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

• 成果类型: Article
• 署名作者: Thalmaier, A
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01192212
• 发表日期: 1996
• 页码: 335-367
• 关键词: finite-time blow; martingales; REGULARITY; EXISTENCE; convexity; mappings

摘要

We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of taking 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.