Abstract
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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