On asymptotic independence of the exit moment and position from a small domain for diffusion processes

Abstract

If ξ(t) is the solution of homogeneous SDE in R ^m, and T _∃ is the first exit moment of the process from a small domain D _∃, then the total expansion for the following functional showing independence of the exit time and exit place is

$$Eexp( - \lambda T_\varepsilon )f(\frac{{\xi (T_\varepsilon )}}{\varepsilon }) - Eexp( - \lambda T_\varepsilon )Ef(\frac{{\xi (T_\varepsilon )}}{\varepsilon }),\varepsilon \searrow 0,\lambda > 0.$$