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Equations driven by fast-oscillating functions of an Itô diffusion process

Tanner Reese, Jan Wehr

TL;DR

We address the weak convergence of Itô SDEs driven by fast-oscillating functions of an underlying diffusion. By analyzing three scaling regimes—amplitude scaling, time scaling, and integrated-noise time scaling—we show that the limits are Stratonovich SDEs driven by a multivariate Wiener process, with covariance matrices given by Gram-type bilinear forms determined either by the periodic forcing (in the amplitude case) or by the invariant measure or integrated-noise structure of the driving process. The proofs rely on convergence of generators and the construction of correctors, with ergodicity-based and spectral methods providing explicit covariance structures. Applied examples to phototactic robots and motility-induced phase separation illustrate the practical relevance and show how the limiting dynamics capture averaged drift corrections and stochastic fluctuations in physically motivated systems.

Abstract

We study Itô SDE systems driven by oscillating functions of a single Itô diffusion process. In the limit when oscillations become fast, we show that the solution process converges in law to the process defined by an SDE system driven by a multivariate Wiener process whose covariance we calculate explicitly. Interestingly, the limiting system of SDEs are most naturally stated using the Stratonovich integral. The problem has been originally motivated by experimental work and special cases of theorems proved here provide a rigorous treatment of equations arising from physics.

Equations driven by fast-oscillating functions of an Itô diffusion process

TL;DR

We address the weak convergence of Itô SDEs driven by fast-oscillating functions of an underlying diffusion. By analyzing three scaling regimes—amplitude scaling, time scaling, and integrated-noise time scaling—we show that the limits are Stratonovich SDEs driven by a multivariate Wiener process, with covariance matrices given by Gram-type bilinear forms determined either by the periodic forcing (in the amplitude case) or by the invariant measure or integrated-noise structure of the driving process. The proofs rely on convergence of generators and the construction of correctors, with ergodicity-based and spectral methods providing explicit covariance structures. Applied examples to phototactic robots and motility-induced phase separation illustrate the practical relevance and show how the limiting dynamics capture averaged drift corrections and stochastic fluctuations in physically motivated systems.

Abstract

We study Itô SDE systems driven by oscillating functions of a single Itô diffusion process. In the limit when oscillations become fast, we show that the solution process converges in law to the process defined by an SDE system driven by a multivariate Wiener process whose covariance we calculate explicitly. Interestingly, the limiting system of SDEs are most naturally stated using the Stratonovich integral. The problem has been originally motivated by experimental work and special cases of theorems proved here provide a rigorous treatment of equations arising from physics.
Paper Structure (13 sections, 8 theorems, 127 equations)

This paper contains 13 sections, 8 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Amplitude Scaling
  4. Time Scaling
  5. Non-Compact Time Scaling using Ergodicity
  6. Time Scaling driven by Integrated Noise
  7. Examples
  8. Motion of a Phototactic Robot
  9. Motility Induced Phase Separation
  10. Proofs
  11. Amplitude Scaling
  12. Non-Compact Time Scaling by Ergodicity
  13. Time Scaling driven by Integrated Noise

Key Result

Theorem 1

Suppose $b$, $\{\phi_\alpha\}_{\alpha=1}^{d_B}$, and $\{\}$ are locally bounded and Borel measurable. Additionally, $\theta \mapsto b(x, \theta)$ and $\phi_\alpha$ are $P$-periodic with $\int_0^P \phi_\alpha(\theta) \; \mathrm{d}\theta = 0$. For every $x \in \mathbb{R}^{d_X}$ and $m \in \mathbb{R}^{ and we require that with probability one $\kappa(M(t)) > 0$ for all $t > 0$. It follows from the le

Theorems & Definitions (18)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • Definition 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Proposition 8
  • Lemma 9
  • proof
  • ...and 8 more