Stochastic flows in the Brownian web and net

172 Pages·2014·2.095 MB·English

by  Emmanuel Schertzer, Rongfeng Sun, Jan M. Swart

MEMOIRS
of the
American Mathematical Society
Volume 227 • Number 1065 (first of 4 numbers) • January 2014
Stochastic Flows
in the Brownian Web and Net
Emmanuel Schertzer
Rongfeng Sun
Jan M. Swart
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 227 • Number 1065 (first of 4 numbers) • January 2014
Stochastic Flows
in the Brownian Web and Net
Emmanuel Schertzer
Rongfeng Sun
Jan M. Swart
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Schertzer,Emmanuel,1978-
Stochasticflows in the Brownianweband net / EmmanuelSchertzer,Rongfeng Sun, Jan M.
Swart.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 227,
number1065)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-9088-2(alk. paper)
1. Brownian motion processes. 2. Stochastic analysis. I. Sun, Rongfeng, 1975- II. Swart,
JanM.,1970- III.Title.
QA274.75.S34 2014
519.2(cid:2)33–dc23 2013035390
DOI:http://dx.doi.org/10.1090/S0065-9266-2013-00687-9
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Contents
Chapter 1. Introduction 1
Chapter 2. Results for Howitt-Warren flows 11
Chapter 3. Construction of Howitt-Warren flows in the Brownian web 23
Chapter 4. Construction of Howitt-Warren flows in the Brownian net 35
Chapter 5. Outline of the proofs 45
Chapter 6. Coupling of the Brownian web and net 47
Chapter 7. Construction and convergence of Howitt-Warren flows 75
Chapter 8. Support properties 87
Chapter 9. Atomic or non-atomic 97
Chapter 10. Infinite starting mass and discrete approximation 111
Chapter 11. Ergodic properties 117
Appendix A. The Howitt-Warren martingale problem 129
Appendix B. The Hausdorff topology 141
Appendix C. Some measurability issues 145
Appendix D. Thinning and Poissonization 149
Appendix E. A one-sided version of Kolmogorov’s moment criterion 151
References 153
Index 159
iii

Abstract
It is known that certain one-dimensional nearest-neighbor random walks in
i.i.d. random space-time environments have diffusive scaling limits. Here, in the
continuum limit, the random environment is represented by a ‘stochastic flow of
kernels’, which is a collection of random kernels that can be loosely interpreted as
the transition probabilities of a Markov process in a random environment. The
theory of stochastic flows of kernels was first developed by Le Jan and Raimond,
who showed that each such flow is characterized by its n-point motions. Our work
focuses on a class of stochastic flows of kernels with Brownian n-point motions
which, after their inventors, will be called Howitt-Warren flows.
Our mainresultgivesagraphical constructionofgeneralHowitt-Warrenflows,
where the underlying random environment takes on the form of a suitably marked
Brownian web. This extends earlier work of Howitt and Warren who showed that
a special case, the so-called ‘erosion flow’, can be constructed from two coupled
‘stickyBrownianwebs’. Our constructionforgeneral Howitt-Warrenflows isbased
onaPoissonmarkingproceduredevelopedbyNewman,RavishankarandSchertzer
fortheBrownianweb. Alternatively,weshowthataspecialsubclassoftheHowitt-
Warren flows can be constructed as random flows of mass in a Brownian net, in-
troduced by Sun and Swart.
Using these constructions, we prove some new results for the Howitt-Warren
flows. Inparticular, weshow thatthekernelsspreadwithafinitespeedandhavea
locallyfinite support atdeterministic timesif and only if theflow is embeddable in
aBrowniannet. Weshowthatthekernelsarealwayspurelyatomicatdeterministic
times, but, with the exception of the erosion flows, exhibit random times when the
ReceivedbytheeditorNovember17,2010.
ArticleelectronicallypublishedonMay24,2013;S0065-9266(2013)00687-9.
2010MathematicsSubjectClassification. Primary82C21;Secondary60K35,60K37,60D05.
Key words and phrases. Brownian web, Brownian net, stochastic flow of kernels, measure-
valued process, Howitt-Warren flow, linear system, random walk in random environment, finite
graphrepresentation.
J.M.SwartissponsoredbyGACˇRgrants201/07/0237and201/09/1931. R.Sunissupported
by grants R-146-000-119-133and R-146-000-148-112from the National University of Singapore.
E.SchertzerandJ.M.SwartthanktheNationalUniversityofSingaporeforhospitalityandsup-
port.
