Random walks and diffusing particles have been a corner stone in modelling the random motion of a varying quantity with applications spanning over many research fields. And in most of the applications one can ask a question related to when something happened for the first time. That is, a first-passage problem. Typical examples include chemical reactions which can not happen until the constituents meet for the first time, a neuron firing when a fluctuating voltage exceeds a threshold value and the triggering of buy/sell orders of a stock option. The applications are many, which is why first-passage problems have attracted researchers for a long time, and will keep doing so. In this thesis we analyse first-passage problems analytically.

A stochastic system can always be simulated, so why bother with analytical solutions? Well, there are many system where the first passage is improbable in a reasonable time. Simulating those systems with high precision is hard to do efficiently. But evaluating an analytical expression happens in a heart beat. The only problem is that the first-passage problem is tricky to solve as soon as you take a small step away from the trivial ones. Consequently, many first-passage problems are still unsolved.

In this thesis, we derive approximate solutions to first-passage related problems for a random walker and a diffusing particle bounded in a potential, which the current methods are unable to handle. We also study a continuous-time random walker on a network and solve the corresponding first-passage problem exactly in way that has not been done before. These results give access to a new set of analytical tools that can be used to solve a broad class of first-passage problems.

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.

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.

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.