Euclidean Quantum Mechanics and Universal Nonlinear Filtering

TL;DR

This paper reveals a deep connection between Euclidean quantum mechanics and the continuous nonlinear filtering problem, using path integral representations and Schrödinger equations to provide new insights into estimation of Langevin states.

Contribution

It establishes a novel theoretical link between quantum mechanics and nonlinear filtering, specifically relating the Yau Equation to a Schrödinger equation framework.

Findings

Path integral representation of the Yau Equation

Equivalence between nonlinear filtering and a time-varying Schrödinger equation

New theoretical insights into continuous nonlinear filtering

Abstract

An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr\"odinger equation.