Persistence distributions for non gaussian markovian processes

We propose a systematic method to derive the asymptotic behaviour of the persistence distribution, for a large class of stochastic processes described by a general Fokker-Planck equation in one dimension. Theoretical predictions are compared to simple solvable systems and to numerical calculations. The very good agreement attests the validity of this approach.Comment: 7 pages, 1 figure, to be published in Europhysics Letter

Persistence in a Stationary Time-series

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times. We then construct a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. We show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice.Comment: 8 pages revte

Persistence in Cluster--Cluster Aggregation

Persistence is considered in diffusion--limited cluster--cluster aggregation, in one dimension and when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. The empty and filled site persistences are defined as the probabilities, that a site has been either empty or covered by a cluster all the time whereas the cluster persistence gives the probability of a cluster to remain intact. The filled site one is nonuniversal. The empty site and cluster persistences are found to be universal, as supported by analytical arguments and simulations. The empty site case decays algebraically with the exponent $\theta_E = 2/(2 - \gamma)$. The cluster persistence is related to the small $s$ behavior of the cluster size distribution and behaves also algebraically for $0 \le \gamma < 2$ while for $\gamma < 0$ the behavior is stretched exponential. In the scaling limit $t \to \infty$ and $K(t) \to \infty$ with $t/K(t)$ fixed the distribution of intervals of size $k$ between persistent regions scales as $n(k;t) = K^{-2} f(k/K)$, where $K(t) \sim t^\theta$ is the average interval size and $f(y) = e^{-y}$. For finite $t$ the scaling is poor for $k \ll t^z$, due to the insufficient separation of the two length scales: the distances between clusters, $t^z$, and that between persistent regions, $t^\theta$. For the size distribution of persistent regions the time and size dependences separate, the latter being independent of the diffusion exponent $\gamma$ but depending on the initial cluster size distribution.Comment: 14 pages, 12 figures, RevTeX, submitted to Phys. Rev.