Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by both long- and short-range-dependent noise. The limit process has both Gaussian and non-Gaussian components. The theorem holds for any $$L^2$$ L 2 functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology, the rough topology for processes in $$C^{\frac{1}{2}+}$$ C 1 2 + . This leads to a ‘rough creation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of random smooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, we obtain effective dynamics for the second-order problem and for the kinetic fractional Brownian motion model.

The local time of a two-dimensional diffusion

When a markovian random process taking values in a continuous state-space, such as R, visits a particular point repeatedly, it is natural to seek some quantity which records how long it spends there. Typically, however, the number of visits made to the point is uncountably infinite, and the (Lebesgue) length of time spent there is zero. One interesting object to consider is the local time, sometimes thought of as the occupation density of the process, which at each point is a random Cantor function that increases only when the process visits the point. The review article by Rogers (1989) contains a good introduction to the local time of a one-dimensional brownian motion and its relevance to the excursions of brownian motion from zero. In two dimensions, a typical diffusion, such as brownian motion in the plane, never revisits a point, so it does not have a local time. In this paper we shall construct the local times of some particular two-dimensional diffusions on a one-dimensional subspace, and show that they are jointly continuous in both time and space.

A GAUSSIAN, NON-MARKOVIAN APPROACH TO LONG MEMORY WITH APPLICATION TO A STOCHASTIC THEORY OF LINE SHAPE

A new approach to long memory effects is suggested. The main assumption is that the regression of fluctuations occurs via a clustering mechanism. The regression of a scalar variable takes place through a succession of linear decay processes, resulting in a long time tail of the autocorrelation function. The number of decay events is a random variable obeying a self-similar probability distribution whose fractal exponent determines the long time behavior of fluctuation regression. The overall fluctuation dynamics is described by a stationary Gaussian and non-Markovian random process. The theory is applied to the stochastic theory of line shape. The relaxation function ϕ(t) can be exactly evaluated. We show that the memory effects lead to a narrowing of the relaxation function for large time. As the time t tends to infinity, ϕ(t) may be approximated by a ‘contracted’ exponential ϕ(t) ~ exp(-const. t2−H) where 1≥H>0 is the fractal exponent describing the clustering process.