Exponential ergodicity of exact and numerical solutions for McKean-Vlasov SDEs driven by Lévy noise

Abstract: This paper investigates the exponential ergodicity of the exact solution and the tamed Euler solution for McKean-Vlasov stochastic differential equations driven by Lévy noise. First, we establish exponential ergodicity for both the original equation and the tamed Euler method. Then we prove the convergence of the numerical invariant measure to the exact invariant measure, which is obtained by combining the propagation of chaos (PoC) result with the strong convergence of the tamed Euler scheme. Furthermore, we derive a convergence rate for the numerical invariant measure by establishing uniform-in-time PoC and uniform-in-time convergence of the tamed Euler method. Finally, numerical experiments are presented to illustrate the theoretical results.

• The paper establishes exponential ergodicity for Lévy-driven MV-SDEs, ensuring unique invariant measures under dissipativity and Lipschitz conditions.
• It demonstrates that a tamed Euler scheme maintains ergodic properties and provides explicit convergence rates in the Wasserstein metric.
• The study quantifies propagation of chaos and strong convergence with detailed error bounds for both particle approximations and numerical invariant measures.

Exponential Ergodicity and Numerical Approximation for Lévy-Driven McKean-Vlasov SDEs

Problem Setting and Motivation

The paper addresses the long-time statistical behavior and numerical approximation of McKean-Vlasov stochastic differential equations (MV-SDEs) driven by Lévy noise (2606.18815). MV-SDEs represent stochastic dynamics whose drift, diffusion, and jump coefficients depend explicitly on the solution's law, encapsulating mean-field interactions. Lévy noise introduces jump discontinuities that cannot be captured by Gaussian processes, modeling real-world phenomena such as abrupt events in finance or neuroscience.

Analytical understanding of ergodic properties—specifically, the existence, uniqueness, and exponential convergence to invariant measures—has been the subject of recent rigorous study. However, explicit forms are usually unavailable, so practical computation relies on numerical schemes, often constructed via interacting particle systems and time discretization methods. The main technical challenge is to preserve ergodicity in the presence of nonlinear law-dependencies and jumps, while deriving strong convergence and uniform-in-time error estimates for the invariant measures obtained numerically.

Main Contributions

The paper systematically develops a rigorous framework for (i) exponential ergodicity of MV-SDEs with Lévy noise, (ii) exponential ergodicity for the tamed Euler scheme, (iii) convergence of numerical invariant measures to the true invariant measure, and (iv) explicit, uniform-in-time convergence rates.

1. Exponential Ergodicity of MV-SDEs

The authors establish exponential contractivity in the 2-Wasserstein metric for the law of the solution to a class of MV-SDEs with Lévy noise, under dissipativity, monotonicity, and Lipschitz-type conditions. Specifically, for coefficient parameters satisfying certain inequalities (e.g., $L_1 - L_5 - 8L_3 - 1 > 0$), they show:

$$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$

where $\mathcal{P}_t$ is the Markov semigroup over measures and $\lambda^*$ is a rate depending on the coefficients. This contractivity yields existence and uniqueness of an invariant measure, together with exponential convergence of solutions (regardless of initial law) to the invariant distribution.

2. Exponential Ergodicity of the Tamed Euler Scheme

For numerical approximation, the paper studies a tamed Euler discretization of the associated interacting particle system. The taming procedure controls super-linear growth in coefficients, especially crucial for Lévy-driven SDEs.

They prove that under modified dissipativity and coercivity conditions, the numerical scheme exhibits exponential contractivity in Wasserstein distance, mirroring the exact SDE behavior:

$$W_2(\mathcal{P}_{A,N}^k \mu_N, \mathcal{P}_{A,N}^k \nu_N) \leq e^{-\lambda^* t_k} W_2(\mu_N, \nu_N)$$

where $t_k = k\Delta$ (with $\Delta$ the step size), and $\mu_N$, $\nu_N$ are $N$-particle tensorized laws. A unique numerical invariant measure exists, and the empirical distributions converge exponentially fast.

