MARTINGALE APPROXIMATION AND SCALING LIMITS FOR ADDITIVE FUNCTIONALS OF MARKOV PROCESSES
A TCC COURSE HELD FOR GRADUATE STUDENTS OF MATHEMATICS AT BATH, BRISTOL, IMPERIAL COLLEGE LONDON, OXFORD AND WARWICK
2020 AUTUMN (OXFORD: MICHAELMAS) TERM
2020 AUTUMN (OXFORD: MICHAELMAS) TERM
The Central Limit Theorem for sums of independent random variables has a long history starting in the early 18th century and completed by the mid-20th – with contributions from some of the best mathematicians of two hundred+ years. In the applied context (among abundantly many other applications) it serves to modelling diffusion in its many disguises. However, from a physical point of view, assuming independence of the increments is unrealistic and prohibitively restrictive. The problem of understanding diffusive behaviour of processes with stationary but far from independent increments is mathematically deep and difficult beside being overly well justified by physical (and many other) applications.
This lecture course will focus on mathematically rigorously understanding the diffusive (and occasionally anomalous, super-diffusive) behaviour of such processes arising in a very natural context, like sums/integrals along the trajectory of a Markov chain/process, of some naturally chosen function of its state. Motivations come from random walk/diffusion in random environment, tagged particle (self-) diffusion in interacting particle systems, self-interacting random walk/diffusion, some problems raised in MCMC, etc. In all cases the long memory is essential and non-classical ideas and methods are unavoidable. The basic approach is a martingale approximation initiated in the celebrated paper of C Kipnis and SRS Varadhan (1986) and further refined and developed during the last decades. Hence the shorter title of this lecture course could be Introduction to Kipnis-Varadhan Theory.
Prerequisites: It is assumed that the attending student has solid knowledge of Markov chains/processes and functional analysis on Hilbert spaces, including spectral calculus.
The baby example: CLT for additive functionals of finite irreducible Markov chains
Observing a random scenery along a random walk trajectory. Does the CLT hold?
Random walk in random environment (random conductances, divergence-free random drift field). Does the CLT hold?
Tagged particle diffusion in interacting particles systems.
MCMC
Analytic approach to stationary Markov chains/processes: L^2 framework, Markov semigroups and their infinitesimal generators.
Dynkin martingales.
The Martingale CLT.
The theorem of Gordin and Lifshitz.
Examples revisited.
Reversible setting: Kipnis-Varadhan theorem.
Extension to non-reversible setting, full proof.
Maxwell-Woodroofe theorem and its reduction to KV.
Application: Tagged particle diffusion in symmetric simple exclusion process [SSEP].
Application: Random walk among random conductances.
Application: Persistent random walk in random environment - a caricature of the Lorentz gas.
The Strong Sector Condition.
Application: Tagged particle diffusion in zero mean asymmetric simple exclusion process [ASEP]
Application: Random walk in random divergence-free drift of finite cycle type.
The Graded Sector Condition
Application: Tagged particle diffusion in asymmetric simple exclusion process [ASEP] in $d \geq 3$.
Application: Diffusion in divergence-free Gaussian drift field.
Relaxation of the sector conditions.
Application: Random walk in div-free drift: $H_{-1}$ suffices.
Here follows an annotated list of some of the original research papers where the basic ideas of Kipnis-Varadhan theory had been developed. The lecture course is mainly based on the content of these papers. The list reflects the developments in chronological order. The list will be extended.
MI Gordin, BA Lifšic: Central limit theorem for stationary Markov processes. (Russian) Dokl. Akad. Nauk SSSR 239 (1978), no. 4, 766–767 [English translation: Soviet Math. Dokl. 19 (1978), no. 2, 392–394.] click here for the original Russian version (English translation not available on-line)
This is a short and technically simple but very innovative paper. Assuming that the observed function is in the range of the infinitesimal generator, the Dynkin-martingale is directly at hand. This is the entry point to the theory.
