Construction of surface measures for Brownian motion

Let $L$L be a submanifold of a Riemannian manifold $M$M. The authors discuss several ways to construct surface measures $\mu$µ on the path space of $L$L as weak limits of path measures $\mu_\varepsilon$µe on $\varepsilon$e-tubular neighbourhoods $L(\varepsilon)$L(e) of $L$L in $M$M, as the radius $\varepsilon$e of the tubular neighbourhoods tends to zero. The measures $\mu_\varepsilon$µe are induced by the Wiener measure on paths in $M$M. For instance, one considers Brownian motion on the ambient space $M$M conditioned to stay within $L(\varepsilon)$L(e) up to some finite time $T$T, or being absorbed at the boundary of the tube along with a proper renormalization. The limit measures on the path space of the submanifold $L$L, as the tube radius tends to zero, are typically absolutely continuous with respect to the intrinsic Brownian motion measure on $L$L, with a density depending on intrinsic (such as the scalar curvature) and extrinsic (such as mean curvature and traces of the Riemann tensor restricted to $TL$TL) properties of the embedded submanifold $L$L. Connections to the dynamics of a quantum particle of bounded energy confined to small tubes around the submanifold by an infinite hard-wall potential are discussed