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 It\^o 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.