Brownian motion and the formation of singularities in the heat flow for harmonic maps (Q1916689)

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.