Stochastic processes in random environment


Ortgiese, M., 2009. Stochastic processes in random environment. Thesis (Doctor of Philosophy (PhD)). University of Bath.

Related documents:

PDF (thesis.pdf) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (1039kB) | Preview


    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.


    Item Type Thesis (Doctor of Philosophy (PhD))
    CreatorsOrtgiese, M.
    Uncontrolled Keywordsrandom environments,parabolic anderson model,polymers,aging
    DepartmentsFaculty of Science > Mathematical Sciences
    ID Code17911


    Actions (login required)

    View Item

    Document Downloads

    More statistics for this item...