Special examples of Markov processes, such as random walks in discrete time and Brownian motion in continuous time, have occurred many times in preceding chapters as illustrative examples of martingales and Markov processes.

1. 1.

   The authors thank our colleague Yevgeniy Kovchegov for suggesting this example to illustrate products of random matrices. Such examples as this, including the positivity constraints, arise naturally in the context of mathematical biology.

2. 2.

   A comprehensive treatment of such Markov processes can be found in Bhattacharya, R., and M. Majumdar (2007). Limit distributions of products of random matrices has been treated in some generality by Kaijser, T.(1978): A limit theorem for Markov chains on compact metric spaces with applications to random matrices,Duke Math. J.45, 311–349; Kesten, H. and F. Spitzer (1984): Convergence in distribution of products of random matrices,Z. Wahrsch. Verw. Gebiete67 363–386.

3. 3.

   The original calculations of the Ehrenfests and Smoluchowski were for the mean recurrence times. Such calculations are easily made from the general mean return-time formula\({\mathbb {E}}_i\tau _i = {1\over \pi _i}\), where\(\tau _i = \inf \{n\ge 1: Y_n = i\}, i\in S\), for irreducible, ergodic Markov chains. In particular, using the formula for\(\pi \) and Stirling’s formula,\({\mathbb {E}}_0\tau _0 \sim 2^{20,000}\),\({\mathbb {E}}_d\tau _d\sim 100\sqrt{\pi }\), for the same numerical values for the number of balls and transition rate; e.g., see Kac (1947): Random walk and the theory of Brownian motion,Am. Math. Monthly,54(7), 369–391. The mean-return time formula and more general theory can be found in standard treatments of discrete parameter Markov processes.

4. 4.

   The invariant distribution of the Ornstein–Uhlenbeck process is referred to as the Maxwell–Boltzmann distribution. The physics of fluids requires that the variance be given by the physical parameter\({\kappa T\over m}\) where\(\kappa \) is Boltzmann constant,T is absolute temperature, andm is the mass of the particle.

5. 5.

   This random walk plays a role in probabilistic analysis of the incompressible Navier–Stokes equations introduced by Y. LeJan, A. S. Sznitman (1997): Stochastic cascades and three-dimensional Navier–Stokes equations.Probab. Theory Related Fields 109, no. 3, 343–366. This particular structure was exploited in Dascaliuc, R., N. Michalowski, E. Thomann, E. Waymire (2015): Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations,Chaos, American Physical Society, 25 (7). The dilogarithmic functions are well-studied and arise in a variety of unrelated contexts.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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