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Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics

Theo Diamantakis, James Woodfield

TL;DR

The paper addresses how noise interpretation, specifically Wong–Zakai anomalies and Lévy-area corrections, impacts deterministic Lagrangian multi-time dynamics in 2D fluid models. It develops a SALT-based, homogenisation-informed framework and uses rough-path theory to identify drift terms that arise from fast-slow interactions, illustrating these with stochastic point-vortex dynamics. The results show that Wong–Zakai anomalies and higher-order Lévy-area terms can destroy invariants such as area and angle, whereas Stratonovich noise often better preserves the geometric structure, guiding the choice of stochastic modelling for SALTalgo in geophysical applications. The findings highlight the practical importance of obstacle-free area/angle conservation in stochastic fluid simulations and inform the design of numerically stable, physically faithful discretisations.

Abstract

Stochastic modelling necessitates an interpretation of noise. In this paper, we describe the loss of deterministically stable behaviour in a fundamental fluid mechanics problem, conditional to whether noise is introduced in the sense of Itô, Stratonovich or a limit of Wong-Zakai type. We examine this comparison in the wider context of discretising stochastic differential equations with and without the Lévy area. From the numerical viewpoint, we demonstrate performing higher order discretisations with the use of a Lévy area can lead to the loss of conserved area and angle quantities. Such behaviour is not physically expected in the Stratonovich model. Conversely, we study Stochastic Advection by Lie Transport and its derivation from homogenisation theory, which introduces drift corrections of the same class naturally. From the viewpoint of homogenisation, the qualitative properties of the Wong-Zakai anomaly are physically motivated as arising due to correlations from a fast and mean scale fluid decomposition.

Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics

TL;DR

The paper addresses how noise interpretation, specifically Wong–Zakai anomalies and Lévy-area corrections, impacts deterministic Lagrangian multi-time dynamics in 2D fluid models. It develops a SALT-based, homogenisation-informed framework and uses rough-path theory to identify drift terms that arise from fast-slow interactions, illustrating these with stochastic point-vortex dynamics. The results show that Wong–Zakai anomalies and higher-order Lévy-area terms can destroy invariants such as area and angle, whereas Stratonovich noise often better preserves the geometric structure, guiding the choice of stochastic modelling for SALTalgo in geophysical applications. The findings highlight the practical importance of obstacle-free area/angle conservation in stochastic fluid simulations and inform the design of numerically stable, physically faithful discretisations.

Abstract

Stochastic modelling necessitates an interpretation of noise. In this paper, we describe the loss of deterministically stable behaviour in a fundamental fluid mechanics problem, conditional to whether noise is introduced in the sense of Itô, Stratonovich or a limit of Wong-Zakai type. We examine this comparison in the wider context of discretising stochastic differential equations with and without the Lévy area. From the numerical viewpoint, we demonstrate performing higher order discretisations with the use of a Lévy area can lead to the loss of conserved area and angle quantities. Such behaviour is not physically expected in the Stratonovich model. Conversely, we study Stochastic Advection by Lie Transport and its derivation from homogenisation theory, which introduces drift corrections of the same class naturally. From the viewpoint of homogenisation, the qualitative properties of the Wong-Zakai anomaly are physically motivated as arising due to correlations from a fast and mean scale fluid decomposition.
Paper Structure (27 sections, 3 theorems, 155 equations, 11 figures)

This paper contains 27 sections, 3 theorems, 155 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Outline and objectives of this paper.
  3. Stochastic models of ideal fluids
  4. Background for SALT
  5. Example: Stochastic Incompressible Euler Equations
  6. Homogenisation of deterministic geometric fluid dynamics
  7. Rough paths theory intepretation of covariance and Lévy area
  8. The Wong-Zakai anomaly for 2D incompressible flows
  9. Stochastic Point Vortex Dynamics in Two Dimensions
  10. Deterministic theory and conserved quantities
  11. Case study : Stochastically stable and unstable configurations of three unit strength point vortices
  12. Stable Stratonovich configurations
  13. Pseudostable Itô configurations
  14. Unstable Wong-Zakai anomaly configuration
  15. Wong-Zakai anomaly from a pure area process
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\mathcal{Y} = M$ any smooth manifold of finite dimension with $\phi_t: M \rightarrow M$ a smooth flow. Let $\Lambda \subset M$ be a hyperbolic basic set of the dynamics Recall a $\phi_t$ invariant set $\Lambda$ is called a hyperbolic basic set if $T_x M$ splits into stable and unstable spaces i Then the following hold: The operator $\mathfrak{B}: C^\kappa_\nu(\Lambda) \times C^\kappa_\nu(\Lam

Figures (11)

  • Figure 1: The radial vector field associated to noise that induces rotation around $\boldsymbol x_c = 0$ times a multiple of $\mathrm{d}W^1_t$ to the velocity field. $r = 1, A = \frac{1}{2}$, giving $\widehat{\boldsymbol \xi}_1 = (y, -x) \frac{1}{2}\exp(-\frac{1}{2}\|\boldsymbol x\|)$.
  • Figure 2: The constant vector field associated to noise that translates in a multiple of $\mathrm{d}W^2_t$ by the direction $\boldsymbol \xi_2 = (-b,a) = (a,b)^\perp$ shown for $a=1, b=-1$.
  • Figure 3: The subtraction of the Itô-Stratonovich correction drift term \ref{['fig:Itô-stratonovich']} (relative to the Stratonovich noise \ref{['stratparticle']}) caused the Itô point vortices to remain circular but spread out into a larger circle. The appearance of the Wong-Zakai drift field \ref{['fig:Wong_Zakai drift Lie Braket']} and the location of the point vortices within it, caused additional dynamical behaviour.
  • Figure 4: The area weighted histograms of particle trajectories of a 100 member ensemble over the time interval $[0,40]$. Values are binned into an array of size $1024 \times 1024$ and the number of occurrences are used to compute the area weighted density.
  • Figure 5: Area between the three point vortices as time evolves, for a 100 member ensemble, for \ref{['Method:type II']}, \ref{['Method:Deterministic']}, \ref{['Method:Stratonovich integration scheme']}, \ref{['Method:NLA']}, \ref{['Method:type ito']}, \ref{['Method:type I']}, and \ref{['Method:FBM']} respectively. We plot the traced paths of ensemble member area in black, also plotted is the mean, the standard deviation, and twice the standard deviation. Data is not necessarily normal and the variance doesn't necessarily provide accurate representation of the ensemble statistics.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Remark A.1
  • Remark A.1
  • Remark A.2
  • Theorem : Deterministic homogenisation theorem. Kelly, Melbourne 2017 kelly2017chaos
  • Theorem : Iterated WIP. Kelly, Melbourne 2016 kelly2016
  • Remark A.1
  • Theorem : Scalar invariance principle for the fractional OU process. Gehringer, Li 2022 Gehringer2022
  • Remark A.2
  • Remark A.3