Statistics of the occupation time for a class of Gaussian Markov processes

Stochastically gated local and occupation times of a Brownian particle

We generalize the Feynman-Kac formula to analyze the local and occupation times of a Brownian particle moving in a stochastically gated one-dimensional domain. (i) The gated local time is defined as the amount of time spent by the particle in the neighborhood of a point in space where there is some target that only receives resources from (or detects) the particle when the gate is open; the target does not interfere with the motion of the Brownian particle. (ii) The gated occupation time is defined as the amount of time spent by the particle in the positive half of the real line, given that it can only cross the origin when a gate placed at the origin is open; in the closed state the particle is reflected. In both scenarios, the gate randomly switches between the open and closed states according to a two-state Markov process. We derive a stochastic, backward Fokker-Planck equation (FPE) for the moment-generating function of the two types of gated Brownian functional, given a particular realization of the stochastic gate, and analyze the resulting stochastic FPE using a moments method recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment-generating function, averaged with respect to realizations of the stochastic gate.

Statistical properties of functionals of the paths of a particle diffusing in a one-dimensional random potential

We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size t. We compute the disorder average distributions of the local time, the inverse local time, the occupation time, and the inverse occupation time and show that in many cases disorder modifies the behavior drastically.

Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes

We consider the persistence probability, the occupation-time distribution, and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian stationary processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the "partial survival" probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2002)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d(n)x/dt(n)=eta(t) with n>/=3, incorporating an extrapolation of the series to the limit of continuous sampling. We then extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d(n)x/dt(n)=eta(t) processes for n=1 (random walk), n=2 (random acceleration), and larger n, and to simple diffusion from random initial conditions in one to three dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.

Exact occupation time distribution in a non-Markovian sequence and its relation to spin glass models

We compute exactly the distribution of the occupation time in a discrete non-Markovian toy sequence that appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function that is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model; at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.

Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process

We introduce a general method, based on a mapping onto quantum mechanics, for investigating the large-T limit of the distribution P(r,T) of the nonlinear functional r[V]=(1/T)integral(T)(0)dT' V[X(T')], where V(X) is an arbitrary function of the stationary Gaussian Markov process X(T). For T-->infinity at fixed r we obtain P(r,T) approximately exp[-theta(r)T], where theta(r) is a large-deviation function. We present explicit results for a number of special cases including V(X)=XH(X) [where H(X) is the Heaviside function], which is related to the cooling and the heating degree days relevant to weather derivatives.

Persistence of a continuous stochastic process with discrete-time sampling: non-Markov processes

We consider the problem of "discrete-time persistence," which deals with the zero crossings of a continuous stochastic process X(T) measured at discrete times T=nDeltaT. For a Gaussian stationary process the persistence (no crossing) probability decays as exp(-theta(D)T)=[rho(a)](n) for large n, where a=exp(-DeltaT/2) and the discrete persistence exponent theta(D) is given by theta(D)=(ln rho)/(2 ln a). Using the "independent interval approximation," we show how theta(D) varies with DeltaT for small DeltaT and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015101(R) (2001)] to determine rho(a) for a two-dimensional random walk and the one-dimensional random-acceleration problem. We also consider "alternating persistence," which corresponds to a<0, and calculate rho(a) for this case.

Critical dimensions of the diffusion equation

We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d(1) approximately 26 and d(2) approximately 46. We also give refined measurements of the persistence exponents for low dimensions.