Taming discrete rough paths via strong Lyapunov functions

Technical Summary: Taming Discrete Rough Paths via Strong Lyapunov Functions

Problem Statement
This working paper addresses the numerical approximation of Rough Differential Equations (RDEs) of the form $dy_t = f(y_t)dt + g(y_t)dx_t$, where $x$ is a rough path (e.g., fractional Brownian motion) with Hölder regularity $\nu \in (1/3, 1/2]$. While the Doss-Sussmann transformation allows for the analysis of continuous RDE solutions, extending these techniques to discrete numerical schemes presents significant challenges.

Specifically, standard Euler-type schemes for RDEs often fail to provide integrable error estimates or guarantee the existence of global attractors when the drift $f$ exhibits super-linear growth. Previous approaches using cut-off techniques (e.g., [4]) yield pathwise convergence but result in constants that are not necessarily integrable. Furthermore, establishing the existence of pullback attractors for the discrete random dynamical systems generated by these schemes, and proving their convergence to the continuous attractor, remains an open gap in the literature under general non-linear drift conditions.

Methodology
The paper employs a "tamed" numerical scheme combined with the recently introduced concept of strong Lyapunov functions [2].

1. Tamed Euler Scheme: Instead of the standard Euler update, the authors utilize a tamed scheme (Equation 1.8) where the drift term is divided by a factor $1 + M\|f(y_{t_k})\|(t_{k+1}-t_k)$. This modification prevents the numerical solution from exploding in finite time, even when $f$ has super-linear growth.
2. Discrete Doss-Sussmann Transformation: The authors extend the Doss-Sussmann technique to the discrete setting. They define a pure rough difference equation and a corresponding flow $\phi$. By transforming the tamed numerical solution $y_{t_k}$ into a new variable $z_{t_k}$ via $y_{t_k} = \phi(t_k, s, x)z_{t_k}$, the problem is reduced to analyzing the growth of $z_{t_k}$ under a modified drift.
3. Strong Lyapunov Functions: The analysis relies on the existence of a strong Lyapunov function $V$ satisfying specific growth and gradient conditions (Assumption $H_V$). Crucially, the paper utilizes a condition (Equation 1.7) that bounds the inner product of the gradient $\nabla V$ with the modified drift, allowing for the control of solution norms without requiring global Lipschitz continuity of $f$.
4. Stopping Times and Induction: To handle the non-uniformity of the rough path increments, the authors construct discrete stopping times based on the variation of the path $x$. Estimates are derived over intervals between these stopping times and then aggregated via induction.

Key Contributions and Results

• Integrable Solution Norm Estimates: Under the strong Lyapunov condition, the paper derives explicit estimates for the norms of the tamed numerical solution (Theorem 7). Unlike previous cut-off methods, these estimates are shown to be integrable ($L^1$) with respect to the noise, provided the Lyapunov function satisfies polynomial growth conditions.
• Convergence in $L^1$: The paper proves that the tamed scheme converges to the continuous RDE solution in both the pathwise sense and the $L^1$ sense (Theorem 9). The pathwise error is of order $|\Pi|^{3\nu-1}$, while the $L^1$ convergence is established using the Lebesgue Dominated Convergence Theorem, leveraging the integrability of the solution norms.
• Existence of Numerical Pullback Attractors: For systems satisfying a negative gradient condition (where the Lyapunov drift coefficient $\delta < 0$), the authors prove the existence of a global numerical pullback attractor $A^\Delta_{C_g}$ for the discrete random dynamical system generated by the tamed scheme (Theorem 20).
• Upper Semi-Continuity: A significant result is the proof of upper semi-continuity of the numerical attractor with respect to two parameters:
  1. The step size $\Delta$ (convergence to the continuous attractor $A^0_{C_g}$).
  2. The noise intensity $C_g$ (convergence to the attractor of the unperturbed drift system).
     These limits hold almost surely and, in the case of Gaussian rough paths, also in the $L^1$ sense.

Significance and Claims
The paper claims to close a specific gap in the numerical analysis of RDEs by demonstrating that tamed schemes, when paired with strong Lyapunov functions, can handle non-linear drifts (such as those in Fitzhugh-Nagumo or Lorenz systems) while maintaining rigorous integrability properties.

The authors emphasize that their approach avoids the non-integrable constants associated with earlier cut-off techniques. By establishing the existence and stability (via upper semi-continuity) of numerical pullback attractors, the work provides a theoretical foundation for the long-term numerical simulation of rough dynamical systems. The results are presented as a direct extension of the continuous theory in [2] to the discrete domain, ensuring that the qualitative properties of the continuous system (specifically the existence of attractors) are preserved in the numerical approximation.