Stationary-Action Stochastic Control Representation of the Schrödinger Initial Value Problem
Ruobing Zhao
Hamilton-Jacobi partial differential equations (HJ PDEs) arise in many scientific fields and applications, especially in mechanics and optimal control. Solutions to second order Hamilton-Jacobi partial differential equations (HJ PDEs) have controlled diffusion process representations. Of particular interest is the diffusion representation of an action functional associated with the solution of Schrödinger initial value problems (IVPs). In existing work that connects stochastic control problems to Schrödinger IVPs, one searches for the minimum of an action functional which is the payoff of the control problem. Issues arise, however, as the action functional loses convexity over longer time durations. The time duration where such representation is valid is in fact infinitesimal when the system dimension is infinite. In this dissertation, we present an approach inspired by the Principle of Stationary Action in physics that removes this limitation, which leads to exceptional computational benefits. Instead of searching for local minima of the action functional as in traditional optimal control problems, we look for its stationary values instead. We introduce the ``staticization" operator and stationary-action control problems. A stationary-action stochastic control representation for the dequantized Schrödinger IVP in a non-inertial frame where the potential field is a polynomial is given. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations. Following this, the staticization operator is studied in detail. A new approach to solving conservative dynamical systems (e.g. paths of objects in gravitational or Coulomb potential field) using stationary-action control-theoretic methods with promising computational benefits is introduced. Two examples of application, the N-body problem and the Schrödinger IVP, are given. Lastly, we demonstrate the existence of strong solutions of a class of degenerate stochastic differential equations (SDEs) that arises in the stationary-action stochastic control representation of Schrödinger IVPs with the Coulomb potential, which is non-smooth and has branch cuts in the complex range. The SDEs we consider have drift terms that have discontinuities and singularities along some manifolds, and diffusion coefficients that are degenerate, and there is no previously existing results regarding the existence of strong solutions for such SDEs.
•Jan 1, 2019•0 citations