Inferring diffusivity from killed diffusion

原文摘要

We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given “binding potential.” The binding process can be modeled by an additive random functional corresponding to the canonical construction of a “killed” diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the “killing” positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a Schrödinger-type equation. The inference problem can then be recast as a nonlinear inverse problem for this partial differential equation, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the system, provided the binding potential is not too “aggressive.” In the proofs, we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. A numerical illustration of the estimator by MCMC methods is also provided.

摘要来源：Crossref