MATH 289A: TOPICS IN PROBABILITY (FALL 2023)

Topic: Mathematics of Stochastic Networks in Biology

Professor: Professor R. J. Williams, AP&M 7161.
Time: Tu, Th 5-6.30pm.
Place: AP&M B412.

Office Hour: TBA (via Zoom).

Background: Stochastic processes of various types are used to model the noisy dynamics of complex networks in a wide variety of applications in science and engineering. There are challenging mathematical problems stemming from the need to analyse and control such networks.

Content: This course will describe mathematical techniques for modeling and analysing stochastic networks, with particular emphasis on models of complex biological systems, which is a growing area of application. Although techniques will be illustrated with biological examples, the techniques themselves are more generally applicable. These techniques will include those for rigorously deriving stochastic process level approximations at various scales (especially deterministic differential equation and diffusion approximations), and for analyzing the behavior of these models will be included. Some open problems will be presented and students will have the opportunity to present current research papers as part of the course assessment.

The mathematical topics covered will include path spaces for stochastic processes, weak convergence of processes, fundamental building block processes and invariance principles, stationary distributions for Markov processes, fluid models and reflected diffusion processes.

Prerequisites: This is an advanced graduate probability course featuring a topic of current research interest. Students enrolling in this course should have a background in probability at the level of Math 280AB (courses such as Math 285, Math 280C, Math 286 provide additional useful background).

References:

Background References:

  • P. Billingsley, Convergence of Probability Measures, Wiley, 1999.
  • S. N. Ethier and T. Kurtz, Markov Processes, Wiley, 1986.
  • J. Jacod and A. Shiryaev, Limit theorems for stochastic processes, 2nd edition, Springer-Verlag, New York, 2003.
  • F. P. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979, reprinted 1987, 1994.
  • W. Whitt, Stochastic Process Limits, Springer, 2002.

    Biochemical Reaction Networks

  • D. F. Anderson and T. G. Kurtz, Continuous time Markov chain models for chemical reaction networks, chapter in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology, H. Koeppl. et al. (eds.), Springer.

    References on Reflected Brownian Motions

  • Harrison, J. Michael; Reiman, Martin I. Reflected Brownian Motion on an Orthant. Ann. Probab. 9 (1981), no. 2, 302--308. doi:10.1214/aop/1176994471. Click here for the paper.
  • Reiman, M. I., and Williams, R. J., A boundary property of semimartingale reflecting Brownian motions, Probability Theory and Related Fields 77 (1988), 87-97. Link to paper. Correctional note link.
  • Taylor, L. M. and Williams, R. J., Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant, Probability Theory and Related Fields 96 (1993), 283-317, Click here to access the paper.
  • Dupuis, Paul; Ishii, Hitoshi. SDEs with Oblique Reflection on Nonsmooth Domains. Ann. Probab. 21 (1993), no. 1, 554--580. doi:10.1214/aop/1176989415. Click here for the paper. Correctional note: Ann. Probab. 36 (2008), no. 5, 1992--1997. doi:10.1214/07-AOP374. Click here for correctional note.
  • M. Shashiashvili, A lemma of variational distance between maximal functions with applications to the skorokhod problem in a nonnegative orthant with state-dependent reflection directions, Stochastics and Stochastics Reports, 48 (1994), 161-194. For the paper, click here.
  • R. J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic Networks, IMA Volumes in Mathematics and Its Applications, Volume 71, eds. F. P. Kelly and R. J. Williams, Springer-Verlag, New York, 1995, pp. 125-137. Survey up through 1995. For a copy click here.
  • S. Ramasubramanian, A subsidy-surplus model and the Skorokhod Problem in an orthant, Math of Operations Research, 25 (2000), 509-538.
  • Kang, W.; Williams, R. J. An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 (2007), no. 2, 741--779. doi:10.1214/105051606000000899. Click here for the paper.

    Questions: Please direct any questions to Professor Williams (rjwilliams at ucsd dot edu).

    Last updated February 17, 2023.