3. Strong Convergence and Propagation of Chaos

The paper quantifies the convergence of the interacting particle system to the mean-field limit, both in finite and infinite time horizons. Propagation of chaos results are given with dimension-dependent rates:

• For $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$0: $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$1
• For $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$2: $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$3
• For $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$4: $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$5

Extending previous work, they show that the marginal law of a single particle (under the invariant measure of the numerical scheme) converges in Wasserstein-2 distance to the invariant law of the MV-SDE as both $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$6 and $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$7. This is achieved by combining uniform-in-time PoC, strong convergence analysis of the discretization, and comparison of numerical and exact invariant measures via tensorization arguments.

4. Explicit Convergence Rates for Invariant Measures

Uniform-in-time convergence is shown via refined moment bounds and contractivity estimates. Under additional assumptions (boundedness for jump and diffusion coefficients), the authors obtain:

$$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$8

for the numerical invariant measure $$W_2(\mathcal{P}_t \mu, \mathcal{P}_t \nu) \leq e^{-\lambda^* t} W_2(\mu, \nu), \qquad t \geq 0$$9 and the tensorized exact invariant measure, with $\mathcal{P}_t$0 depending on dimension. This quantifies the approximation error in terms of both discretization and particle number, and covers the marginal law as well.

5. Numerical Experiments

Empirical validation is provided for several SDE examples, demonstrating:

• Convergence of empirical densities (from various initial distributions) toward the invariant measure;
• Agreement between numerical invariant measures (for tamed Euler) and proxy exact measures (via smaller $\mathcal{P}_t$1 and larger $\mathcal{P}_t$2);
• Numerical convergence rates consistent with theoretical predictions (time-step error $\mathcal{P}_t$3, particle-number error $\mathcal{P}_t$4 for 1D and 2D examples).

Strong Numerical Results and Claims

• Uniform exponential ergodicity in Wasserstein-2 distance for both MV-SDEs and the tamed Euler scheme, with explicit rates independent of initial law.
• Unique existence and uniqueness of invariant measure, in both exact and numerical settings, under general super-linear growth and Lévy noise.
• Quantitative convergence rates for numerical approximation: The error in invariant measures can be made arbitrarily small by increasing $\mathcal{P}_t$5 and decreasing $\mathcal{P}_t$6, with rates matching theoretical results.
• Numerical demonstrations confirm ergodicity and convergence rates on nontrivial test problems, including multidimensional settings.

Implications and Future Directions

Practical Implications

These results provide rigorous justification for employing tamed Euler discretizations to study stationary behavior of Lévy-driven MV-SDEs, including those with non-globally Lipschitz and super-linear coefficients frequently encountered in applied contexts (finance, biology, neural networks). The explicit error bounds offer guidance for step-size and particle number selection in simulations, ensuring accuracy in invariant measure computation even for systems with jumps.

Theoretical Advancements

The analysis advances understanding of ergodicity—particularly exponential rates and invariant measure approximation—for law-dependent SDEs with jumps, bridging a gap between theory and practice. It extends prior work on Gaussian-driven systems to the broader Lévy setting, with nontrivial results for propagation of chaos, contractivity, and uniform-in-time convergence for both exact and numerical solutions.

Potential for AI and Stochastic Modeling

The theoretical machinery may impact stochastic modeling in machine learning, e.g., mean-field inference, reinforcement learning with jumps, and training of ensemble-based stochastic neural networks. As MV-SDEs emerge in distributed learning, understanding ergodic properties and invariant measure approximation is critical for robustness and stability analysis. Methodologically, precise error bounds and ergodic preservation in numerical schemes pave the way for trustworthy simulation-based statistical estimation in high-dimensional, jump-driven systems.

Future Developments

• Extension to higher-order numerical schemes (Milstein, adaptive, etc.), with improved convergence rates.
• Analysis under relaxed measure-dependency (e.g., weaker Lipschitz or non-Lipschitz coefficients).
• Application to control problems, mean-field games with jump dynamics, and complex networks.
• Investigation of dimension-independent convergence rates for PoC and discretization.
• Integration into AI model architectures utilizing stochastic differential modeling.

Conclusion

The paper delivers a comprehensive, rigorous treatment of exponential ergodicity and numerical invariant measure approximation for Lévy-driven McKean-Vlasov SDEs. It establishes strong theoretical foundations for ergodic behavior, explicit convergence rates for tamed Euler discretizations, and propagation of chaos in both finite and infinite time. The practical numerical guidance substantiates the theoretical claims and opens avenues for further research in mean-field modeling, ergodicity, and stochastic simulation in complex AI systems and applications (2606.18815).