C Kipnis, SRS Varadhan: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1–19 click here for open access
This is the celebrated Kipnis-Varadhan paper. A complete (if-and-only-if) characterization of the efficient martingale approximation is provided in the reversible case (with self-adjoint infinitesimal generator). Applications to tagged particle diffusion in symmetric simple exclusion process and to random walk among random conductances are the motivating applications.
B Tóth: Persistent random walks in random environment. Probab. Theory Relat. Fields 71 (1986), no. 4, 615–625 click here for on-line access
In this paper the Kipnis-Varadhan approach is extended to the non-reversible (non-self-adjoint) setting, replacing spectral calculus with resolvent calculus. The extension is primarily motivated by application to a random walk in random environment problem.
A De Masi, PA Ferrari, S Goldstein, WD Wick: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989), no. 3-4, 787–855. click here for on-line access
This paper provides a survey of Kipnis-Varadhan theory in the reversible setting with many relevant applications.
SRS Varadhan: Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 1, 273–285 click here for open access
The strong sector condition is introduced for handling some non-reversible settings. Applied to tagged particle diffusion in non-symmetric but zero mean simple exclusion.
S Sethuraman, SRS Varadhan,, H-T Yau: Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Comm. Pure Appl. Math. 53 (2000) 972–1006 click here for a copy available at the author's web page
In this paper the so-called graded sector condition is introduced in order to handle some more complex non-reversible cases where the strong sector condition does not hold. The method is applied to tagged particle diffusion in non-symmetric simple exclusion with possibly non-zero mean, in dimension 3 or more.
M Maxwell, M Woodroofe: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000).713–724 click here for open access
This paper provides an alternative approach to control the martingale approximation in general (non-reversible) setting. It is mainly motivated by MCMC.
T Komorowski, S Olla: On the sector condition and homogenization of diffusions with a Gaussian drift. J. Funct. Anal. 197 (2003), no. 1, 179–211. click here for open access
In this paper the Graded Sector Condition method is applied to diffusion in a divergence-free Gaussian random drift field. CLT is obtained under the $H_{-1}$-condition
I Horváth, B Tóth, B Vető: Relaxed sector condition. Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 4, 463–476 click here for a copy available at the author's web page
A weaker (than the graded) sector condition is formulated and proved to be sufficient to control the martingale approximation. It is shown that this is indeed weaker than the gsc.
B Tóth: Comment on a theorem of M. Maxwell and M. Woodroofe. Electron. Commun. Probab. 18 (2013), no. 13, 4 pp click here for open access
It is shown that the theorem of Maxwell-Woodroofe (see above) is a direct consequence of the non-reversible extension (as in Toth (1986)) of Kipnis-Varadhan.
G Kozma, B Tóth: Central limit theorem for random walks in doubly stochastic random environment: $H_{-1}$ suffices. Ann. Probab. 45 (2017), no. 6B, 4307–4347 click here for a copy available at the author's web page
Diffusive CLT is established for random walks in divergence-free random drift field. This is an application of the relaxed sector condition.
TEXTBOOK ABOUT CONTINUOUS TIME MARKOV CHAINS AND PROCESSES
TM Liggett: Continuous Time Markov Processes - An Introduction. Graduate Studies in Mathematics, 113. American Mathematical Society, Providence, RI, 2010. click here for MathSciNet review of the book
TEXTBOOK ON FUNCTIONAL ANALYSIS
M Reed, B Simon: Methods of Modern Mathematical Physics. vol. 1. Functional Analysis, vol. 2. Self-Adjointness. Academic Press New York, 1972-1975.. click here for MathSciNet review of the books
MONOGRAPH ABOUT THE SUBJECT OF THIS COURSE
T Komorowski, C Landim, S Olla: Fluctuations in Markov Processes. Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften vol. 345. Springer, Heidelberg, 2012. click here for MathSciNet review of the book
Huge technical monograph, covering the state of the art at the time of publication.