Asymptotic Survival Probability Research Articles

Error-induced extinction in a multi-type critical birth–death process

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth–death processes with n distinct types as a birth–death model of EEX in a growing population. Each type-i cell divides independently (i)→(i)+(i)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(i)\\rightarrow (i)+(i)$$\\end{document} or mutates (i)→(i+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(i)\\rightarrow (i+1)$$\\end{document} at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-n appears, which cell type can only divide or die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general n-type critical birth–death process. We find that the mass function of the number of cells of type-k has algebraic and stationary tail (size)-1-χk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {size})^{-1-\\chi _k}$$\\end{document}, with χk=21-k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _k=2^{1-k}$$\\end{document}, for k=2,⋯,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=2,\\dots ,n$$\\end{document}, in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability (time)-ξk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {time})^{-\\xi _k}$$\\end{document}. We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.

Sisyphus random walks in the presence of moving traps

It has recently been proved that, in the presence of a static absorbing trap, Sisyphus random walkers with a restart mechanism are characterized by exponentially decreasing asymptotic survival probability functions. Interestingly, in the present compact paper we prove analytically that, in the presence of a moving trap whose velocity approaches zero asymptotically in time as vtrap∼1/t, the survival probabilities of the Sisyphus walkers are dramatically changed into inverse power-law decaying tails.

Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties.

The paper addresses Brownian motion in the logarithmic potential with time-dependent strength, U(x, t) = g(t)log(x), subject to the absorbing boundary at the origin of coordinates. Such model can represent kinetics of diffusion-controlled reactions of charged molecules or escape of Brownian particles over a time-dependent entropic barrier at the end of a biological pore. We present a simple asymptotic theory which yields the long-time behavior of both the survival probability (first-passage properties) and the moments of the particle position (dynamics). The asymptotic survival probability, i.e., the probability that the particle will not hit the origin before a given time, is a functional of the potential strength. As such, it exhibits a rather varied behavior for different functions g(t). The latter can be grouped into three classes according to the regime of the asymptotic decay of the survival probability. We distinguish 1. the regular (power-law decay), 2. the marginal (power law times a slow function of time), and 3. the regime of enhanced absorption (decay faster than the power law, e.g., exponential). Results of the asymptotic theory show good agreement with numerical simulations.

Asymptotic survival probability of a particle in reaction–diffusion process with exclusion in presence of traps

Reaction-diffusion process with exclusion in the presence of traps has been studied. The asymptotic survival probability for the case of uniformly distributed random traps shows a stretched e\ xponential behavior. We show that additional correction terms appear in the stretched exponent when exclusion is taken into account. Analytically it is shown to be $\sim t^{1/6}$ which is ver\ ified by numerical simulations.

Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: A fractional equation approach

We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability ∝ t − α , but independent of the dimension, where α is the anomalous diffusion exponent.

Critical branching random walk in an IID environment

Using a high performance computer cluster, we run simulations regarding an open problem aboutd -dimensional critical branching random walks in a random IID en- vironment The environment is given by the rule that at every site independently, with probability p 2 Œ0;1� , there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability inn steps is the same in the annealed and the quenched case; it is 2 qn ,w hereq WD1� p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probabilityq for every particle at every iteration.

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

Target annihilation by diffusing particles in inhomogeneous geometries

The survival probability of immobile targets annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks, and gels, is analytically investigated. It is shown that, while it cannot in general be related to the number of distinct visited points as in the case of homogeneous lattices, in the case of bounded coordination numbers its asymptotic behavior at large times can still be expressed in terms of the spectral dimension d and its exact analytical expression is given. The results show that the asymptotic survival probability is site-independent of recurrent structures (d<or=2) , while on transient structures (d>2) it can strongly depend on the target position, and such dependence is explicitly calculated.

Capture of particles undergoing discrete random walks

It is shown that particles undergoing discrete-time jumps in three dimensions, starting at a distance r(0) from the center of an adsorbing sphere of radius R, are captured with probability (R-c sigma)/r(0) for r(0)>>R, where c is related to the Fourier transform of the scaled jump distribution and sigma is the distribution's root-mean square jump length. For particles starting on the surface of the sphere, the asymptotic survival probability is nonzero (in contrast to the case of Brownian diffusion) and has a universal behavior sigma/(R square root(6)) depending only upon sigma/R. These results have applications to computer simulations of reaction and aggregation.

