Diffusions, exit time moments and Weierstrass theorems

Let $X_t$ be a one-dimensional diffusion with infinitesimal generator given by the operator $L = \frac12 (a(x) \frac{d}{dx})^2 + b(x) \frac{d}{dx}$ where $a(x)$ is a smooth, positive real-valued function and the ratio of $a(x)$ and $b(x)$ is a constant. Given a compact interval, we prove a Weierstrass-type theorem for the exit time moments of $X_t$ and their corresponding (naturally weighted) first derivatives, and we provide an algorithm that produces uniform approximations of arbitrary continuous functions by exit time moments. We investigate analogues of these results in higher-dimensional Euclidean spaces. We give expansions for several families of special functions in terms of exit time moments.