Accelerating diffusions

الأصل · EN

Let U be a given function defined on Rᵈ and π(x) be a density function proportional to -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 πin variational norm (ρ(C)), are used to describe the convergence rate, where λ(C)= Supreal part of μμis in the spectrum of LC, μis not zero, -2.8cmρ(C) = Infρ∫ | p(t,x,y) -π(y)| dy ≤ g(x) eρᵗ.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.