On Some Results of the Nonuniqueness of Solutions Obtained by the Feynman–Kac Formula

Abstract

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation.

Keywords

• Feynman–Kac formula
• Schrödinger equation
• partial differential equation
• Brownian motion
• uniqueness