On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in \mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+\frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A stochastic model for microtubule length dynamics

In this paper, we use random walk theory to describe the length dynamics of microtubules, one of the principal components of the cytoskeleton. We present a simple two-state model (growing and shrinking) of microtubule length evolution that incorporates a variable rate of switching between the states. Using the generating function technique, we calculate the mean length of microtubule, its variance and diffusion coefficient. We also report analytical and computer simulation results for the mean number of positive monomers in microtubule, and find good qualitative agreement with experimental data.     

A random walker on a ratchet potential: effect of a non Gaussian noise

We analyze the effect of a colored non Gaussian noise on a model of a random walker moving along a ratchet potential. Such a model was motivated by the transport properties of motor proteins, like kinesin and myosin. Previous studies have been realized assuming white noises. However, for real situations, in general we could expect that those noises be correlated and non Gaussian. Among other aspects, in addition to a maximum in the current as the noise intensity is varied, we have also found another optimal value of the current when departing from Gaussian behavior. We show the relevant effects that arise when departing from Gaussian behavior, particularly related to current's enhancement, and discuss its relevance for both biological and technological situations.     

Critical comparison of several order-book models for stock-market fluctuations

Far-from-equilibrium models of interacting particles in one dimension are used as a basis for modelling the stock-market fluctuations. Particle types and their positions are interpreted as buy and sel orders placed on a price axis in the order book. We revisit some modifications of well-known models, starting with the Bak-Paczuski-Shubik model. We look at the four decades old Stigler model and investigate its variants. One of them is the simplified version of the Genoa artificial market. The list of studied models is completed by the models of Maslov and Daniels et al. Generically, in all cases we compare the return distribution, absolute return autocorrelation and the value of the Hurst exponent. It turns out that none of the models reproduces satisfactorily all the empirical data, but the most promising candidates for further development are the Genoa artificial market and the Maslov model with moderate order evaporation.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Study on nitridation processes of β-Ga2O3 single crystal

Nitridated /3-Ga2Os (100) substrate was investigated as the substrate for CaN epitaxial growth.The effects of nitridation temperature and surface roughness of -Ga2O3 wafers on the formation of CaN were studied.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process X _t as t0 or t, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t0; for example, we may have X _t /t ^P + (t0) (weak divergence to +), whereas X _t /t a.s. (t0) is impossible (both are possible when t), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. Almost sure stability of X _t , i.e., X _t tending to a nonzero constant a.s. as t or as t0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to strong law behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.     

Blow-up of semilinear pde's at the critical dimension. A probabilistic approach

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation in the critical dimension . By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as . In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of -Laplacians with possibly different parameters .