A Stochastic Porous Media Schr{\"o}dinger Equation: Feynman-type Motivation, Well-Posedness and Control Interpretation

TL;DR

This paper introduces a new class of Schr{"o}dinger-type equations derived from Feynman's path approach, establishing their well-posedness and linking solutions to control problems, with applications to complex porous media operators.

Contribution

It derives a novel Schr{"o}dinger-type equation with complex porous media operators and provides existence, uniqueness, and a control-based characterization of solutions.

Findings

Established well-posedness of the new equation.

Linked solutions to a control problem using maximal monotone operator theory.

Extended the Schr{"o}dinger equation framework to complex porous media settings.

Abstract

This paper's aim is threefold. First, using Feynman's path approach to the derivation of theclassical Schr{\"o}dinger's equation in [6] and by introducing a slight path (or wave) dependency ofthe action, we derive a new class of equations of Schr{\"o}dinger type where the driving operatoris no longer the Laplace one but rather of complex porous media-type. Second, using suitableconcepts of monotonicity in the complex setting and on appropriate functional spaces, we showthe existence and uniqueness of the solution to this type of equation. In the formulation of ourequation, we adjoin possible measurement absolute errors translating in an additive Brownianperturbation and interactions between different waves translating in a mean-field (or McKean-Vlasov) dependency of drift coefficient. Finally, using Fitzpatrick's characterization of maximalmonotone operators (cf. [7]), we propose a Br{\'e}zis-Ekeland type characterization of the solutionof the deterministic equation via a control problem. This is envisaged as a possible way toovercome strict monotonicity requirements in the complex setting.