Accelerating Brownian motion on NN-torus

On NN-torus, we consider antisymmetric perturbations of Laplacian of the form LC≐Δ+C⋅∇LC≐Δ+C⋅∇, where CC is a divergence free vector field. The spectral gap, denoted by λ(C)λ(C), of L(C)L(C) is defined by −sup{real part of μ,μ is in the spectrum ofLC, μ≠0}μ≠0}. We characterize for a certain class of CC’s, the limit of λ(kC)λ(kC) as kk goes to infinity and prove that sup{λ(C),C is divergence free}=∞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 NN-torus the acceleration of Brownian motion could be infinitely fast.