier
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This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.

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Thalmaier, A. Brownian motion and the formation of singularities in the heat flow for harmonic maps. Probab. Th. Rel. Fields 105, 335–367 (1996). https://doi.org/10.1007/BF01192212

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  • Received: 03 March 1995

  • Revised: 12 February 1996

  • Issue Date: September 1996

  • DOI: https://doi.org/10.1007/BF01192212

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Mathematics Subject Classification (1991)

  • 58G32
  • 58G11
  • 60H30
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