The long time behavior of Brownian motion in tilted periodic potentials

2015 ◽  
Vol 297 ◽  
pp. 1-32 ◽  
Author(s):  
Liang Cheng ◽  
Nung Kwan Yip
Keyword(s):  
Brownian Motion ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Periodic Potentials ◽  
Long Time

scholarly journals On fractional heat equation

2021 ◽  
Vol 24 (1) ◽  
pp. 73-87
Author(s):  
Anatoly N. Kochubei ◽  
Yuri Kondratiev ◽  
José Luís da Silva
Keyword(s):  
Brownian Motion ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Convolution Operators ◽  
Fractional Heat Equation ◽  
Long Time ◽  
Time Changes ◽  
Distributed Order

Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

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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