The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold. (Q1423438)

Let \(M\subset \mathbb{R}^n\) be a smooth \(m\)-dimensional compact Riemannian manifold and \(M_\varepsilon\) be its tubular neighborhood. Denote by \(W_\varepsilon\) the conditional law of a Brownian particle conditioned not to leave the \(\varepsilon\)-neighborhood of the manifold. The authors prove that \(W_\varepsilon\) converges weakly to a probability measure \(W_0\) on \(C([0,1], M)\) which is equivalent to the Wiener measure \(W_M\) on the manifold and give the explicit formula for the density \[ \frac{dW_0}{dW_M}(\omega)=\frac{\exp\{-\frac{1}{4}\int_{0}^{1}R(\omega_t) \,dt+\frac{1}{8} \int_{0}^{1}\| \sigma\| ^2(\omega_t) \,dt\}} {E_{W_M}\exp\{-\frac{1}{4}\int_{0}^{1}R(\omega_t) \,dt+\frac{1}{8} \int_{0}^{1}\| \sigma\| ^2(\omega_t) \,dt\}} \] where \(R(x)\) is the scalar curvature and \(\sigma(x)\) is the tension vector of \(M\) at \(x\in M\).