Fractional Non-linear Quantum Analysis, Probability, Discretization, and Limits

Non-linear quantum equations with a fractional Laplacian arise from limits of models for biophysical systems with self-interactions and long-range interactions, such as organic semiconductors or other biopolymers with quantum activity including an electron moving along DNA. There, a one-dimensional lattice describes it on the local microscopic level, but the macroscopic folding of DNA into chromatin and chromosomes means that the electron can jump a short distance in three-dimensional space which is actually a long distance if measured along the one-dimensional lattice. These long-distance jumps, scaled appropriately in a limit, result in a stable Lévy stochastic process whose infinitesimal generator is a fractional Laplacian. We work with the fractional Laplacian through two definitions that are equivalent, as well as expanded notions of harmonic functions and stochastic processes similar to Brownian motion. We also examine some geometric, probabilistic, and analytic results about this operator, ending with its role in non-linear quantum equations, relationships between discrete and continuous quantum equations, and some mysteries that need to be resolved.