Semi-Markov approach to continuous time random walk limit processes

Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^d$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.Comment: Published in at http://dx.doi.org/10.1214/13-AOP905 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit Theorems and Governing Equations for Levy Walks

The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of beta-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Levy Walk and its limit process are continuous and ballistic in the case beta in (0,1). In the case beta in (1,2), the scaling limit of the process is beta-stable and hence discontinuous. This case exhibits an interesting situation in which scaling exponent 1/beta on the process level is seemingly unrelated to the scaling exponent 3-beta of the second moment. For beta = 2, the scaling limit is Brownian motion

Reflected Spectrally Negative Stable Processes and their Governing Equations

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time

The effect of cave illumination on bats

Artificial light at night has large impacts on nocturnal wildlife such as bats, yet its effect varies with wavelength of light, context, and across species involved. Here, we studied in two experiments how wild bats of cave-roosting species (Rhinolophus mehelyi, R. euryale, Myotis capaccinii and Miniopterus schreibersii) respond to LED lights of different colours. In dual choice experiments, we measured the acoustic activity of bats in response to neutral-white, red or amber LED at a cave entrance and in a flight room – mimicking a cave interior. In the flight room, M. capaccinii and M. schreibersii preferred red to white light, but showed no preference for red over amber, or amber over white light. In the cave entrance experiment, all light colours reduced the activity of all emerging species, yet red LED had the least negative effect. Rhinolophus species reacted most strongly, matching their refusal to fly at all under any light treatment in the flight room. We conclude that the placement and light colour of LED light should be considered carefully in lighting concepts for caves both in the interior and at the entrance. In a cave interior, red LED light could be chosen – if needed at all – for careful temporary illumination of areas, yet areas important for bats should be avoided based on the precautionary principle. At cave entrances, the high sensitivity of most bat species, particularly of Rhinolophus spp., towards light sources almost irrespective of colour, calls for utmost caution when illuminating cave entrances