Subdiffusive target problem: Survival probability

The asymptotic survival probability of a spherical target in the presence of a single subdiffusive trap or surrounded by a sea of subdiffusive traps in a continuous Euclidean medium is calculated. In one and two dimensions the survival probability of the target in the presence of a single trap decays to zero as a power law and as a power law with logarithmic correction, respectively. The target is thus reached with certainty, but it takes the trap an infinite time on average to do so. In dimensions higher than two a single trap may never reach the target and so the survival probability is finite and, in fact, does not depend on whether the traps move diffusively or subdiffusively. When the target is surrounded by a sea of traps, on the other hand, its survival probability decays as a stretched exponential in all dimensions (with a logarithmic correction in the exponent for d=2). A trap will therefore reach the target with certainty, and will do so in a finite time. These results may be directly related to enzyme binding kinetics on DNA in the crowded cellular environment.

Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents

A number of results for reactions involving subdiffusive species, all with the same anomalous exponent , have recently appeared in the literature and can often be understood in terms of a subordination principle whereby time in ordinary diffusion is replaced by tgamma. However, very few results are known for reactions involving different species characterized by different anomalous diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive particle surrounded by a sea of (sub)diffusive traps in one dimension. We find rigorous results for the asymptotic survival probability of the particle in most cases, with the exception of the case of a particle that diffuses normally while the anomalous diffusion exponent of the traps is smaller than 2/3.

Time-dependent perturbation theory for diffusive non-equilibrium lattice models

The authors recently presented a perturbation theory for the asymptotic survival probability of an interacting particle system which can become trapped in an absorbing state. They extend the method to a simple diffusive model. Analysis of the resulting series shows that diffusion is an irrelevant perturbation, i.e. it does not change the critical behaviour. Quantitative predictions of the phase boundary are confirmed by results of Monte Carlo simulations.

Asymptotic survival probability on a one-dimensional random chain.

We consider a one-dimensional random chain where the probabilities of taking steps to the right (or left) at any site are identically distributed independent random variables, having the distribution p(${p}_{i}$)=0.5\ensuremath{\delta}(${p}_{i}$-\ensuremath{\alpha}) +0.5\ensuremath{\delta}(${p}_{i}$-1+\ensuremath{\alpha}). Furthermore, this chain contains a random distribution of traps. We compute upper and lower bounds for the asymptotic survival probability. We show that this follows a power law and that the bounds agree as far as the exponent of t is concerned.

Asymptotic survival probability of random walks on a semi-infinite line

Symmetric and asymmetric random walks on a segment (−∞,T>0) of the real line are considered. There is a non-zero probability for the random walk to get absorbed at a site it visits. We derive for such random walks, expressions for survival probabilities in the asymptotic limit ofT→∞. An application of this asymptotic formulation to the problem of radiation transport through thick shields is presented.

Trapping of random walks on the line

Several features of the trapping of random walks on a one-dimensional lattice are analyzed. The results of this investigation are as follows: (1) The correction term to the known asymptotic form for the survival probability ton steps is O((λ2n)−1/3), where λ=−ln(1−c), andc is the trap concentration. (2) The short time form for the survival probability is found to be exp[−a(c)n1/2], wherea(c) is given in Eq. (21). (3) The mean-square displacement of a surviving random walker is found to go liken2/3for largen. (4) When the distribution of trap-free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn. One such model is studied.

Extreme values and crossings for the X 2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks thesurvivalprobabilitywhereX(t) = (X1(t), …,Xn(t)) is a stationary, multivariate Gaussianloadprocess, andSis asafe region.In general, the asymptotic survival probability for largeT-values is the most interesting quantity.By considering the point process formed by the extreme points of the vector processX(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x1, …,xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf&amp;gt;x∉S||x||, i.e. the smallest distance from the origin to an unsafe point.

Extreme values and crossings for theX2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks thesurvivalprobabilitywhereX(t) = (X1(t), …,Xn(t)) is a stationary, multivariate Gaussianloadprocess, andSis asafe region.In general, the asymptotic survival probability for largeT-values is the most interesting quantity.By considering the point process formed by the extreme points of the vector processX(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x1, …,xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf&gt;x∉S||x||, i.e. the smallest distance from the origin to an unsafe point.

A calculation of photon decay

We present a calculation of the decay of the photon to lowest order in the gravitational coupling constant. The decay is highly non-exponential, and in addition is characterized by a finite asymptotic survival probability. The results are sensitive to the choice of the photon form factor. Depending on how it behaves, one gets: (a) no visible effects (b) the possibility of performing the gravito-electric effect (c) astronomical intensity-frequency correlations, or (d) some combination of the above.