Применение стохастических уравнений к изучению второй краевой задачи для параболических дифференциальных уравнений с малым параметром

A. Ya. Kogan

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 2, pp. 301-309

Résumé

In this paper the investigation of the limit behaviour of the solution of the boundary value problem on ring $D$ with boundaries $s$ and $S$ \begin{gather*} \nu(\varepsilon)\frac{\partial v^\varepsilon}{\partial t}=\frac12\biggl[a_{11}(r,\varphi)\frac{\partial^2v^\varepsilon}{\partial r^2}+2a_{12}(r,\varphi)\frac{\partial^2v^\varepsilon}{\partial r\partial\varphi}+a_{22}(r,\varphi)\frac{\partial^2v^\varepsilon}{\partial\varphi^2}\biggr]+ \\ +b_1(r,\varphi)\frac{\partial v^\varepsilon}{\partial r}+b_2(r,\varphi)\frac{\partial v^\varepsilon}{\partial\varphi}+\frac1{\varepsilon^2}\biggl[B_1(r,\varphi)\frac{\partial v^\varepsilon}{\partial r}+B_2(r,\varphi)\frac{\partial v^\varepsilon}{\partial\varphi}\biggr], \\ (r,\varphi)\in D,\quad t>0;\quad v^\varepsilon(0,r,\varphi)=f(r,\varphi),\quad\frac{\partial v^\varepsilon}{\partial r}\biggr|_S=\frac{\partial v^\varepsilon}{\partial r}\biggr|_s=0 \end{gather*} ($r,\varphi$ are polar coordinates) when $\varepsilon\to0$ and $B_1(r,\varphi)>\theta>0$ is reduced to investigating the limit behaviour of the trajectories of the corresponding Markov diffusion process. This enables us to get the results using the probability methods.