Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and to establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the $M$-th moment of a solution to a switching PDE corresponds to exit statistics for $M$ particles following a switching SDE. We note that though the particles are non-interacting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynamics of these systems.
Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
Universität Potsdam（波茨坦大学）；Freie Universität Berlin（柏林自由大学）；Brandenburgische Technische Universität Cottbus-Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格）
A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers
本文将随机常微分方程的扩散逼近理论推广到非线性随机薛定谔方程。该系统是在研究随机双折射光纤中脉冲传输时产生的。我们首先证明了随机偏微分方程和极限方程解的存在唯一性。我们遵循Garnier和Marty [Wave Motion 43（2006）544-560]，Marty [Problèmes d 'évolution en milieux aléatoires：Théorèmes limites，schémas numériques et applications en optique（2005）Univ. Paul Sabatier]的工作，其中考虑线性电场，并且我们得到非线性电场的渐近动力学。
