Stochastic stability of master-slave synchronization for dissipative PDEs with Burgers-type nonlinearity and application to data assimilation

We investigate the stochastic stability of master--slave synchronization for a class of nonlinear dissipative evolution equations with a Burgers-type convective nonlinearity and a polynomial linear differential operator.

The family includes the Burgers, Kuramoto--Sivashinsky, Kawahara, Benney--Lin, and Nikolaevskiy equations.

Under periodic boundary conditions, each equation is represented through a finite-dimensional Fourier truncation, yielding a complex state vector coupled to a slave system driven by observations of the master.

We first establish local exponential stability of the deterministic synchronization manifold under a simple condition on the coupling strength.

We then introduce observational noise in the coupling signals, which transforms the slave system into an Itô diffusion and prevents exact synchronization.

Our analysis focuses on the deviation $\Delta(t)$ between the stochastic synchronization error and the exponentially stable deterministic reference error.

We prove an $\mathcal{O}(\sigma^2)$ finite-time mean-square bound on $\|\Delta(t)\|^2$, together with a corresponding tail-probability estimate.

Under a global one-sided dissipativity assumption, the localization is removed and an $\mathcal{O}(\sigma^2)$ time-uniform bound is obtained.

These estimates are derived in Fourier space and translated to physical space via Parseval's relation.

Finally, we interpret the stochastic slave dynamics as a continuous-time synchronization-based data-assimilation scheme and compare its structure with the ensemble Kalman--Bucy filter, emphasizing the difference between prescribed stability-oriented and adaptive covariance-based gains.