The fractional Laplacian operator $-{\left(-\Delta \right)}^{\alpha ∕2}$ appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.

We propose to model the stochastic dynamics of a polymer passing through a pore (translocation) by means of a fractional Brownian motion and study its behavior in the presence of an absorbing boundary. Based on scaling arguments and numerical simulations, we present a conjecture that provides a link between the persistence exponent $\theta$ and the Hurst exponent $H$ of the process, thus shedding light on the spatial and temporal features of translocation. Furthermore, we show that this conjecture applies more generally to a broad class of self-affine processes undergoing anomalous diffusion in bounded domains, and we discuss some significant examples.

We present the universal features of the hitting probability Q(x,L), the probability that a generic stochastic process starting at x and evolving in a box [0, L] hits the upper boundary L before hitting the lower boundary at 0. For a generic self-affine process, we show that Q(x,L)=Q(z=x/L) has a scaling Q(z)$\sim${}z$\phi${} as z$\rightarrow${}0, where $\phi${}=$\theta${}/H, H, and $\theta${} being the Hurst and persistence exponent of the process, respectively. This result is verified in several exact calculations, including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical support for our analytical results.

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, and therefore, fluctuations around their average are important: then, it is commonly assumed that the susceptible–infected–recovered mechanism can be described by a stochastic birth–death process of Galton–Watson type. The displacements of the infected individuals can be modeled by resorting to Brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, like in the case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and we compare them with Monte Carlo simulations.