We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence.
We then apply this coupling to resolve two conjectures of Darling and Erd\H{o}s (1956).
Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.]

Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the
classical law of the iterated logarithm of Khintchine for Brownian motion (1933).
[This portion is joint work with David Levin and Zhan Shi, and has recently appeared in
the Electr. Comm. of Probab. (2005)]

We study the asymptotic shift for principal eigenvalue for a
large class of second order elliptic operators on bounded domains subject
to perturbations known as obstacles. The results extend the well-studied
self-adjoint case. The approach is probabilistic.

We study the regularity for the tangential Cauchy-Riemann equations and the associated Laplacian on CR manifolds with minimal smoothness assumption. One application is to extend the embedding theorem of Boutet De Monvel
to strongly pseudoconvex CR manifolds of class C^2.

(Joint work with Lihe Wang).