Wiener functionals associated with joint distributions of exit time and position from small geodesic balls

Abstract

Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.

Stochastic analysis in a tubular neighborhood or Onsager-Machlup functions revisited

“…In Takahashi-Watanabe [6], this part (and only this part) was proved by appealing to an analytical method, i.e., the singular perturbation technique, using which Fujita-Kotani [1] gave another whole proof. After these two proofs, the authors gave a purely probabilistic method by "smooth Besselization technique" [4], which was introduced in [3] by the first author. However, it lost the concrete study of the ergodic effect in return for simplicity.…”

“…The drift c(t, x) has the coordinate symmetry, but it has also the singularity at x = 0. On the other hand, in [4] (originated in [3]) we broke the symmetry to choose another Besselization drift, which is totally smooth without any singularity.…”

On the Onsager-Machlup functional for elliptic diffusion processes

“…All the O q > 0, which will appear in the developments of the functions of the form h(t, z) will be uniform in t. in [0, 1]. We will use the following developments (see [5] , [6] , [7] , [15] l;(t, z) = f;(t, 0) -03A6i(t) + £ ( §(t, °) -jR;;(t, °)) z; + 0 3 B 4 i = fi(t,0) -…”