Numerous applications all the way from biology and physics to economics depend on the density of first crossings over a boundary. Motivated by the lack of general purpose analytical tools for computing first-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the independent interval approximation (IIA). We generalise previous formulations of the IIA to include arbitrary initial conditions as well as to deal with discrete time and non-smooth continuous time processes. Wederive a closed form expression for the FPTD in z and Laplace-transform space to a boundary in one dimension. Two classes of problems are analysed in detail: discrete time symmetric random walks (Markovian) and continuous time Gaussian stationary processes (Markovian and non-Markovian). Our results are in good agreement with Langevin dynamics simulations.

In applications spanning from image analysis and speech recognition to energy dissipation in turbulence and time-to failure of fatigued materials, researchers and engineers want to calculate how often a stochastic observable crosses a specific level, such as zero. At first glance this problem looks simple, but it is in fact theoretically very challenging, and therefore few exact results exist. One exception is the celebrated Rice formula that gives the mean number of zero crossings in a fixed time interval of a zero-mean Gaussian stationary process. In this study we use the so-called independent interval approximation to go beyond Rice's result and derive analytic expressions for all higher-order zero-crossing cumulants and moments. Our results agree well with simulations for the non-Markovian autoregressive model.

The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time n. Few results are known for the persistence P0(n) in discrete time, except the large time behavior which is characterized by the nontrivial constant θ through P0(n)∼θn. Using a modified version of the independent interval approximation (IIA) that we developed before, we are able to calculate P0(n) analytically in z-transform space in terms of the autocorrelation function A(n). If A(n)→0 as n→∞, we extract θ numerically, while if A(n)=0, for finite n>N, we find θ exactly (within the IIA). We apply our results to three special cases: the nearest-neighbor-correlated "first order moving average process", where A(n)=0 for n>1, the double exponential-correlated "second order autoregressive process", where A(n)=c1λn1+c2λn2, and power-law-correlated variables, where A(n)∼n−μ. Apart from the power-law case when μ<5, we find excellent agreement with simulations.

Several processes in the cell, such as gene regulation, start when key proteins recognize and bind to short DNA sequences. However, as these sequences can be hundreds of million times shorter than the genome, they are hard to find by simple diffusion: diffusion-limited association rates may underestimate in vitro measurements up to several orders of magnitude. Moreover, the rates increase if the DNA is coiled rather than straight. Here we model how this works in vivo in mammalian cells. We use chromatin-chromatin contact data from Hi-C experiments to map the protein target-search onto a network problem. The nodes represent DNA segments and the weight of the links are proportional to measured contact probabilities. We then put forward a diffusion-reaction equation for the density of searching protein that allows us to calculate the association rates across the genome analytically. For segments where the rates are high, we find that they are enriched with active gene starts and have high RNA expression levels. This paper suggests that the DNA's 3D conformation is important for protein search times in vivo and offers a method to interpret protein-binding profiles in eukaryotes that cannot be explained by the DNA sequence itself.