稳态对流扩散方程解的概率表示

Representation of Solutions for Steady state Convection diffusion Equation

给出了有界区域Ｄ 上的稳态对流扩散方程Δｕ／２ ＋ ｂ ·ｕ ＝ ０ 的解ｕ 可表成如下形式ｕ（ｘ）＝ Ｅｘ φ（Ｂτ）ｅｘｐ （∫τ０ｂ（Ｂｓ）ｄＢｓ－ １２∫τ０｜ｂ（Ｂｓ）｜２ｄｓ） ｘ∈Ｄ的两个等价条件（这里Ｂｔ 是Ｂｒｏｗ ｎ 运动，τ是Ｄ的首出时，φ是Ｄ上可测函数）。证明了稳态对流扩散方程随机Ｄｉｒｉｃｈｌｅｔ 问题解的存在唯一性。

In this paper,two equivalent conditions for a solution u in a domain D of the steady state convection diffusion equation Δu/2+b·u=0 to be expressed as follow: u(x)=E xφ(B τ) exp (∫ τ 0b(B s) d B s-12∫ τ 0｜b(B s)｜ 2 d s) x∈D (where B τ is a Brownian motion, τ is the exit time of D , φ is a measurable function on D ) are given. The exist and uniqueness of solution of the stochastic problem for the equation is proved.