Stochastic Foundations of the Fractal Schrödinger Equation

Khalili Golmankhaneh A., Isayeva E.

International Journal of Theoretical Physics, vol.65, no.2, 2026 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 65 Issue: 2
  • Publication Date: 2026
  • Doi Number: 10.1007/s10773-026-06250-3
  • Journal Name: International Journal of Theoretical Physics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, MathSciNet, zbMATH
  • Keywords: Fractal calculus, Fractal Nelson’s stochastic mechanics, Fractal Schrödinger equation, Fractal stochastic differential equations
  • Van Yüzüncü Yıl University Affiliated: Yes

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