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.

Date of Award | 1 Sept 2009 |
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Original language | English |
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Awarding Institution | |
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Supervisor | Peter Morters (Supervisor) |
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- random environments
- parabolic Anderson model
- Polymers
- aging

Stochastic processes in random environment

Ortgiese, M. (Author). 1 Sept 2009

Student thesis: Doctoral Thesis › PhD