In both modern stochastic analysis and more traditional probability and statistics, one way of characterizing a static or dynamic probability distribution is through its quantile function. This paper is focused on obtaining a direct understanding of this function via the classical approach of establishing and then solving differential equations for the function. We establish ordinary differential equations and power series for the quantile functions of several common distributions. We then develop the partial differential equation for the evolution of the quantile function associated with the solution of a class of stochastic differential equations, by a transformation of the Fokker–Planck equation. We are able to utilize the static formulation to provide elementary time-dependent and equilibrium solutions.

Such a direct understanding is important because quantile functions find important uses in the simulation of physical and financial systems. The simplest way of simulating any non-uniform random variable is by applying its quantile function to uniform deviates. Modern methods of Monte–Carlo simulation, techniques based on low-discrepancy sequences and copula methods all call for the use of quantile functions of marginal distributions. We provide web resources for prototype implementations in computer code. These implementations may variously be used directly in live sampling models or in a high-precision benchmarking mode for developing fast rational approximations also for use in simulation.

A model focusing on key components involved in tumour invasion is studied. Tumour cell migration is based on cell motility and haptotaxis, i.e., the directed migratory response of tumour cells up gradients of cell-adhesion molecules. Individual cell processes are modelled according to cell age and several tumour phenotypes are incorporated. Global existence and uniqueness of nonnegative solutions to the corresponding coupled system of nonlinear partial differential equations are shown.

In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4), 897–936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography.