Martingales on Riemannian manifolds
and the nonlinear heat equation
by Anton Thalmaier
Solutions to the nonlinear heat equation for maps
between Riemannian manifolds are studied by determining starting points
for martingales on Riemannian manifolds with prescribed terminal values.
Monotonicity properties of the Riemannian quadratic variation for these
martingales allow to explain blow-up of the heat flow in finite time.
Moreover, the probabilistic construction of martingales with given
terminal state is discussed, and partial regularity results
for the heat flow are established.
In: I.M. Davies, A. Truman and K.D. Elworthy (Eds.)
Stochastic Analysis and Applications.
Proc. of the Fifth Gregynog Symposium, Gregynog, 1995.
Singapore: World Scientific Press, 1996, 429-440.
The paper is available here:
Anton Thalmaier <anton.thalmaier@uni.lu>
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