Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean--Vlasov SDEs and associated particle systems

TL;DR

The paper investigates a broad class of McKean–Vlasov SDEs with fully nonlinear, convolution-based drift and diffusion that exhibit super-linear growth in both space and law. A novel additional symmetry condition enables rigorous L^p moment bounds and enables wellposedness and propagation of chaos results, while a separate dissipativity framework yields exponential ergodicity and the existence of an invariant distribution. The authors introduce a particle-system Euler-type split-step method (SSM) tailored to these nonlinear features, proving C-stability and B-consistency with strong convergence order 1/2, plus mean-square contractivity for long-time simulations. Numerical experiments illustrate PoC-rate behavior across dimensions, invariant distribution estimation, and the preservation of phase-space geometry, highlighting SSM’s robustness relative to taming schemes. Overall, the work provides a unified treatment of wellposedness, long-time behavior, and efficient numerical approximation for MV-SDEs with super-linear, non-Lipschitz measure interactions, with implications for high-dimensional mean-field modeling and inference.

Abstract

We study a class of McKean--Vlasov Stochastic Differential Equations (MV-SDEs) with drifts and diffusions having super-linear growth in measure and space -- the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift's super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient requires novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC), and under further assumptions on the model parameters, we show an exponential ergodicity property alongside the existence of an invariant distribution. No differentiability or non-degeneracy conditions are required. Further, we present a particle system based Euler-type split-step scheme (SSM) for the simulation of this type of MV-SDEs. The scheme attains, in stepsize, the strong error rate $1/2$ in the non-path-space root-mean-square error metric and we demonstrate the property of mean-square contraction. Our results are illustrated by numerical examples including: estimation of PoC rates across dimensions, preservation of periodic phase-space, and the observation that taming appears to be not a suitable method unless strong dissipativity is present.

Figures (5)

• Figure 3.1: Simulation of the double-well model \ref{['eq:example:toy1']} with $N=1000$ particles. All schemes are initialized on the exact same samples. (a) and (c) show the density map for Taming-out (left), Taming-in (middle) and SSM (right) with $h=0.01$ at times $T\in\{1,3,10\}$ seen top-to-bottom and with different initial distribution. (b) Strong error (rMSE) of SSM and Taming with $X_0 \sim \mathcal{N}(3,9)$ in log-scale. (d) Strong error (Path) of SSM and Taming with $X_0 \sim \mathcal{N}(3,9)$ in log-scale.
• Figure 3.2: Approximation of the invariant distribution of \ref{['eq:example:toy222']} with $N=1000$ particles. The simulated Brownian motion paths and initial distribution are the same for all schemes. (a) and (c) show the distribution for Taming-out (left), Taming-in (middle) and SSM (right) with $h=0.01$ at times $T\in\{1,3,10\}$ seen top-to-bottom and with different initial distribution; $x$- and $y$-scales are fixed. (b) Strong error (rMSE) of SSM and Taming with $X_0 \sim \mathcal{N}(2,16)$. (d) Expected distance (in log-scale) between particles under different initial distributions with $h=10^{-3}$ for the SSM.
• Figure 3.3: Simulation of the Vdp model \ref{['eq:example:vdp']} with a different number of particles and $h=10^{-2}$, $T=12$, $X_{1,0}\sim \mathcal{N}(2,16),X_{2,0}\sim \mathcal{N}(0,16)$. (a)(b)(c)(d)(e) are phase portraits of the Taming-out method with different choices of $N$. (f)(g)(h)(i)(j) are phase portraits of the Taming-in method with different choices of $N$. (k)(l)(m)(n)(o) are phase portraits of the SSM with different choices of $N$.
• Figure 3.4: Approximation of \ref{['eq:example:supermeasurediff']} with $N=1000$ particles. The simulated Brownian motion sample paths and initial distribution are the same for all schemes. (a) and (c) show the distribution for Taming-out (left), Taming-in (middle) and SSM (right) with $h=0.01$ at times $T\in\{1,3,10\}$ seen top-to-bottom and with different initial distribution; $x$- and $y$-scales are fixed. (b), (d) and (e) show the strong error (rMSE) of SSM and Taming with $X_0 \sim \mathcal{N}(1,1)$ for different cases. The changed Case 1 in (d) is Case 1 with $v(x,\mu)=-x^3/4.$
• Figure 3.5: Estimation of PoC rate for equation \ref{['Eq:General MVSDE']}-\ref{['Eq:General MVSDE shape of v']} under \ref{['eq:example:poc']} using SSM \ref{['eq:SSTM:scheme 0']}-\ref{['eq:SSTM:scheme 2']} with fixed stepsize $h=10^{-3}$, $T=1$ and number of particles $N\in \{40,80,160,320,640,1280,2560\}$. In all figures the reference rate $0.5$ and the upper bound rate from Theorem \ref{['theorem:Propagation of Chaos']} are displayed.

Theorems & Definitions (45)

• Remark 2.2: Time dependency for $u$
• Example 2.3
• Remark 2.4: Implied properties
• Theorem 2.5: Wellposedness
• Remark 2.6: On the 'additional symmetry' restriction
• Lemma 2.7: Properties of the particle system as a system in $\mathbb{R}^{Nd}$
• proof
• Theorem 2.8: Propagation of Chaos
• Theorem 2.9: Contraction, exponential ergodicity property and invariance
• Definition 2.10