# Brownian Motion

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## Surface Measures

Therefore, I have investigated for quite some time the following question which can be seen as an analogue of the first construction described above to infinite dimensions: If we condition Brownian motion on a Riemannian manifold M to stay within the $\epsilon$- tube $L(\epsilon)$ around a closed submanifold $L\subset M$ up to some fixed time $T > 0$, and we consider the behavior of this family of conditional measures as $\epsilon$ tends to zero -- will the measures converge weakly, and, if this is the case, what will be the limit measure ? We try to give a preliminary answer to these questions in the recent preprints arXiv:1908.01385 (joint work with Vera Nobis) and arXiv:1908.01387.