An analog of Andre's reflection principle for diffusion processes with a small parameter (Q1603164)

This paper deals with the \(\mathbb{R}^N\)-valued diffusion process \((X^h_t,P_x)\) with uniformly parabolic infinitesimal generator \[ L_{h,x}= {h\over 2}\sum^N_{i,j=1} a_{ij}(x){\partial^2 \over\partial x_i\partial x_j}, \tag{1} \] where the coefficients \(a_{ij}(x)\) \((x\in\mathbb{R}^N)\) are smooth functions and \(h\in(0,1]\). The author presents a formula expressing the asymptotics of the solution of the first boundary value problem (with zero initial condition) in a bounded domain \(\Omega\) via the asymptotics of the solution of the Cauchy problem in the exterior of \(\Omega\).