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I have been recently trying to find context and motivation for my Ph.D. Thesis in probability theory. I have not much background on the history of my theme. I just studied it aiming at solving a particular problem involved.

So my understanding of the matter is very poor. I lack context. I've been looking at the PAM survey from König and Wolff. But despite their effort to approach the theme in a pedagogical way, I feel that context is not very well established in those notes and the whole subject is much more devoted to abstract notions and mathematical motivations that are alien to my knowledge.

So I come to you hopping that you might be able to help me in this quest for historical meaning.

I've looked in Wikipedia, and as far as I see, the work of Phillip W. Anderson is related to the study of particle physics. This particle physics area, however demands quite some knowledge on quantum and relativistic physics. I am willing to learn those subjects (There is the classes from Leonard Susskind on YouTube) but I am afraid I might miss the connection it has with this equation.

Basically, the stochastic differential equation I am studying comes from scaling limits of reaction diffusion models in particle systems and have the form: $$ dX_t(p) = (\Delta X_t(p) - X_t^k(p)) d t + X_t^{l/2} dB_t\,. $$

When $k = 0$ and $l = 2$ this looks a lot with the PAM with the difference that the random field is induced by the state of the system.

The first phrase of the intro of the survey is

Random motions in random media are an important subject in probability theory since there have a lot of applications to real-world problems in the sciences, like astrophysics, magnetohydrodynamics, chemical reactions. For these reasons and also because of their mathematical interest, they have been studied a lot for decades, with a particular intensity in the last twenty years. There is a number of different models of random motions in random media, like random walk in random environment, the random conductance model, random walk in random scenery, random walk in a random potential. Because of the variety of models, there is also a variety of questions and of mathematical methods to solve the questions, like homogenisation, subadditive ergodic theorems, Lyapunov exponents.

Can you help me find a way to understand those examples and motivations?

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    $\begingroup$ The best way to "find motivations and historical meaning" for your thesis is to talk to your thesis adviser. $\endgroup$ – Alexandre Eremenko Nov 30 '15 at 19:33
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    $\begingroup$ I did some editing, but the trouble I see with this question is that parabolic Anderson model is too special a subject likely unfamiliar to most users (people familiar with it are more likely to be on Physics SE), and the long list of related topics is too broad, and " find a way to understand", too vague for a meaningful answer. Perhaps try to select a specific issue about the equation that obstructs you the most and try to focus the question more narrowly on it? $\endgroup$ – Conifold Nov 30 '15 at 21:32
  • $\begingroup$ The part about "understanding examples" seems to be off-topic, as it is a pure science/math/etc. question. If you focused on the historical motivations (in addition to what @Conifold said), then this might be better focused. $\endgroup$ – HDE 226868 Dec 2 '15 at 21:14

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