On asymptotic independence of the exit moment and position from a small domain for diffusion processes (Q1856357)

Let \(\xi(t)\) be a diffusion process in \(\mathbb{R}^m\) and \(D\) be a bounded domain in \(\mathbb{R}^m\) such that \(0\in D\). Define \[ T_\varepsilon= \inf\{t> 0; \xi(t)\not\in D\}. \] The purpose of this paper is to study the behaviour of \[ I_\varepsilon(\lambda, f)= \mathbb{E}\Biggl(e^{-\lambda T_\varepsilon} f\Biggl({\xi(T_\varepsilon)\over \varepsilon}\Biggr)\Biggr)- \mathbb{E}(e^{-\lambda T_\varepsilon}) \mathbb{E}\Biggl(f\Biggl({\xi(T_\varepsilon)\over \varepsilon}\Biggr)\Biggr) \] when \(\varepsilon\to 0\), which measures the independence of the time and the place of exit from \(\varepsilon D\). This kind of problem was first arised by \textit{M. Liao} [Ann. Probab. 16, No. 3, 1039-1050 (1988; Zbl 0651.58037)].