Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. By considering a sequence of regeneration times, we will establish an annealed invariance principle under a second moment condition on the trapping times. In the quenched setting, an environment dependent centring is necessary to achieve a central limit theorem. We determine a suitable expression for this centring. As our main motivation, we apply these results to biased walks on subcritical Galton–Watson trees conditioned to survive for a range of bias values.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Polarized radiation transfer in finite binary Markovian random media

The problem of polarized radiation transfer in a finite plane-parallel binary Markovian random medium is studied. Pomraning-Eddington approximation is used for both the total intensity and the difference function of the polarized radiation. The formalism obtained by Levermore et al. and Pomraning is used to average the obtained solution in the deterministic case. The random medium is assumed to have specular-reflecting boundaries with angular-dependent externally incident flux. Numerical results of the average reflectivity and the average transmissivity are calculated for the different degrees of polarization.     

A study on correlated exponential random walks

Based on the linear Boltzmann transport formulation, we investigate the statistics of correlated exponential random walks that are continuous in space and discrete in time. We show that asymptotically, the correlated random walk process is diffusive and derive an effective diffusion constant. We investigate the power spectral characteristics of the associated random forces. We also present some results on the first passage time distribution and establish that asymptotically it reduces to that associated with simple Gaussian walks.     

Ultrasound-enhanced diffusion through isolated frog skin

The effect of ultrasound on the transport of oxygen across excised frog abdominal skin has been studied. Samples were mounted in an exposure chamber in which the Ringer's solution on one side was saturated with oxygen while the other side of the skin had a low initial oxygen concentration. They were treated with ultrasound at 1, 1.5 and 2 W cm⁻² SATA c.w., respectively, and increases in the rate of oxygen transport were observed at all intensities. These increases ranged from 38 ± 4% at 1 W cm⁻² to 55 ± 8% at 2 W cm⁻². Variation in the pulse lengths from 25 to 200 ms and a constant average intensity did not affect the rate of transport significantly provided that the temporal intensity was constant. Since the peak acoustic pressure within the pulse increased with decreasing pulse length and increasing acoustic pressure increases the probability of cavitation occurring, the mechanism responsible for this phenomenon is probably not cavitation.     

A measure of the symmetry of random walks

We derive the probability density for a simple measure of the asymmetry of a one-dimensional random walk, namely the ratio of the minimum of the two maximum displacements in the positive and negative directions, to the maximum. We show that in the diffusion limit the asymmetry is independent of time. These results are shown to apply to random walks in which individual steps have a stable law distribution as well as to multidimensional random walks.     

Renewal analysis of a replacement process

We consider a unit with a random lifetime which is replaced at renewal times by a new identical one regardless of whether it has failed before or not. For this random periodic replacement policy, we derive exact formulas for the cycle length, defined as the time between the replacements of two successive failed units, the stationary probability of the current unit to have failed, and the stationary and the transient distributions of the residual lifetime of the current unit.     

The Effect of Rock Matrix Heterogeneities Near Fracture Walls on the Residence Time Distribution (RTD) of Solutes

For interpreting solute transport experiments within rocks, the Residence Time Distributions (RTDs) of solutes in the rock matrix were either derived from the assumption of a homogeneous matrix, or were considered through analytical distributions of diffusion coefficients. A numerical approach based on a Lagrangian framework was developed in order to investigate the effect of spatial heterogeneities on RTDs. The matrix diffusion was simulated over two-dimensional computation grids, representing virtual or real digitized porosity maps. First, virtual porosity maps were used to mimic porous features linked to conductive fractures, such as (i) low-porosity coatings on fracture walls, and (ii) porosity gradients within the rock matrix. Furthermore, RTD was calculated for the real pore network of the Palmottu granite (Finland). It was shown that the arrangement of spatial heterogeneities located in the immediate vicinity of a conductive fracture modifies the RTD of solutes within the rock matrix. Porous zones located near fractures are of particular importance, because they generate anomalies on the RTD.     

Lower bounds for randomized direct search with isotropic sampling

Randomized direct-search methods for the optimization of a function f:Rⁿ→R that is given by a black box for f-evaluations are investigated. These iterative methods generate new candidate solutions by adding isotropically distributed vectors to the current candidate solution. Lower bounds on the number of f-evaluations necessary for reducing the approximation error in the search space are proved.     

M/M/∞ queues in semi-Markovian random environment

In this paper we investigate an M/M/∞ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed. Part of this research took place while the author was still post-doc at EURANDOM, Eindhoven, The Netherlands. The work was supported by the Spanish Ministry of Education and Science by the Grant MTM2007-63140.