Abstract
In this article we examine some properties of the solutions of the parabolic Anderson model. In particular we discuss intermittency of the field of solutions of this random partial differential equation, when it occurs and what the field looks like when intermittency doesn't hold. We also explore the behavior of a polymer model created by a Gibbs measure based on solutions to the parabolic Anderson equation.
Highlights
It has been twenty years since the publication of the seminal work “Parabolic Anderson Problem and Intermittency” by Carmona and Molchanov
In this article we examine some properties of the solutions of the parabolic Anderson model
We explore the behavior of a polymer model created by a Gibbs measure based on solutions to the parabolic Anderson equation
Summary
It has been twenty years since the publication of the seminal work “Parabolic Anderson Problem and Intermittency” by Carmona and Molchanov. The equation satisfied by a magnetic field generated by a turbulent flow leads naturally to an analogous parabolic equation with a time varying random field {v(t, x) : t ∈ [0, ∞), x ∈ R3} as opposed to the time stationary field {ξx : x ∈ Zd} which arises in the original localization question. The difference is that the random medium changes rapidly in the case of the magnetic field generated by turbulent flow whereas the impurities in the localization can be considered to be unchanging in time. In the latter case, the fluctuations in the medium are slow compared to the phenomena of interest, capture of electrons. These questions have an affirmative answer in the scalar case and remain difficult open problems in the multidimensional model just discussed
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