By Wolfgang König
This is a entire survey at the examine at the parabolic Anderson version – the warmth equation with random power or the random stroll in random capability – of the years 1990 – 2015. The research of this version calls for a mix of instruments from likelihood (large deviations, extreme-value concept, e.g.) and research (spectral concept for the Laplace operator with strength, variational research, e.g.). We clarify the heritage, the purposes, the questions and the connections with different types and formulate the main appropriate effects at the long-time habit of the answer, like quenched and annealed asymptotics for the complete mass, intermittency, confinement and focus houses and mass circulate. in addition, we clarify the main profitable evidence tools and provides a listing of open examine difficulties. Proofs aren't unique, yet concisely defined and commented; the formulations of a few theorems are a bit simplified for greater comprehension.
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Additional info for The Parabolic Anderson Model: Random Walk in Random Potential
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We refer to Sect. 4 for some asymptotic results for the moments for this model. 3 Finite-Space Feynman-Kac Formulas For many proofs, it will be important later to approximate the PAM with finite boxes. Luckily, the two most important types of boundary conditions turn out to serve for very useful lower and upper bounds, respectively. , the zero boundary condition is translated into the condition that the random walk does not leave B by time t. , restricted to not leaving B. 8) satisfies UB Ä U. Now let B D .
This estimate turns out to be very satisfactory, as it matches with a corresponding upper bound. t/ so well that the sum coming from the complement of this island is negligible with respect to the sum from that island. In order to achieve this, we a priori need to collect much more such islands, whose family is then called the intermittent islands in the sense of Sect. 4. Such assertions are handled with the help of spatial extreme-value analysis, which is used to identify the number, location, size and form of such islands.
This can be seen, with the help of the central limit theorem, as follows. Chop the random walk path into t=rt2 pieces of length rt2 to see that staying t time units in a ball with radius r is equivalent to each of these t=rt2 pieces staying in that ball. 0; 1/, according to the central limit theorem. t=rt2 /g. 2r C 1/d=2 . Π1; 1d / as r ! Π1; 1d / is the principal eigenvalue of the Laplace operator in Π1; 1d with Dirichlet boundary condition. 3 First Heuristic Observations 39 1; 1d / rt2 .