The parabolic Anderson model. Random walk in random potential (Q289673)

The purpose of this book is to survey the main results in the field of the parabolic Anderson model (PAM) and its applications. Chapters 1 and 2 are dedicated to the introduction of the main subject and mathematical tools of the book. In Section 1.2 the author introduces the PAM in the spatially discrete and continuous cases. In the discrete case the PAM is represented by the Cauchy problem for the heat equation with random coefficients (potential) and the discrete Laplace operator on the integer lattice. Under the integrability condition on the potential this Cauchy problem has a unique non-negative solution almost surely. In the continuous case for the probabilistic investigation of the Cauchy problem, the Feynman-Kac formula is used. In Section 1.3, the following main questions regarding the PAM are formulated: 1) What is the asymptotic behavior of the total mass of the solutions as time goes to infinity? 2) What are the regions that contribute most to the total mass of the solutions? 3) What do the typical shapes of the potential and the solutions look like in these regions? 4) What is the behavior of the entire process of the mass flow? Chapters 3, 5 and 6 of the book give answers to the above questions. In Chapter 7 the author surveys generalizations of the results of previous chapters using refined techniques. Chapter 8 deals with the dynamic case of the PAM where the random potential is allowed to be time-dependent.