Affiliations at time of publication: Emmanuel Schertzer, LPMA, Universit´e Pierre et
Marie Curie, France, [email protected]; Rongfeng Sun, Department of Mathemat-
ics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore, email:
[email protected]; and Jan M. Swart, Institute of Information Theory and Automation of the
ASCR(U´TIA),Podvod´arenskouvˇeˇz´ı4,18208Praha8,CzechRepublic,email: [email protected].
(cid:3)c2013 American Mathematical Society
v
vi ABSTRACT
kernels are purely non-atomic. We moreover prove ergodic statements for a class
of measure-valued processes induced by the Howitt-Warren flows.
Our work also yields some new results in the theory of the Brownian web and
net. In particular, we prove several new results about coupled sticky Brownian
webs and about a natural coupling of a Brownian web with a Brownian net. We
also introduce a ‘finite graph representation’ which gives a precise description of
how paths in the Brownian net move between deterministic times.
CHAPTER 1
Introduction
1.1. Overview. In[LR04a], LeJanandRaimondintroducedthenotionofa
stochastic flow of kernels, which is a collection of random probability kernels that
can be loosely viewed as the transition kernels of a Markov process in a random
space-time environment, where restrictions of the environment to disjoint time in-
tervals are independent and the environment is stationary in time. For suitable
versions of such a stochastic flow of kernels (when they exist), this loose interpre-
tation is exact, see Definition 2.1 below and the remark following it. Given the
environment, onecansamplenindependentcopiesoftheMarkov processandthen
average over the environment. This defines the n-point motion for the flow, which
satisfies a natural consistency condition: namely, the marginal distribution of any
k components of an n-point motion is necessarily a k-point motion. A fundamen-
tal result of Le Jan and Raimond [LR04a] shows that conversely, any family of
Feller processes that is consistent in this way gives rise to an (essentially) unique
stochastic flow of kernels.
As an example, in [LR04b], the authors used Dirichlet forms to construct a
consistentfamilyofreversiblen-pointmotionsonthecircle,whichareα-stableL´evy
processes with some form of sticky interaction characterized by a real parameter
θ. In particular, for α = 2, these are sticky Brownian motions. Subsequently,
Howitt and Warren [HW09a] used a martingale problem approach to construct a
much larger class of consistent Feller processes on R, which are Brownian motions
withsomeformofstickyinteractioncharacterizedbyafinitemeasureν on[0,1]. In
particular,ifν isamultipleoftheLebesguemeasure,thesearethestickyBrownian
motions of Le Jan and Raimond. From now on, and throughout this paper, we
specialize to the case of Browian underlying motions. By the general result of Le
Jan and Raimond mentioned above, the sticky Brownian motions of Le Jan and
Raimond, resp. Howitt and Warren, are the n-point motions of an (essentially)
unique stochastic flow of kernels on R, which we call a Le Jan-Raimond flow, resp.
Howitt-Warren flow (the former being a special case of the latter). It has been
shown in [LL04,HW09a] that these objects can be obtained as diffusive scaling
limits of one-dimensional random walks in i.i.d. random space-time environments.
ThemaingoalofthepresentpaperistogiveagraphicalconstructionofHowitt-
Warrenflowsthatfollowsascloselyaspossible thediscreteconstructionofrandom
walks in an i.i.d. random environment. In particular, we want to make explicit
what represents the random environment in the continuum setting. The original
construction of Howitt-Warren flows using n-point motions does not tell us much
about this. In [HW09b], it was shown that the Howitt-Warren flow with ν =
δ +δ , knownasthe erosion flow, canbeconstructedusing twocoupledBrownian
0 1
webs,whereoneBrownianwebservesastherandomspace-timeenvironment,while
1
2 EMMANUELSCHERTZER,RONGFENGSUN,andJANM.SWART
the conditional law of the second Brownian web determines the stochastic flow of
kernels.
We will extend this construction to general Howitt-Warren flows, where in the
general case, the random environment consists of a Brownian web together with
a marked Poisson point process which is concentrated on the so-called points of
type (1,2) of the Brownian web. A central tool in this construction is a Poisson
marking procedure invented by Newman, Ravishankar and Schertzer in [NRS10].
Ofcourse, wealsomakeextensiveuseofthetheoryoftheBrownianwebdeveloped
in [TW98,FINR04]. For a special subclass of the Howitt-Warren flows, we will
show that alternatively the random space-time environment can be represented
as a Brownian net, plus a countable collection of i.i.d. marks attached to its so-
called separation points. Here, we use the theory of the Brownian net, which was
developed in [SS08] and [SSS09].
