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Stationary properties of the Gauss-Galerkin QMoM truncation of MV-SDEs

Alexander Alecio

TL;DR

This work investigates stationary properties of the Gauss–Galerkin QMOM (GG-QMoM) closure for McKean–Vlasov SDEs with polynomial drift and diffusion. It recasts the infinite moment hierarchy into an $n$-point mass approximation, enforcing positivity via a Hankel–PSD constraint and proving convergence of the $n$-approximant $\tilde\rho_n$ to the true law as $n$ grows. The paper analyzes stationary solutions of the closed GG-QMoM equations, revealing a scaling property with diffusion strength and showing that, in the deterministic limit, stationary states align with extrema of the effective potential, not only its minima, thereby preserving the full bifurcation structure. The Dawson–Shiino bistable example is used to illustrate these properties, demonstrating symmetry preservation, stability changes, and the concentration of stationary measures at drift roots as noise vanishes. Overall, GG-QMoM provides a robust, positivity-preserving, closure approach that accurately captures stationary behavior and bifurcations in MV-SDEs.

Abstract

The Gauss Galerkin Method/Quadrature method of moments (GG-QMoM) closure scheme, introduced by Dawson, closes a truncated set of moment equations of an SDE by a Galerkin approximation of its law in the space of probability measures. Here, results are presented on stationary solutions of the closed equations, irrespective of the number of moments retained (hence not dependent on the convergence theorem). These are applied to polynomial MV-SDEs, with explicit dependence on its moments in the drift, which can possess multiple stationary solutions. Particularly, we show in the deterministic limit, there are as many stationary solutions as extrema of the potential (critically, not just the minima), preserving the bifurcation diagram of the full equations. Further, a scaling property of solutions is proven, allowing changes of stability to be directly probed. Finally, this is applied to the GG-QMoM truncation of Dawson-Shiino model, whose stationary measures and change in stability has been previously studied.

Stationary properties of the Gauss-Galerkin QMoM truncation of MV-SDEs

TL;DR

This work investigates stationary properties of the Gauss–Galerkin QMOM (GG-QMoM) closure for McKean–Vlasov SDEs with polynomial drift and diffusion. It recasts the infinite moment hierarchy into an -point mass approximation, enforcing positivity via a Hankel–PSD constraint and proving convergence of the -approximant to the true law as grows. The paper analyzes stationary solutions of the closed GG-QMoM equations, revealing a scaling property with diffusion strength and showing that, in the deterministic limit, stationary states align with extrema of the effective potential, not only its minima, thereby preserving the full bifurcation structure. The Dawson–Shiino bistable example is used to illustrate these properties, demonstrating symmetry preservation, stability changes, and the concentration of stationary measures at drift roots as noise vanishes. Overall, GG-QMoM provides a robust, positivity-preserving, closure approach that accurately captures stationary behavior and bifurcations in MV-SDEs.

Abstract

The Gauss Galerkin Method/Quadrature method of moments (GG-QMoM) closure scheme, introduced by Dawson, closes a truncated set of moment equations of an SDE by a Galerkin approximation of its law in the space of probability measures. Here, results are presented on stationary solutions of the closed equations, irrespective of the number of moments retained (hence not dependent on the convergence theorem). These are applied to polynomial MV-SDEs, with explicit dependence on its moments in the drift, which can possess multiple stationary solutions. Particularly, we show in the deterministic limit, there are as many stationary solutions as extrema of the potential (critically, not just the minima), preserving the bifurcation diagram of the full equations. Further, a scaling property of solutions is proven, allowing changes of stability to be directly probed. Finally, this is applied to the GG-QMoM truncation of Dawson-Shiino model, whose stationary measures and change in stability has been previously studied.
Paper Structure (5 sections, 18 theorems, 42 equations, 2 figures)

This paper contains 5 sections, 18 theorems, 42 equations, 2 figures.

Table of Contents

  1. MV-SDEs and Outline
  2. Properties of GG-QMoM
  3. Example: Dawson-Shiino Model - Bistable Potential with Quadratic Interaction
  4. Langrange Interpolating Polynomials
  5. A Priori Estimates for MV-SDE (\ref{['sdep']})

Key Result

Theorem 1

Suppose there exists constants $\{k_n:n\geq1\}$ and $\theta_0>0$ such that for the MEEs associated to process $X$. Further, for suitable initial probability measure suppose process $X$ possesses a unique law. Then $\tilde{\rho}_n(t)\Rightarrow\rho(t)$ as $N\rightarrow\infty$, where $\rho_t=\mathrm{Law}(X_t)$, in the sense of weak convergence of probability measures.

Figures (2)

  • Figure 1: Gauss Galerkin Moment Closure for (\ref{['sdep']}): Initialised with $\mathcal{N}(-0.1,0.1)$, $n=8$ (and changing $y$-axis scale). The equilibrium $m_1$ is approaching zero and there is a change in stability between $\sigma=0.9$ and $1.25$. The critical temperature $\sigma_c$ was determined to be $\sim0.97$
  • Figure 2: Gauss Galerkin Moment Closure for (\ref{['sdep']}): Initialised with $\exp{(-2(x^4-0.1x))}$$n=8$$\sigma=0.5$ (left), $\sigma=1.25$ (right)

Theorems & Definitions (38)

  • Theorem 1: Convergence of GG-QMoM approximates: Campillo, Dawson CampDA
  • proof
  • Definition 1.1: Gauss Galerkin Equations
  • Lemma 1.2: Hajjafar Hajj
  • proof
  • Proposition 1.3: Ordering of $x_i$
  • Remark 1.4
  • Proposition 1.5: Parity of $\tilde{\rho}_n$
  • proof
  • Theorem 1.6: Limit of Stationary Solutions, $\sigma\downarrow 0$
  • ...and 28 more