Two identical dynamical systems started from different initial conditions will generally diverge if the system is chaotic or converge if it's dissipative. Add noise — the same noise to both copies — and something unexpected happens: the systems synchronize. Different initial conditions converge to the same trajectory, driven to agreement by the shared randomness.
This phenomenon, synchronization by noise, is well-understood for standard Brownian motion (Markov noise: each increment independent of history). Gess & Kadhim (arXiv:2603.12774) extend it to fractional Brownian motion with Hurst parameter H ∈ (0,1) — noise with memory. When H > 1/2, the noise is positively correlated: a large increment makes the next increment likely to be large too. When H < 1/2, the noise is anti-correlated: large follows small.
The result: synchronization occurs for all H provided the deterministic dynamics has a negative top Lyapunov exponent (the system is contractive on average). The memory in the noise doesn't prevent synchronization — the contraction overwhelms the correlation structure.
The technical challenge is non-Markovianity. For Markov noise, you can characterize the invariant measure (the long-run distribution) using standard ergodic theory. Fractional Brownian motion isn't Markov — the future depends on the entire past, not just the present state. The authors develop a support theorem for the invariant measure in this non-Markovian setting, showing that the support (the set of states the system actually visits) is determined by the Lyapunov structure, not the correlation structure.
The principle: synchronization is a property of the dissipative dynamics, not the noise statistics. The noise provides the perturbation; the contraction provides the convergence. As long as the perturbation is shared and the contraction is strong enough, memory in the noise is irrelevant to the qualitative outcome. The quantitative convergence rate may depend on H, but the existence of synchronization does not.