Stochastic processes in random environment

Research output: ThesisDoctoral Thesis

Abstract

We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer, represented as the path of a simple random walk on a lattice, interacts with an environment given by a collection of time-dependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropy-dominated regime at high temperatures, to a localised regime at low temperatures. Our main result shows that at high temperatures, even though a central limit theorem holds, we can identify a set of paths constituting a vanishing fraction of all paths that supports the free energy. We compare the situation to a mean-field model defined on a regular tree, where we can also describe the situation at the critical temperature. Secondly, we consider the parabolic Anderson model, which is the Cauchy problem for the heat equation with a random potential. Our setting is continuous in time and discrete in space, and we focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.
LanguageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Award date1 Sep 2009
StatusPublished - 2009

Fingerprint

Random Environment
Stochastic Processes
Anderson Model
Path
Directed Polymers
Random Potential
Simple Random Walk
Mean-field Model
Time Windows
Critical Temperature
Time Constant
Probabilistic Model
Central limit theorem
Heat Equation
Cauchy
Envelope
Free Energy
Tail
Polymers
Linearly

Keywords

  • random environments
  • parabolic Anderson model
  • Polymers
  • aging

Cite this

Stochastic processes in random environment. / Ortgiese, Marcel.

2009.

Research output: ThesisDoctoral Thesis

Ortgiese, M 2009, 'Stochastic processes in random environment', Ph.D., University of Bath.
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N2 - We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer, represented as the path of a simple random walk on a lattice, interacts with an environment given by a collection of time-dependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropy-dominated regime at high temperatures, to a localised regime at low temperatures. Our main result shows that at high temperatures, even though a central limit theorem holds, we can identify a set of paths constituting a vanishing fraction of all paths that supports the free energy. We compare the situation to a mean-field model defined on a regular tree, where we can also describe the situation at the critical temperature. Secondly, we consider the parabolic Anderson model, which is the Cauchy problem for the heat equation with a random potential. Our setting is continuous in time and discrete in space, and we focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

AB - We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer, represented as the path of a simple random walk on a lattice, interacts with an environment given by a collection of time-dependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropy-dominated regime at high temperatures, to a localised regime at low temperatures. Our main result shows that at high temperatures, even though a central limit theorem holds, we can identify a set of paths constituting a vanishing fraction of all paths that supports the free energy. We compare the situation to a mean-field model defined on a regular tree, where we can also describe the situation at the critical temperature. Secondly, we consider the parabolic Anderson model, which is the Cauchy problem for the heat equation with a random potential. Our setting is continuous in time and discrete in space, and we focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

KW - random environments

KW - parabolic Anderson model

KW - Polymers

KW - aging

M3 - Doctoral Thesis

ER -