Solutions Of Linear And Semilinear Distributed Parameter Equations With A Fractional Brownian Motion

摘要

In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parameter H ∈ (0, 1). For H = 1/2 the process is Brownian motion. Solutions of these linear and semilinear equations are given for each H ∈ (0, 1) with the assumptions differing for the cases H ∈ (0, 1/2) and H ∈ (1/2, 1). For the linear equations, the solutions are mild solutions and limiting Gaussian measures are characterized. For the semilinear equations, the solutions are either mild or weak. The weak solutions are obtained by transforming the measure of the associated linear equation by a Radon-Nikodym derivative (likelihood function). An application to identification is given by obtaining a strongly consistent family of estimators for an unknown parameter in a linear equation with distributed noise or boundary noise.