scholarly journals Long time behavior of the transition probability of a random walk with drift on an abelian covering graph

2003 ◽  
Vol 55 (2) ◽  
pp. 255-269 ◽  
Author(s):  
Tomoyuki Shirai
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Covering Graph ◽  
Random Walk With Drift ◽  
Long Time ◽  
Abelian Covering

scholarly journals A memory-based random walk model to understand diffusion in crowded heterogeneous environment

2018 ◽  
Vol 32 (16) ◽  
pp. 1850193
Author(s):  
Sabeeha Hasnain ◽  
Upendra Harbola ◽  
Pradipta Bandyopadhyay
Keyword(s):  
Monte Carlo ◽  
Random Walk ◽  
Analytical Solution ◽  
Heterogeneous Environments ◽  
Time Behavior ◽  
Heterogeneous Environment ◽  
Long Time Behavior ◽  
Current Time ◽  
Long Time ◽  
Mc Simulation

We study memory-based random walk models to understand diffusive motion in crowded heterogeneous environments. The models considered are non-Markovian as the current move of the random walk is determined by randomly selecting a move from history. At each step, particle can take right, left or stay moves which is correlated with the randomly selected past step. There is a perfect stay–stay correlation which ensures that the particle does not move if the randomly selected past step is a stay move. The probability of traversing the same direction as the chosen history or reversing it depends on the current time and the time or position of the history selected. The time- or position-dependent biasing in moves implicitly corresponds to the heterogeneity of the environment and dictates the long-time behavior of the dynamics that can be diffusive, sub or superdiffusive. A combination of analytical solution and Monte Carlo (MC) simulation of different random walk models gives rich insight on the effects of correlations on the dynamics of a system in heterogeneous environment.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

scholarly journals Stochastic Models for a Chemostat and Long-Time Behavior

2013 ◽  
Vol 45 (03) ◽  
pp. 822-836 ◽  
Author(s):  
Pierre Collet ◽  
Servet Martínez ◽  
Sylvie Méléard ◽  
Jaime San Martín
Keyword(s):  
Population Size ◽  
Nutrient Concentration ◽  
Bacterial Population ◽  
Fixed Number ◽  
Death Process ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nutrient Distribution ◽  
Long Time ◽  
Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

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