A Stochastic Porous Media Schrödinger Equation: Feynman-type Motivation, Well-Posedness and Control Interpretation
Résumé
This paper’s aim is threefold. First, using Feynman’s path approach to the derivation of the classical Schrödinger’s equation in [6] 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 (cf. [7]), 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.
Mots clés
- MSC2020: 35J10
- 81-10
- 76S99
- secondary 47H05 49J20 Classification: 40.09 Mathematical physics 40.12 Non-linear PDE 60.07 Quantum dynamics
- control problem
- Fitzpatrick's characterization of complex maximal monotone operators
- complex stochastic porous media equation
- Feynman's approach
- equations of Schrödinger type
- 60.07 Quantum dynamics
- 40.12 Non-linear PDE
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