We consider stochastic reaction–diffusion equations on a finite network represented by a finite graph. On each edge in the graph, a multiplicative cylindrical Gaussian noise-driven reaction–diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so, we generalize existing results on abstract stochastic reaction–diffusion equations in Banach spaces.

我们考虑由有限图表示的有限网络上的随机反应扩散方程。在图中的每条边上，给出了乘法圆柱高斯噪声驱动的反应扩散方程，并辅以动态基尔霍夫型定律，该定律受顶点中乘法标量高斯噪声的干扰。假设每条边上的反应项是奇次多项式，不一定每条边上的次数都相同，可能具有随机系数和负前导项。我们利用巴拿赫空间中的随机演化方程的半群方法来获得图上连续函数空间中具有样本路径的解的存在性和唯一性。为了做到这一点，我们概括了巴拿赫空间中抽象随机反应扩散方程的现有结果。