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<p>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 Schrödinger equation, revealing how quantum behavior can emerge from stochastic dynamics in fractal time.</p>

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Stochastic Foundations of the Fractal Schrödinger Equation

  • 期刊论文

  • 发表日期：

    2026年1月20日

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  • Alireza Khalili Golmankhaneh,
  • Elmira Isayeva

摘要

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 Schrödinger equation, revealing how quantum behavior can emerge from stochastic dynamics in fractal time.