Uniform-in-Time Convergence Bounds

• Uniform-in-time convergence bounds are quantitative guarantees ensuring that approximation errors between stochastic processes and their discrete approximations remain uniformly bounded over the entire time horizon.
• They rely on key conditions including contractivity, finite-time local error control, and uniform moment bounds to maintain stability regardless of time progression.
• This framework is applied in averaging techniques, numerical SDE discretizations, and mean-field particle systems to secure robust long-term convergence guarantees.

Uniform-in-time convergence bounds are quantitative guarantees asserting that the approximation error between a stochastic process and its discretized, numerical, or mean-field surrogate remains uniformly controlled over the entire time horizon, rather than growing with time or deteriorating asymptotically. Such bounds are essential for the paper of long-term accuracy and stability in the approximation of stochastic systems, averaging methods, and propagation of chaos in interacting particle systems. Recent work has clarified a set of sufficient and, to a degree, necessary conditions under which uniform-in-time convergence holds, and provided a unifying proof methodology with wide-ranging applicability (Schuh et al., 6 Dec 2024).

1. Mathematical Formulation and Main Theorem

Assume two processes or evolution systems, with law at time $t$ denoted as $\nu p_t$ (exact process) and $\nu p_t^{(\delta)}$ (approximated or discretized process, with discretization parameter $\delta$). Uniform-in-time convergence asks for a bound of the form

$$\sup_{t\geq 0} \text{dist}(\nu p_t, \nu p_t^{(\delta)}) \leq C\, \delta^\alpha$$

for some metric (e.g., Wasserstein, total variation), constant $C$, exponent $\alpha > 0$, and all sufficiently small $\delta$. The key result is that under three conditions—contractivity, finite time local error, and uniform control—the above bound holds, with $C$ independent of $t$.

The proof architecture relies on decomposing the trajectory into intervals of length $\tau > 0$ and telescoping the local finite time errors, which are geometrically damped by the contractivity of the reference process.

2. Sufficient Conditions for Uniform-in-Time Convergence

Uniform-in-time convergence is guaranteed if the following conditions are satisfied for an appropriate choice of distance function:

Condition	Mathematical Formulation	Role
Contractivity	For some $\lambda>0$, $$\text{dist}(\nu p_t, \eta p_t) \leq e^{-\lambda t}\text{dist}(\nu,\eta)$$	Ensures memory loss and convergence to invariant measure
Local Error Bound	$$\sup_{t\leq \tau} \text{dist}(\nu p_t, \nu p_t^{(\delta)}) \leq \delta^\alpha M(\nu)$$, $\forall \nu$	Controls one-step (finite time) error
Uniform Control	$\sup_{t\geq 0} M(\nu p_t^{(\delta)}) \leq C(\nu)$	Prevents growth of process “size” or moments

Here, $M(\nu)$ is typically a moment or Lyapunov functional on $\nu$. The parameter $\tau$ is a fixed time window independent of $\delta$.

3. General Proof Sketch

The uniform-in-time error bound follows by a geometric series argument:

1. Decompose the time interval $[0, t]$ into $k$ subintervals of length $\tau$.
2. Use contractivity to show that errors from earlier intervals are exponentially damped: at the $i$-th subinterval, the local error is multiplied by $e^{-\lambda(k-i-1)\tau}$.
3. Sum the $k$ local errors, each $$\leq \delta^\alpha M(\nu^{(\delta)}_{i\tau}) \leq \delta^\alpha C(\nu)$$.
4. The sum converges to $\delta^\alpha C(\nu)/(1-e^{-\lambda \tau})$, giving independence from $t$.

No additional time-splitting is needed for $t$ not a multiple of $\tau$ because the bounds hold uniformly.

4. Applications Across Problem Classes

This methodology has direct applicability to a wide spectrum of stochastic modeling paradigms:

• Averaging and Multiscale Systems: For slow–fast SDE systems where the averaged limit is exponentially stable, and the local error in the Wasserstein distance is $O(\delta^\alpha)$, the averaged process approximates the slow dynamics uniformly in time (Schuh et al., 6 Dec 2024).
• Numerical Discretization for SDEs: Euler and higher order schemes for contractive SDEs, with local weak or strong error rates $O(\delta^\alpha)$, yield uniform-in-time convergence if moments of the numerical scheme are bounded (Schuh et al., 6 Dec 2024).
• Mean-Field Particle Systems: Propagation of chaos is extended to a global-in-time context provided contractivity of the (limiting) McKean–Vlasov SDE, finite time de Finetti-type fluctuation estimates of order $O(N^{-\alpha})$, and uniform moment bounds on the particle process hold.

The implementation of condition verification differs by context; for numerical methods, $M(\nu)$ typically bounds the $p$-th moment, while in mean-field settings it may involve entropy or other collective statistics.

5. Comparative Examples and Boundary Cases

Examples illustrate the necessity of each condition:

• If contractivity fails (e.g., a deterministic flow or conservative system), local errors can accumulate linearly or worse, causing global errors to become unbounded in $t$.
• Local error alone is insufficient: without moment control, discretization errors may be amplified by growth in tails over time, particularly for systems with unbounded coefficients or explosive solutions.
• Examples in (Schuh et al., 6 Dec 2024) demonstrate the approach for slow–fast SDEs, discretized Langevin dynamics, and mean-field McKean–Vlasov particle approximations, verifying exponential contractivity, explicit local error, and uniform moment bounds for each.

6. Implications and Extensions

The general framework for uniform-in-time convergence has significant implications:

• Stability and Invariant Measure Approximation: Uniform-in-time bounds imply that approximations not only converge to the stationary law but provide controlled error for finite-time dynamics, allowing accurate computation of ergodic averages.
• Optimization and Sampling: For stochastic optimization methods (e.g., SGD), uniform-in-time error guarantees underpin the validity of anytime-estimates and adaptive stopping rules.
• Complex and Noncontractive Systems: The theory identifies a sharp dichotomy; relaxation of contractivity or failure of uniform moment control destroys the uniform-in-time bound. In systems with only polynomial decay, the global error can scale with time, so the exponential contractivity is critical.

Recent research continues to extend these principles to non-Lipschitz systems via taming, stochastic PDEs via maximal inequalities, and interacting particle systems with singular or non-convolution drifts, but the fundamental structure of contractivity plus local error plus uniform control persists (Bao et al., 7 May 2024, Klioba et al., 2023, Angeli et al., 2023).

7. Summary Table: Schematic of the Uniform-in-Time Framework

Step	Process A (Exact or Contractive)	Process B (Approximate/Discrete/Particle)	Role
Contractivity	$$\text{dist}(p_t \nu, p_t \eta) \le e^{-\lambda t}\,\text{dist}(\nu, \eta)$$	Not required	Damps errors over time
Local finite-time error	--	$$\sup_{t \le \tau}\text{dist}(p_t \nu, p_t^{\delta}\nu) \le \delta^\alpha M(\nu)$$	Controls immediate error
Uniform-in-time control	--	$\sup_{t \ge 0} M(p_t^{\delta}\nu) \le C(\nu)$	Prevents blow-up
Global uniform bound	\multicolumn{2}{c}{$$\sup_{t \ge 0}\text{dist}(p_t \nu, p_t^{\delta} \nu) \le C\,\delta^\alpha$$}	Target conclusion

This approach provides a unified and explicit recipe for deriving uniform convergence in time for a large class of stochastic approximation and numerical schemes, making it central to contemporary stochastic analysis and simulation methodology (Schuh et al., 6 Dec 2024).