Random Walks on Random Lattices and Their Applications (May 2015) [PDF] [Defense Slides]
Analysis of multidimensional stochastic processes with mutually dependent components and their fluctuations about thresholds. The process experiences multidimensional increments of random magnitude (for each component) upon random times, while the values of the components are viewed by an independent delayed random process rather than in real-time. I consider these types of processes particularly in the context of successive losses of weighted nodes and edges in random graphs as well as more general marked point processes.
Consider some counters that increase over time, each with a threshold. The counters increase at random times by random amounts such that each is related to one another.
We answer questions such as
Which threshold will be crossed first?
How long will it likely take for this first threshold crossing to occur?
By how much will it likely surpass its threshold?
What can we adjust to effectively increase the time until a threshold crossing?