# Tools and Concepts

• Wolfgang König
Chapter
Part of the Pathways in Mathematics book series (PATHMATH)

## Abstract

One of the most interesting features of the PAM is that, being a partial differential equation with random coefficients, it lies in the intersection of probability and functional analysis, which opens up exciting possibilities for combining tools from these two different parts of mathematics. Furthermore, there are classic and well-developed mathematical theories that enable explicit solution formulas and the application of further techniques to the study of the PAM. In this chapter, we give an account on these tools and pave the way for a deep understanding and a powerful analysis of the PAM. We bring the probabilistic side in Sect. 2.1 and the functional analytic side in Sect. 2.2. In Sect. 2.3, we discuss a number of aspects and conclusions that immediately follow from a combination of these tools; a panorama of precise conjectures arises.

## Keywords

Random Walk Principal Eigenvalue Anderson Localisation Instructive Explanation Path Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Wolfgang König
• 1
• 2
1. 1.für Angewandte Analysis und StochastikWeierstraß-InstitutBerlinGermany
2. 2.Institute for MathematicsTU BerlinBerlinGermany