Accelerating Brownian motion on N-torus

On N-torus, we consider antisymmetric perturbations of Laplacian of the form LC≐Δ+C⋅∇, where C is a divergence free vector field. The spectral gap, denoted by λ(C), of L(C) is defined by −sup{real part of μ,μ is in the spectrum ofLC, μ≠0}. We characterize for a certain class of C’s, the limit of λ(kC) as k goes to infinity and prove that sup{λ(C),C is divergence free}=∞. This problem is motivated by accelerating diffusions. By adding a weighted antisymmetric drift to a reversible diffusion, the convergence to the equilibrium is accelerated using the spectral gap as a comparison criterion. However, how good can the improvement be is yet to be answered. In this paper, we demonstrate that on N-torus the acceleration of Brownian motion could be infinitely fast.