Accelerating diffusions

Let U be a given function defined on ℝd and π (x) be a density function proportional to exp−U (x). The following diffusion X (t) is often used to sample from π (x), dX (t) =- U (X (t) ) \, dt+2\, dW (t), X (0) =x₀. To accelerate the convergence, a family of diffusions with π (x) as their common equilibrium is considered, dX (t) = (- U (X (t) ) +C (X (t) ) ) \, dt+2\, dW (t), X (0) =x₀. Let LC be the corresponding infinitesimal generator. The spectral gap of LC in L2 (π) (λ (C) ), and the convergence exponent of X (t) to π in variational norm (ρ (C) ), are used to describe the convergence rate, where λ (C) =Sup real part of μ: μ is in the spectrum of LC, μ is not zero, (C) = Inf\: p (t, x, y) - (y) \, dy g (x) e^{ t\}. Roughly speaking, LC is a perturbation of the self-adjoint L0 by an antisymmetric operator C⋅∇, where C is weighted divergence free. We prove that λ (C) ≤λ (0) and equality holds only in some rare situations. Furthermore, ρ (C) ≤λ (C) and equality holds for C=0. In other words, adding an extra drift, C (x), accelerates convergence. Related problems are also discussed.