Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond

We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of L\'evy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also L\'evy processes, of which Brownian Motion is a special case, have become increasingly popular. L\'evy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include L\'evy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.Comment: 22 pages, including 4 figures and 8 pages of source code for the generation of sample paths of Feller processe

Analysis of Polyethylene Glycol in the α-Hemolysin Nanopore

Nanopores have been shown to be a useful analytical tool for single molecule detection. They have been used to study the composition of DNA and other molecules of interest. These pores are usually α-hemolysin which is a toxin from Staphylococcus aureus or more recently nanoscale synthetic solid state pores. Now we are beginning to look at other molecules or proteins by sending them into the nanopores and measuring a characteristic partial current blockade. In this thesis we look at polyethylene glycol (PEG) as it enters and blocks current through a single alpha hemolysin pore. We report the effects of ionic strength, PEG size, and applied voltage on the depth and duration of the current blockades. We also apply autocorrelation analysis on the arrival times of PEG molecules to the pore see if we can identify if the PEG is translocating through the pore or escaping from the same side it enters. This suggests a new approach to current blockade analysis