Abstract
A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson–Mehl–Avrami–Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.
Reference
Abstract
This paper deals with $2\times 2$ conservation laws at a junction. For the Cauchy problem, existence, uniqueness, and Lipschitz continuous dependence of the solution from the initial data as well as from the conditions at the junction are proved. The present construction comprehends the case of the $p$-system used to describe gas flow in networks and hereby unifies different approaches present in the literature. Furthermore, different models for water networks are considered.