Establish stochastic well-posedness under measure-of-noncompactness conditions

Establish well-posedness for a stochastic counterpart of the Volterra sweeping process in which the set-valued perturbation is replaced or augmented by a random forcing, under assumptions involving a measure of noncompactness.

Background

The paper treats uncertain forcing through a measurable, upper semicontinuous, convex-valued perturbation and proves deterministic existence under a measure-of-noncompactness condition. The applications to viscoelastic contact and fishery dynamics motivate extending the framework to genuinely random inputs.

The authors explicitly identify well-posedness for such a stochastic model under measure-of-noncompactness assumptions as unresolved. This would require adapting the existence and compactness arguments to random forcing while preserving the memory-dependent and state-constrained structure.