A Mini-Max Variational Formula Giving Necessary and Sufficient Conditions for Recurrence or Transience of Multidimensional Diffusion Processes

Abstract

Let $L = \frac{1}{2} \nabla \cdot a\nabla + b \cdot \nabla$ generate a diffusion process on $R^d$. An expression involving $a$ and $b$ on $1 \leq |x| \leq n$ and two functions $g$ and $h$, varied over suitable domains, attains its mini-max value at $\lambda_n$. It is shown that $\lim_{n\rightarrow\infty}\lambda_n = 0$ or $\lim_{n\rightarrow\infty} \lambda_n > 0$ according to whether the process is recurrent or transient.