Stochastic Foundations of the Fractal Schrödinger Equation

Abstract

In this paper, we extend Nelson's stochastic mechanics to a fractal time framework using fractal calculus. By modeling quantum particle motion with fractal stochastic differential equations, we define forward, backward, and osmotic velocities and derive the fractal Fokker-Planck equation. Introducing a fractal complex wave function, we rigorously derive the fractal Schr & ouml;dinger equation, revealing how quantum behavior can emerge from stochastic dynamics in fractal time.