Using our graphical construction, we prove a number of new properties for the
Howitt-Warren flows. In particular, we give necessary and sufficient conditions in
termsofthemeasure ν fortherandomkernelstospreadwithfinitespeed, fortheir
supporttoconsist ofisolatedpointsatdeterministictimes, andfortheexistenceof
random times when the kernels are non-atomic (Theorems 2.5, 2.7 and 2.8 below).
We moreover use our construction to prove the existence of versions of Howitt-
Warren flows with nice regularity properties (Proposition 3.8 below), in particular,
versionswhichcanbeinterpretedasbonafidetransitionkernelsinarandomspace-
timeenvironment. Lastly,westudytheinvariantlawsformeasure-valuedprocesses
associated with the Howitt-Warren flows (Theorem 2.11).
Our graphical construction of the Howitt-Warren flows is to a large extent
motivatedbyitsdiscretespace-timecounterpart,i.e.,randomwalksini.i.d.random
space-time environments on Z. Many of our proofs will also be based on discrete
approximation. Therefore, in the rest of the introduction, we will introduce a
class of random walks in i.i.d. random space-time environments and some related
objectsofinterest,andsketchheuristicallyhowtheBrownianwebandtheBrownian
net will arise in the representation of the random space-time environment for the
Howitt-Warren flows. An outline of the rest of the paper will be given at the end
of the introduction.
Incidentally, we note that random walks in i.i.d. random space-time environ-
ments have been used in the physics literature to model the flow of stress in a
granular medium, called the q model, see e.g. [LMY01,JM11] and the references
therein. The Howitt-Warren flows we consider are effectively scaling limits of so-
called near-critical q models.
1.2. Discrete Howitt-Warren flows. Let Z2 := {(x,t) : x,t ∈ Z, x+
even
t is even}be theevensublattice of Z2. Weinterpret the first coordinatex as space
and the second coordinate t as time, which is plotted vertically in figures. Let
ω := (ωz)z∈Z2 be i.i.d. [0,1]-valued random variables with common distribution
even
μ. We view ω as a random space-time environment for a random walk, such that
conditional on the environment ω, if the random walk is at time t at the position
x, then in the next unit time step the walk jumps to x+1 with probability ω
(x,t)
and to x−1 with the remaining probability 1−ω (see Figure 1).
(x,t)
Toformalizethis,letPdenotethelawoftheenvironmentωandforeach(x,s)∈
Z2 , let Qω denote the conditional law, given the random environment ω, of
even (x,s)
the random walk in random environment X = (X(t))t≥s we have just described,
1. INTRODUCTION 3
0.96 1 1 0.81 0.99
0.93 0.74 0.01 0 0.99
0.68 0.86 0.56 0.01 0
0.02 1 0 0 0.58
0.99 0.94 ω(x,t) 0.95 0.85
(x,t)
Figure 1. Random walk on Z2 in a random environment ω.
even
started at time s at position X(s) = x. Since parts of the random environment
belonging to different times a(cid:2)re independent, it is not hard to see that under the
averaged (or ‘annealed’) law P(dω)Qω , the process X is still a Markov chain,
(x,s) (cid:2)
whichineachtimestepjumpst(cid:2)otherightwithprobability μ(dq)q andtotheleft
withtheremainingprobability μ(dq)(1−q). Notethatthisisquitedifferentfrom
the usual random walk in random environment (RWRE) where the randomness is
fixed for all time, and the averaged motion no longer has the Markov property.
We will be interested in three objects associated with the random walks in
the i.i.d. random space-time environment ω, namely: random transition kernels,
n-point motions, and a measure-valued process. The law of each of these objects is
uniquely characterized by μ and, conversely, uniquely determines μ.
Firstofall,therandomenvironmentωdeterminesafamilyofrandomtransition
probability kernels,
(cid:3) (cid:4) (cid:5) (cid:6)
(1.1) Kω (x,y):=Qω X(t)=y s≤t, (x,s),(y,t)∈Z2 ,
s,t (x,s) even
which satisfy
(cid:7) (cid:5)
(i) Kω (x,y)Kω (y,z) = Kω (x,z) s ≤ t ≤ u, (x,s),(z,u) ∈
s,t t,u s,u
y:(y,t)∈Z2
even
Z2 ).
even
(ii) For each t < ··· < t , the random variables (Kω ) are inde-
0 n ti−1,ti i=1,...,n
pendent.
(iii) Kω and Kω are equal in law for each u∈Z :={2x:x∈Z}.
s,t s+u,t+u even
We call the collection of random probability kernels (Ksω,t)s≤t the discrete Howitt-
Warren flow with characteristic measure μ. Such a collection is a discrete time
analogue of a stochastic flow of kernels as introduced by Le Jan and Raimond in
[LR04a] (see Definition 2.1 below).