Non-Asymptotic Exponential Convergence

• Non-Asymptotic Exponential Convergence is a framework that provides explicit, time-uniform exponential rates for synchronization and stability in stochastic systems with irregular drifts.
• The approach leverages Zvonkin’s transform to regularize non-smooth drift, enabling rigorous quantitative bounds in both almost sure and Lp senses.
• These non-asymptotic bounds offer practical tools for analyzing synchronization-by-noise across various applications including control, biology, and finance.

Non-Asymptotic Exponential Convergence

Non-asymptotic exponential convergence describes the phenomenon wherein a sequence (typically solutions to differential equations, stochastic processes, or iterates of numerical algorithms) converges to its long-time limit at an explicit, uniform exponential rate, with finite-time, non-asymptotic error bounds that are valid for all time—rather than only in limiting or average senses. In contemporary mathematical analysis and applied probability, such results provide crucial quantitative information for synchronization, stability, and mixing properties in a broad range of domains, especially where classical regularity or convexity assumptions are relaxed or violated.

1. Exponential Synchronization in Non-Regular Stochastic Systems

A prototypical setting is the synchronization of trajectories for one-dimensional stochastic differential equations (SDEs) featuring highly irregular, merely measurable drift. Consider

$$dX_t = \left(-\lambda X_t + \beta(X_t) + \alpha(X_t)\right)dt + \sigma(X_t) dW_t,$$

with $\lambda>0$ (dissipativity), $\beta$ Lipschitz, $\alpha$ bounded measurable (possibly discontinuous), and $\sigma$ Lipschitz, uniformly non-degenerate.

The core result establishes that, for sufficiently large $\lambda$, the distance between two trajectories starting at arbitrary initial positions contracts exponentially fast, both almost surely and in $L_p$: $$\lim_{t \to \infty} |X_t^x - X_t^y|\, e^{c_\lambda t} = 0 \quad \text{a.s.}$$ with explicit exponential rate

$$c_\lambda = \frac{\lambda}{2} - L_{\tilde{\beta}} - L_{\tilde{\gamma}},$$

where $L_{\tilde{\beta}}$ and $L_{\tilde{\gamma}}$ are computable from the coefficients, and explicit formulas are provided for cases where $\alpha$ is either compactly supported or bounded by a global Lipschitz function.

In $L_p$,

$$\|X_t^x - X_t^y\|_p \leq C |x-y|\, e^{-c_{\lambda,p} t}, \qquad c_{\lambda,p} = p c_\lambda - \frac{p(p-1)}{2} L_{\tilde{\sigma}}^2,$$

with $L_{\tilde{\sigma}}$ an explicit Lipschitz constant of the transformed diffusion.

<table> <tr><th>Case for $\alpha$</th><th>Sufficient $\lambda$</th><th>Exponential rate $c_\lambda$</th><th>$L_p$-rate $c_{\lambda,p}$</th></tr> <tr><td>Bounded, compact support</td><td>$2(L_{\tilde{\beta}}+L_{\tilde{\gamma}})$</td><td>$$\frac{\lambda}{2} - L_{\tilde{\beta}} - L_{\tilde{\gamma}}$$</td><td>$p c_\lambda - \frac{p(p-1)}{2}L_{\tilde{\sigma}}^2$</td></tr> <tr><td>Bounded by global Lipschitz</td><td>as above, different $L$'s</td><td>as above</td><td>as above</td></tr> </table>

All constants in the rates are fully explicit in the coefficients’ parameters and universal numerical constants.

2. Methodology: Zvonkin’s Transform and Noise Regularization

Fundamental to this theory is the use of Zvonkin's transform, which remaps the SDE via an explicitly constructed change of variables to eliminate the irregular drift $\alpha$, regularizing the system sufficiently to enable comparison, contraction, and Itô calculus methods. The transform allows the inherited dissipativity of $-\lambda X$ and Lipschitz regularity of $\beta$ and $\sigma$ to dominate, even after incorporating the convoluted effects of a non-smooth $\alpha$.

Key estimates on the transformed drift and diffusion are derived, bounding the propagation of the non-regularity under the transformation, and enabling global dissipation estimates suitable for non-asymptotic, time-uniform bounds.

3. Non-Asymptotic Exponential Convergence: Quantitative Bounds

All synchronization and contraction bounds are genuinely non-asymptotic, delivering explicit uniform inequalities for all $t\geq0$, not merely statements about the limit $t\to\infty$. In the regular drift case (continuous $b$ with global dissipativity), for $Y^x$, $Y^y$ solutions,

$$|Y_t^x - Y_t^y|e^{ct} \to 0~\text{a.s. for any}~c < D_b,$$

and in $L_p$,

$$\|Y_t^x - Y_t^y\|_p \leq |x-y|\,e^{-(D_b - \frac{p-1}{2}L_\sigma^2)t},$$

but these results extend, by way of the Zvonkin transform, to the fully singular drift case. Thus, for processes with only measurable, bounded drift irregularities, the ambient noise still regularizes, enforcing exponential mutual synchronization at explicitly computed rates for any time horizon.

4. Significance and Generalizations

The pivotal advance is the removal of any continuity or finite-variation assumption on the non-Lipschitz drift component. Classical synchronization and stability results for SDEs require, at minimum, continuous or bounded variation drift. Here, only bounded measurability is imposed, encompassing SDEs with discontinuous, oscillatory, or even "jump-like" drift behavior otherwise considered analytically intractable.

The work demonstrates that noise can enforce pathwise exponential stabilization—even when drift components fluctuate arbitrarily and instantaneously—provided sufficient overall dissipativity and mild regularity of the stochastic forcing. This phenomenon is now called synchronization-by-noise.

5. Practical Implications and Computability

All rate constants and dissipativity thresholds are explicit, facilitating engineering and scientific applications where precise quantification of time-to-synchrony or stability margins is necessary. Given empirical or prescribed bounds on $\|\alpha\|_\infty$, $L_\beta$, $L_\sigma$, and the ellipticity lower bound $c_\sigma$, the required dissipativity $\lambda$ and the resulting $c_\lambda$ are calculable directly.

Practically, this provides tools for rigorous design and robustness analysis of stochastic dynamical systems encountered in control, population biology, finance, and related fields, where drift irregularities are common and unremovable.

6. Relation to Broader Synchronization Phenomena and Extensions

Synchronization phenomena governed by non-asymptotic exponential convergence appear across systems—from deterministic dissipative PDEs to stochastic optimization and consensus algorithms in networked settings. The explicit time-uniform exponential bounds developed here are related in form and content to exponential ergodicity in stochastic processes, quantitative stability for infinite-dimensional nonlinear SPDEs (Wang, 2013), and Lyapunov function approaches for networks and consensus protocols with possibly non-symmetric or non-gradient couplings (Boudin et al., 2021).

The general method—leveraging coordinate transformation to embed irregular systems into regular ones, then applying quantitative dissipativity—has found increased utility in nonlinear stochastic analysis, Markovian consensus theory, and synchronization-by-noise in neuroscience and random dynamical systems.

7. Key References and Historical Context

• Zvonkin, A.K., "A transformation of the phase space of a diffusion process that removes the drift", Mathematics of the USSR-Sbornik, 1974, established foundational results on existence/uniqueness for SDEs with measurable drift.
• Recent work on noise-induced synchronization in rough-drift systems and explicit $L_p$ quantitative stability, as well as synchronization-by-noise in nonlinear or high-dimensional SPDEs.

The results synthesize these strands, providing sharp, readily applicable, non-asymptotic exponential rates for synchronization in the broadest possible class of one-dimensional stochastic systems.