| Abstract: |
| This papers aim is threefold. First, using Feynmans path approach to the derivation of the
classical Schrodingers equation in Feynman (Rev Mod Phys, 1948) and by introducing a slight path (or wave) dependency of the action, we derive a new class of equations of Schrodinger 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 Fitzpatricks characterization of maximal monotone operators (Schrodinger Phys Rev, 1926) we propose a Brezis-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. |
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