Schrodinger Operator with Random Potential in One Space Dimension
Sergey V. Lototsky, University of Southern California
While many types of stochastic equations with multiplicative Gaussian noise are well-studied, the particular case of space-only white noise has yet to receive the attention it deserves. In addition to a long history in physics, equations with multiplicative space-only white noise have recently appeared in connection with some problems in random matrix theory.
From mathematical point of view, there are two main types of problems (evolution and stationary), and two distinct interpretations (in the Wick-Ito-Skorokhod and Stratonovich sense). Interpretation as the Wick-Ito-Skorokhod integral provides a complete description of the evolution equation, including construction of the fundamental solution. The Stratonovich, or path-wise, interpretation leads to a variety of different formulations and an interesting problem of reconciliation.