Brownian Motion and the Distance to a Submanifold

Abstract

This is a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. It contains a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. The concentration inequality is derived using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficiently sharp to imply a comparison theorem. We provide numerous examples throughout. Further applications will feature in a subsequent article, where we see how the main results and methods presented here can be applied to certain study objects which appear naturally in the theory of submanifold bridge processes.

Brownian Bridges to Submanifolds

“…satisfying the usual conditions. Note that, by [46,Theorem 5], if N is compact then inequality (15) implies the non-explosion of X(x). For and note that from [16] there is the formula…”

“…This allows us in Section 5 to use Theorem 4.1 and the estimate on A given in [46] to deduce a lower bound on the integrated heat kernel. This bound is stated in Theorem 5.1.…”

“…a horizontal lift of X(x) to the orthonormal frame bundle with antidevelopment B. It follows from[46, Proposition 3] and Novikov's criterion that M is a martingale up to time t for each t ∈ [0, T ). For each t ∈ [0, T ) we can therefore define a new probability measure Q t on F t by dQ t = M t dP.…”

Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

“…Then, given either a suitable Kolmogorov equation or, better yet, an Itô formula for the corresponding diffusion X, one should expect to be able to calculate various estimates on the moments of the random variable φ(X t ). In the recent article [14], the second author considered the case of a complete Riemannian manifold with L = 1 2 ∆ + Z where Z is a smooth vector field and ∆ the Laplace-Beltrami operator. The function φ was taken to be the square of the distance to either a fixed point or, more generally, a submanifold.…”

“…It would be desirable to find a general framework ecompassing all three settings mentioned above (namely, the distance to a submanifold, considered in [14], the sub-Riemannian distance, considered in Section 1 and the distance in an RCD * (K, N ) space, considered in Section 2). However, for the time being, no all-encompasing Itô formula nor comparison theorem is to be found in the literature.…”