Accelerating diffusions

Let U be a given function defined on .d and .(x) be a density function proportional to exp.U(x). The following diffusion X(t) is often used to sample from .(x), dX(t)=..U(X(t))dt+.2dW(t),X(0)=x0. To accelerate the convergence, a family of diffusions with .(x) as their common equilibrium is considered, dX(t)=(..U(X(t))+C(X(t)))dt+.2dW(t),X(0)=x0. Let LC be the corresponding infinitesimal generator. The spectral gap of LC in L2(.) (.(C)), and the convergence exponent of X(t) to . in variational norm (.(C)), are used to describe the convergence rate, where .(C)=Sup.{real part of .:. is in the spectrum of LC, . 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 .(C)..(0) and equality holds only in some rare situations. Furthermore, .(C)..(C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.