RESEARCH

My primary research interests lie in probability theory, analysis of stochastic processes, and the underlying mathematics: in short, the study of randomness. My past work focuses on the reliability of stochastic networks, renewal theory, random walks, stochastic game theory, stochastic measure theory, and particularly the application of discrete and continuous operational calculus to stochastic processes. I continue research in these areas, but my work is currently focused on queueing theory, mathematical biology (stochastic analysis of cell dynamics in certain cancers) and probabilistic potential theory in parabolic PDEs.

Dissertation

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.

Non-technical description

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.

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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 most effectively increase (or decrease) the time until a threshold crossing?

Presentations
  • Fluctuation Analysis in Parallel Queues with Hysteretic Control. AMS Fall Southeastern Sectional Meeting, Gainesville, Florida. 2-3 Nov 2019 [Slides]

  • On Exits of Oscillating Random Walks Under Delayed Observation. AMS/MAA Joint Mathematical Meetings, San Diego, California, 10-13 Jan 2018 [Slides]

  • Time Sensitive Analysis of d-dim Independent and Stationary Increment Processes. AMS Fall Southeastern Sectional Meeting, Orlando, Florida. 23-24 Sept 2017. [Slides]

  • Random Walks on Random Lattices and Their Applications. FIT MTH/ORP Graduate Student Seminar. 21 Apr 2015 [Slides]

  • Time Sensitive Analysis of Multivariate Marked Random Walks. SIAM Conference on Computational Science and Engineering. Salt Lake City, UT. 14-18 Mar 2015 [Slides]

  • Time Sensitive Analysis of ISI Processes. FIT Mathematical Sciences Symposium. 8 Jan 2015 [Slides]

  • Random Walks on Random Lattices: An Operational Calculus Approach. FIT MTH/ORP Graduate Student Seminar. 30 Sept 2014 [Slides]

  • Stochastic Analysis of Strategic Networks. 38th Annual SIAM Southeastern Atlantic Section Conference. Melbourne, FL. 29-30 Mar, 2014 [Slides]

Academic Papers
(expand for details and PDFs as publishing agreements allow)
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