The Triality of Radial Nonlinear Dynamics: Analysis of Riccati, Schr\"{o}dinger, and Hamilton--Jacobi--Bellman Equations

TL;DR

This paper introduces a unified mathematical framework linking Riccati, Schrödinger, and Hamilton--Jacobi--Bellman equations, providing new insights into their solutions, asymptotics, and stochastic control applications.

Contribution

It establishes the existence, uniqueness, and asymptotic behavior of solutions across these equations, revealing a fundamental triality and duality in stochastic dynamics.

Findings

Validated the theoretical growth rates and asymptotic plateaus through numerical simulations.

Identified the transition between deterministic and diffusion regimes via sensitivity analysis.

Confirmed the stability and structure of feedback laws in stochastic control scenarios.

Abstract

This study develops a unified mathematical framework for the analysis of radial differential equations, revealing a fundamental connection between three distinct classes of problems: the nonlinear Riccati equation, the linear Schr\"odinger equation, and the Hamilton--Jacobi--Bellman equation for stochastic control. We establish the existence and uniqueness of regular solutions on both bounded and unbounded domains, deriving sharp growth rates and exact asymptotic plateaus through a general barrier theory. A detailed sensitivity analysis of the noise intensity parameter identifies the transition between deterministic and diffusion-dominated regimes via singular perturbation methods. These theoretical results are reinforced by numerical simulations that validate the predicted feedback laws, confirm the convexity--concavity structure of the triality, and illustrate the stability of the system. The resulting framework clarifies the duality between global wave functions and local dynamical drifts, providing a rigorous foundation for the study of multidimensional stochastic processes under central potentials.