Accelerating diffusions

Abstract

Let U be a given function defined on R^d 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)=-\nabla U(X(t)) dt+\sqrt2 dW(t),\qquad X(0)=x_0. To accelerate the convergence, a family of diffusions with π(x) as their common equilibrium is considered, dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr) dt+\sqrt2 dW(t),\qquad X(0)=x_0. Let L_C be the corresponding infinitesimal generator. The spectral gap of L_C in L^2(π) (λ(C)), and the convergence exponent of X(t) to πin variational norm (ρ(C)), are used to describe the convergence rate, where λ(C)= Sup{real part of μ\dvtxμis in the spectrum of L_C, μis not zero}, {-2.8cm}ρ(C) = Inf\biggl{ρ\dvtx\int | p(t,x,y) -π(y)| dy \le g(x) e^{ρt}\biggr}.Roughly speaking, L_C is a perturbation of the self-adjoint L_0 by an antisymmetric operator C\cdot\nabla, where C is weighted divergence free. We prove that λ(C)\le λ(0) and equality holds only in some rare situations. Furthermore, ρ(C)\le λ(C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.