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), $$dX(t)=-\nabla U(X(t))\,dt+\sqrt{2}\,dW(t),\qquad X(0)=x_{0}.$$ To accelerate the convergence, a family of diffusions with π(x) as their common equilibrium is considered, $$dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr)\,dt+\sqrt{2}\,dW(t),\qquad X(0)=x_{0}.$$ Let L_C be the corresponding infinitesimal generator. The spectral gap of L_C 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 L_C, μ is not zero}, $$\rho(C)=\operatorname {Inf}\biggl\{\rho: \int \vert p(t,x,y)-\pi(y)\vert \,dy\le g(x)e^{\rho t}\biggr\}.$$ Roughly speaking, L_C 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.