Tamed Euler Approximation of Invariant Measures for Weakly Interacting Particle Systems and McKean--Vlasov Equations

Abstract

We analyze explicit tamed Euler approximations for weakly interacting particle systems and McKean--Vlasov diffusions with additive noise and polynomially weighted locally Hölder, possibly superlinear drift.Under dissipativity at infinity and suitable stability assumptions, we establish time-uniform $W_1$ bounds by reflection coupling, separating relaxation, time-discretization, and finite-particle errors.For small step sizes, we prove existence and uniqueness of numerical invariant laws and bound their error relative to the continuous-time particle invariant law uniformly in $N$ in a normalized joint $W_1$ distance. We further bound the error of the one-particle numerical invariant marginal relative to the McKean--Vlasov invariant law by $O(θ^{\min\{α,β/2\}}+N^{-1/2})$, where $θ$ is the step size, $N$ the particle number, $α\in(0,1/2]$ the taming exponent, and $β\in(0,1]$ the Hölder exponent. Analytical examples illustrate the framework for smooth and non-Lipschitz drifts.