Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function
Abstract
For one-dimensional diffusion processes with arbitrary diffusion coefficient D(x) and Ito force FI(x), we analyze the statistical properties of the renormalized trajectories at scale Γ, where one keeps only the alternating sequence of local minima and local maxima that are separated by drawups and drawdowns of amplitude bigger than Γ. The elementary RG-step between Γ and Γ+d Γ thus amounts to eliminate the drawups and drawdowns of amplitude Γ, a real-space-RG-procedure that is well-known for Brownian trajectories in the context of the Strong-Disorder-RG approach applied to the disorder landscape of random models. The main result is that the universality classes are determined by the scale function s(x) satisfying D(x) s''(x)+FI(x) s'(x)=0 that governs the exit probabilities from an interval : all the diffusions sharing the same scale function s(x) have the same statistics of renormalized trajectories, even if their initial microscopic trajectories are completely different. The simplest universality classes for diffusions on the whole line x ∈ ]−∞,+∞[ and on the half-line x ∈ ]0,+∞[ are illustrated with several examples of processes, with or without steady states: Cauchy, Student, Bessel, Fisher-Snedecor, Geometric Brownian. This framework concerning diffusion processes is then adapted to discrete-time and to continuous-time random walks on the one-dimensional lattice with arbitrary nearest-neighbor jump probabilities or rates.
Related
- Renormalization Group on Wiener Space: Spectral Theory and Universality
- Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions I: Self-energy Renormalization
- The Renormalization Group as a Stochastic Exploration Process
- A universality result for the critical 2d stochastic heat flow
- Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime
BibTeX
@misc{monthus2026realspace,
title = {{Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function}},
author = {Cecile Monthus},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04853},
eprint = {2611.04853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04853},
}
RIS
TY - UNPB
TI - Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function
AU - Cecile Monthus
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04853
UR - https://arxiv.org/abs/2611.04853
AB - For one-dimensional diffusion processes with arbitrary diffusion coefficient D(x) and Ito force F_I(x), we analyze the statistical properties of the renormalized trajectories at scale Γ, where one keeps only the alternating sequence of local minima and local maxima that are separated by drawups and drawdowns of amplitude bigger than Γ. The elementary RG-step between Γ and Γ+d Γ thus amounts to eliminate the drawups and drawdowns of amplitude Γ, a real-space-RG-procedure that is well-known for Brownian trajectories in the context of the Strong-Disorder-RG approach applied to the disorder landscape of random models. The main result is that the universality classes are determined by the scale function s(x) satisfying D(x) s''(x)+F_I(x) s'(x)=0 that governs the exit probabilities from an interval : all the diffusions sharing the same scale function s(x) have the same statistics of renormalized trajectories, even if their initial microscopic trajectories are completely different. The simplest universality classes for diffusions on the whole line x ∈ ]−∞,+∞[ and on the half-line x ∈ ]0,+∞[ are illustrated with several examples of processes, with or without steady states: Cauchy, Student, Bessel, Fisher-Snedecor, Geometric Brownian. This framework concerning diffusion processes is then adapted to discrete-time and to continuous-time random walks on the one-dimensional lattice with arbitrary nearest-neighbor jump probabilities or rates.
ER -