Lagrangian Chaos in Randomly Shifted Cellular Flows

This paper studies the Lagrangian flow of a randomly shifted cellular flow on the 2D torus driven by Brownian shifts. We prove that, for every positive shift diffusivity, the top Lyapunov exponent of the Lagrangian flow is strictly positive. The proof uses the explicit geometric structure of the model and directly verifies the Hörmander condition for the projective process and the nondegeneracy of the support semigroup, thereby making the mechanism for a positive exponent transparent. We also prove that the top Lyapunov exponent is continuous with respect to the shift diffusivity and converges to zero in both endpoint limits, namely the weak shift limit and the fast shift limit. To quantify the decay in the fast shift regime, we present a formal large diffusivity expansion and finite-time numerical calculations. Our results show that Lagrangian chaos in randomly shifted cellular flows is generated jointly by the geometry of the cellular flow and the accessibility induced by random shifts.