Section 13.1. Rigorous setup for discrete-time continuous-space Markov chains is given concisely in Durrett [133] section 5.6 and in detail in Meyn and Tweedie [263]. For the more sophisticated continuous-time setting see e.g. Rogers and Williams [295]. Aldous et al [10] prove some of the Chapter 4 mixing time inequalities in the discrete-time continuous-space setting.
The central limit theorem (for sums of functions of a Markov chain) does not automatically extend from the finite-space setting (Chapter 2 Theorem 17) to the continuous-space setting: regularity conditions are required. See [263] Chapter 17. But a remarkable result of Kipnis - Varadhan [217] shows that for stationary reversible chains the central limit theorem remains true under very weak hypotheses.
Sections 13.1.1 - 13.1.3. The eigenvalue analysis is classical. The reflection coupling goes back to folklore; see e.g. Lindvall [234] Chapter 6 for applications to multidimensional diffusions and Matthews [259] for Brownian motion in a polyhedron. Burdzy and Kendall [82] give a careful study of coupling for Brownian motion in a triangle. Chen [89] surveys use of coupling to estimate spectral gaps for diffusions on manifolds.
Here is a more concise though less explicit expression for at (13.6) (and hence for at (13.1)). Consider Brownian motions on the circle started at and at . At any time , the former distribution dominates the latter on the interval only, and so
where has Normal law. We quoted this expression in the analysis of Chapter 5 Example 7
Section 13.1.5. Janvresse [195], Porod [285] and Rosenthal [298] study mixing times for other flights on matrix groups involving rotations and reflections; Porod [284] also discusses more general Lie groups.
Section 13.1.6. The mathematical theory has mostly been developed for classes of nested fractals, of which the Sierpinski gasket is the simplest. See Barlow [40], Lindstrøm [233], Barlow [41] for successively more detailed treatments. Closely related is Brownian motion on the continuum random tree, mentioned in section 13.3.4.
One-dimensional diffusions. The continuous-space analog of a birth-and-death process is a one-dimensional diffusion , described by a stochastic differential equation
where is standard Brownian motion and and are suitably regular specified functions. See Karlin and Taylor [209] for non-technical introduction. Theoretical treatments standardize (via a one-to-one transformation ) to the case , though for our purposes the standardization to is perhaps more natural. In this case, if the formula
can give a density function then is the stationary density. Such diffusions relate to two of our topics.
(i) For MCMC, to estimate a density , one can in principle simulate the diffusion with and . This idea was used in Chapter MCMC section 5.
(ii) Techniques for bounding the relaxation time for one-dimensional diffusions parallel techniques for birth-and-death chains [90].
Section 13.2. We again refer to Woess [339] for systematic treatment of random walks on infinite graphs.
Our general theme of using the infinite case to obtain limits for finite chains goes back at least to [14], in the case of ; similar ideas occur in the study of interacting particle systems, relating properties of finite and infinite-site models.
Section 13.2.2. There is a remarkable connection between recurrence of reversible chains and a topic in Bayesian statistics: see Eaton [138]. Properties of random walk on fractal-like infinite subsets of are studied by Telcs [322, 323].
Section 13.2.9. One view of is as one of several “toy models” for the notion of random walk on fractional-dimensional lattice. Also, when we seek to study complicated variations of random walk, it is often simpler to use the hierarchical lattice than itself. See for instance the sophisticated study of self-avoiding walks by Brydges et al [76]; it would be interesting to see whether direct combinatorial methods could reproduce their results.
Section 13.2.10.
Another class of sequences could be defined as follows.
There are certain continuous-time, continuous-space reversible
processes on compact spaces which “hit points” and for which
; for example
(i) Brownian motion on the circle (section 13.1.1)
(ii) Brownian motion on certain fractals (section 13.1.6)
(iii) Brownian motion on the continuum random tree (section 13.3.4).
So for a sequence of finite-state chains one can define the property
as the finite analog of “diffusions which hit points”.
This property holds for the discrete approximations to the examples above:
(i) random walk on the -cycle
(i) random walk on graphs approximating fractals (section 13.1.6)
(iii) random walk on random -vertex trees (section 13.3.4).
Equivalence (13.58) is hard to find in textbooks. The property “trivial boundary” is equivalent to “no non-constant bounded harmonic functions” ([339] Corollary 24.13), which is equivalent ([328] Theorem 6.5.1) to existence of successful shift-coupling of two versions of the chain started at arbitrary points. The property (13.58) is equivalent ([328] Theorem 4.9.4) to existence of successful couplings. In the setting of interest to us (continuized chains on countable space), existence of a shift-coupling (a priori weaker than existence of a coupling) for the discrete-time chain implies existence of a coupling for the continuous-time chain, by using independence of jump chain and hold times.
Section 13.3. Grimmett [177] surveys “random graphical networks” from a somewhat different viewpoint, emphasising connections with statistical physics models.
Section 13.3.1. More precise variants of Proposition 3.35 were developed in the 1970s, e.g. [281, 94]. Lubotzky [243], who attributes this method of proof of Proposition 13.11 to Sarnak [305], asserts the result for but our own calculations give only . Note that Proposition 13.11 uses the permutation model of a -regular random graph. In the alternative uniform model we put balls labeled , balls labeled , and balls labeled into a box; then draw without replacement two balls at a time, and put an edge between the two vertices. In both models the graphs may be improper (multiple edges or self-loops) and unconnected, but are in fact proper with probability and connected with probability as for fixed . Behavior of in the uniform model is implicitly studied in Bollobás [54]. The ideas underlying the proof of Proposition 13.12 were used by Broder and Shamir [66], Friedman [156] and Kahn and Szemerédi [155] in the setting of the permutation model of random -regular graphs. One result is that with probability . Further results in the “random Cayley graph” spirit of Proposition 13.12 can be found in [27, 130, 271].
Section 13.3.2. The monograph of Lyons and Peres [249] contains many more results concerning random walks on infinite deterministic and Galton–Watson trees. A challenging open problem noted in [247] is to prove that has absolutely continuous distribution when is non-constant. The method of fictitious roots used in Proposition 13.14 is also an ingredient in the analysis of cover times on trees [21].
Section 13.3.4.
Moon [264] gives further results in the spirit of (13.70),
e.g. for variances of hitting times.
The fact that random walk on rescales to Brownian motion
on a “continuum random tree” was outlined in
Aldous [22] section 5 and proved in Krebs [218].
While this makes the “order ” property (13.71) of the parameters
essentially obvious, it is still difficult to get explicit information
about the limit distributions .
What’s known [22] is
(a) , as suggested by (13.70);
(b) ,
from (13.69) and the known asymptotics for the diameter of ;
(c) The “cover and return” time
appearing in Chapter 6
satisfies
,
modulo some technical issues.