Random Set Solutions to Stochastic Wave Equations

TL;DR

This paper develops a framework for analyzing solutions to stochastic wave equations as random sets, proving measurability in complex spaces, extending to arbitrary dimensions, and computing probability bounds in one dimension.

Contribution

It introduces a measurability theorem for multifunctions in non-metrizable spaces and applies it to stochastic wave equations across various dimensions.

Findings

Solutions to stochastic wave equations can be characterized as random sets.

Measurability of solutions is established in non-metrizable spaces.

Upper and lower probabilities of solutions are computed in one-dimensional cases.

Abstract

This paper is devoted to three topics. First, proving a measurability theorem for multifunctions with values in non-metrizable spaces, which is required to show that solutions to stochastic wave equations with interval parameters are random sets; second, to apply the theorem to wave equations in arbitrary space dimensions; and third, to computing upper and lower probabilities of the values of the solution in the case of one space dimension.