Ergodic SDEs on submanifolds and related numerical sampling schemes

Abstract

In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measureμon the level set of a smooth functionξ: ℝd→ ℝk, 1 ≤k<d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff’s ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure isμ. Motivated by the previous work of Ciccottiet al.(Commun. Pur. Appl. Math.61(2008) 371–408), as well as the work of Leliévreet al.(Math. Comput.81(2012) 2071–2125), in this paper we construct a family of ergodic diffusion processes on the level set ofξwhose invariant measures coincide with the given one. For the conditional measure, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn’t require computing the second derivatives ofξand the error estimates of its long time sampling efficiency are obtained.