Accelerating Brownian motion on N-torus

On N-torus, we consider antisymmetric perturbations of Laplacian of the form L-C (=) over dot Delta + C center dot del, where C is a divergence free vector field. The spectral gap, denoted by lambda(C), of L(C) is defined by -sup{real part of mu, mu is in the spectrum of L-C, mu not equal 0}. We characterize for a certain class of C's, the limit of lambda(kC) as k goes to infinity and prove that sup {lambda(C), C is divergence free} = infinity. 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. (C) 2013 Elsevier B.V. All rights reserved.