Part V: Stochastic Dynamics
From microscopic noise to macroscopic probability evolution — the Langevin-Fokker-Planck formalism applied to cities, culminating in Schrödinger Bridges for optimal urban transport.
Part Overview
From microscopic noise to macroscopic probability evolution — the Langevin-Fokker-Planck formalism applied to cities, culminating in Schrödinger Bridges. The Langevin equation is derived from jump processes, compared across Itô and Stratonovich conventions, then lifted to the Fokker-Planck PDE solved numerically via the Chang-Cooper scheme. The Schrödinger Bridge problem and Sinkhorn algorithm connect to Nelson stochastic mechanics.
Key Topics
- • Langevin from jump processes
- • Itô vs Stratonovich
- • Fokker-Planck equation
- • Chang-Cooper scheme
- • Schrödinger Bridge
- • Sinkhorn algorithm
- • Nelson stochastic mechanics
3 chapters | Noise-driven urban evolution | From Langevin to optimal transport
Chapters
Chapter 1: Langevin Equation
Deriving the Langevin stochastic differential equation from microscopic jump processes. Itô vs Stratonovich interpretations, drift and diffusion coefficients, and the connection to urban noise sources.
Chapter 2: Fokker-Planck Equation
The probability density evolution equation dual to the Langevin SDE. Stationary solutions, detailed balance, and the Chang-Cooper numerical scheme that preserves positivity and conservation.
Chapter 3: Schrödinger Bridge
The Schrödinger Bridge problem finds the most likely stochastic evolution between two observed distributions. The Sinkhorn algorithm for entropic optimal transport and Nelson stochastic mechanics as a physical interpretation.