A random walk model related to the clustering of membrane receptors

In a cellular medium, the plasmic membrane is a place of interactions between the cell and its direct external environment. A classic model describes it as a fluid mosaic. The fluid phase of the membrane allows a lateral degree of freedom to its constituents: they seem to be driven by random motions along the membrane. On the other hand, experimentations bring to light inhomogeneities on the membrane; these micro-domains (the so-called rafts) are very rich in proteins and phospholipids. Nevertheless, few functional properties of these micro-domains have been shown and it appears necessary to build appropriate models of the membrane for recreating the biological mechanism. In this article, we propose a random walk model simulating the evolution of certain constituents-the so-called ligands-along a heterogeneous membrane. Inhomogeneities-the rafts-are described as being still clustered receptors. An important variable of interest to biologists is the time that ligands and receptors bind during a fixed amount of time. This stochastic time can be interpreted as a measurement of affinity/sentivity of ligands for receptors. It corresponds to the sojourn time in a suitable set for a certain random walk. We provide a method of calculation for the probability distribution of this random variable and we next determine explicitly this distribution in the simple case when we are dealing with only one ligand and one receptor. We finally address some further more realistic models.Comment: 35 page

A trick around Fibonacci, Lucas and Chebyshev

In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of this trick to more general sequences. This study leads us to interesting connections between Fibonacci, Lucas sequences and Chebyshev polynomials.Comment: 23 page

First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂/∂t=±∂N/∂xN\partial/\partial t = \pm\partial^N/\partial x^N

Consider the high-order heat-type equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $\tau_a^+$ and $\tau_a^-$ of the half lines $(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well as those of the vectors $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X(\tau_a^-))$.Comment: 51 page

From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval

Let $N$ be a positive integer, $c$ be a positive constant and $(U_n)_{n\ge 1}$ be a sequence of independent identically distributed pseudo-random variables. We assume that the $U_n$'s take their values in the discrete set $\{-N,-N+1,...,N-1,N\}$ and that their common pseudo-distribution is characterized by the \textit{(positive or negative) real} numbers $$\mathbb{P}\{U_n=k\}=\delta_{k0}+(-1)^{k-1} c\binom{2N}{k+N}$$ for any $k\in\{-N,-N+1,...,N-1,N\}$. Let us finally introduce $(S_n)_{n\ge 0}$ the associated pseudo-random walk defined on $\mathbb{Z}$ by $S_0=0$ and $S_n=\sum_{j=1}^n U_j$ for $n\ge 1$. In this paper, we exhibit some properties of $(S_n)_{n\ge 0}$. In particular, we explicitly determine the pseudo-distribution of the first overshooting time of a given threshold for $(S_n)_{n\ge 0}$ as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudo-random walk to the pseudo-Brownian motion driven by the high-order heat-type equation $\partial/\partial t=(-1)^{N-1} c\;\partial^{2N}/$ $\partial x^{2N}$. We retrieve the corresponding pseudo-distribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\frac{\partial}{\partial t}=\pm \frac{\partial^N}{\partial x^N}$. Electron. J. Probab. 12 (2007), 300--353 [MR2299920]). In the same way, we get the pseudo-distribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudo-process.Comment: 69 page

Sojourn time in Z+\mathbb{Z}^+ for the Bernoulli random walk on Z\mathbb{Z}

Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementioned article, only the symmetric case ($p=q=1/2$) is considered. This is the discrete counterpart to the famous Paul L\'evy's arcsine law for Brownian motion.Comment: 44 page

Entrance and sojourn times for Markov chains. Application to (L,R)(L,R)-random walks

In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.Comment: 30 page

Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation

Consider the high-order heat-type equation $\partial u/\partial t=\pm \partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$

Joint distribution of the process and its sojourn time on the positive half-line for pseudo-processes governed by high-order heat equation.

Consider the high-order heat-type equation ∂u/∂t = ±∂Nu/∂xN for an integer N > 2 and introduce the related Markov pseudo-process (X(t))t≥0. In this paper, we study the sojourn time T(t) in the interval [0, +∞) up to a fixed time t for this pseudo-process. We provide explicit expressions for the joint distribution of the couple (T(t),X(t))

Joint distribution of the process and its sojourn time in a half-line [a,+∞)[a,+\infty) for pseudo-processes driven by a high-order heat-type equation.

Let (X(t))t≥0 be the pseudo-process driven by the high-order heat-type equation ∂u = ± ∂Nu , ∂t ∂xN where N is an integer greater than 2. We consider the sojourn time spent by (X(t))t≥0 in [a,+∞) (a ∈ R), up to a fixed time t > 0: Ta(t) = 0t 1l[a,+∞)(X(s)) ds. The purpose of this paper is to explicit the joint pseudo-distribution of the vector (Ta(t),X(t)) when the pseudo-process starts at a point x ∈ R at time 0. The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution

Some Darling-Siegert relationships connected with random flights

We derive in detail four important results on integrals of Bessel functions from which three combinatorial identities are extracted. We present the probabilistic interpretation of these identities in terms of different types of random walks, including asymmetric ones. This work extends the results of a previous paper concerning the Darling-Siegert interpretation of similar formulas emerging in the analysis of random flights.Comment: 16 page