A stochastic porous media Schrödinger equation: Feynman-type motivation, well-posedness and control interpretation

This paper’s aim is threefold. First, using Feynman’s path approach to the derivation of the classical Schrödinger’s equation in Feynman (Rev Mod Phys 20:367-387, 1948) and by introducing a slight path (or wave) dependency of the action, we derive a new class of equations of Schrödinger type where the driving operator is no longer the Laplace one but rather of complex porous media type. Second, using suitable concepts of monotonicity in the complex setting and on appropriate functional spaces, we show the existence and uniqueness of the solution to this type of equation. In the formulation of our equation, we adjoin possible measurement absolute errors translating in an additive Brownian perturbation and interactions between different waves translating in a mean-field (or McKean–Vlasov) dependency of drift coefficient. Finally, using Fitzpatrick’s characterization of maximal monotone operators (Schrödinger Phys Rev 28:1049-1070, 1926) we propose a Brézis–Ekeland-type characterization of the solution of the deterministic equation via a control problem. This is envisaged as a possible way to overcome strict monotonicity requirements in the complex setting.