Oxford

explain recent work by Peter Friz, Dan Stroock and myself proving that a

Brownian path conditioned to be uniformly close to a given smooth path

converges in distribution to that path in the Rough Path metric. The Stroock

Varadhan support theorem is an immediate consequence.

The novel part of the argument is to

obtain the estimate in a way that is independent of the particular norm used

in the Euclidean space when one defines the uniform norm on path space.

distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with

boundary condition zero. Functional u is also conjectured (in some cases

proved) to be the "limiting minimiser" of various functionals that

arise models of blistering and micro magnetics. The precise formulation of

these problems involves the notion of gamma convergence. The Aviles Giga

functional is a natural "second order" generalisation of the Cahn

Hilliard model which was one of the early success of the theory of gamma

convergence. These problems turn out to be surprisingly rich with connections

to a number of areas of pdes. We will survey some of the more elementary

results, describe in detail of one main problems in field and state some

partial results.

paths in a compact Riemannian manifold embedded into the Euclidean

space. The idea of the constructions is to force a Brownian particle

in the ambient space to stay in a small neighbourhood of the manifold

and then to pass to the limit. Finally, we compare these surface

measures with the Wiener measure on the space of paths in the

manifold.