Diffusion Estimation from Multiscale Data by Operator Eigenpairs

In this paper we present a new procedure for the estimation of diffusion processes from discretely sampled data. It is based on the close relation between eigenpairs of the diffusion operator L and those of the conditional expectation operator Pt, a relation stemming from the semigroup structure Pt = exp(tL) for t ≥ 0. It allows for estimation without making time discretization errors, an aspect that is particularly advantageous in the case of data with low sampling frequency. After estimating eigenpairs of L via eigenpairs of Pt, we infer the drift and diffusion functions that determine L by fitting L to the estimated eigenpairs using a convex optimization procedure. We present numerical examples in which we apply the procedure to one- and two-dimensional diffusions, reversible as well as nonreversible. In the second part of the paper, we consider estimation of coarse-grained (homogenized) diffusion processes from multiscale data. We show that eigenpairs of the homogenized diffusion operator are asymptotically close to eigenpairs of the underlying multiscale diffusion operator. This implies that we can infer the correct homogenized process from data of the multiscale process, using the estimation procedure discussed in the first part of the paper. This is illustrated with numerical examples.