Brownian motion and the formation of singularities
in the heat flow for harmonic maps
by Anton Thalmaier
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.
Probab. Theory Relat. Fields 105 (1996) 335-367
The paper is available here:
Anton Thalmaier <anton.thalmaier@uni.lu>
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