Abstract
Language  English 

Qualification  Ph.D. 
Awarding Institution 

Award date  1 Sep 2009 
Status  Published  2009 
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Keywords
 random environments
 parabolic Anderson model
 Polymers
 aging
Cite this
Stochastic processes in random environment. / Ortgiese, Marcel.
2009.Research output: Thesis › Doctoral Thesis
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TY  THES
T1  Stochastic processes in random environment
AU  Ortgiese,Marcel
PY  2009
Y1  2009
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 timedependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropydominated 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 meanfield 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 timeconstant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the longterm 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 timedependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropydominated 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 meanfield 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 timeconstant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the longterm 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 