In this paper, we deal with the reaction-diffusion equation subject to Dirichlet and Neumann boundary conditions where the input function on the boundary is randomly fluctuated. First we study the fundamental case when this function is a white noise. Explicit form of the correlation function is derived for the reaction-diffusion equation in a half-plane. In this case we obtain the Karhunen–Loève expansion (KL) of the solution which is a partially homogeneous random field, i.e., it is homogeneous along the horizontal direction, and is inhomogeneous in the vertical direction. Then, based on this representation, we extend this result to the general case when the function prescribed on the boundary is an arbitrary homogeneous random field.

在本文中，我们处理受狄利克雷和诺依曼边界条件影响的反应扩散方程，其中边界上的输入函数是随机波动的。首先我们研究该函数是白噪声时的基本情况。针对半平面中的反应扩散方程导出了相关函数的显式形式。在这种情况下，我们获得解的 Karhunen-Loève 展开式 (KL)，它是部分均匀随机场，即沿水平方向均匀，在垂直方向不均匀。然后，基于这种表示，我们将这个结果扩展到当边界上规定的函数是任意齐次随机场时的一般情况。