Lectures on Stochastic Flows and Applications: Lectures delivered at the Indian Institute of Science, Bangalore und the T.I.F.R. - I.I.Sc. Programme ... Lectures on Mathematics and Physics)

130 Pages·1987·0.603 MB·English

by  H. Kunita, M.K. Ghosh

Lectures on
Stochastic Flows And Applications
By
H.Kunita
Lecturesdelivered atthe
IndianInstituteOfScienceBangalore
underthe
T.I.F.R.–I.I.Sc. Programme
InApplicationsOfMathematics
Notesby
M.K. Ghosh
Published forthe
TataInstituteOfFundamentalResearch
Springer-Verlag
BerlinHeidelberg NewYorkTokyo
1986
Author
H.Kunita
FacultyofEngineering 36
DepartmentofAppliedSciences
KyushuUniversity
Hakozaki, Fukuoka812
JAPAN
cTataInstituteOfFundamentalResearch, 1986
(cid:13)
ISBN3-540-12878-6 Springer-verlag, Berlin,Heidelberg,
NewYork. Tokyo
ISBN0-387-12878-6 Springer-verlag, NewYork. Heidelberg.
Berlin. Tokyo
No part of this book may be reproduced in any form by print,
microfilm or any other means without written permission from
the Tata Institute of Fundamental Research, Colaba, Bombay-
400005.
Printed by K.M.Aarif at paper print India, Worli, Bombay 400
018 and published by H.Goetze, Springer-Verlag, Heidelberg,
WestGermanyPrintedInIndia
Preface
These notes are based on a series of lectures given at T.I.F.R. Centre,
Bangalore during November and December 1985. The lectures con-
sisted of two parts. In the first part, I presented basic properties of
stochastic flows, specially of Brownian flows. Their relations with lo-
calcharacteristicsandwithstochasticdifferentialequationswerecentral
problems. Iintendedtoshowthehomeomorphismpropertyoftheflows
without using stochastic differential equations. In the second part, as
an application of the first part, I presented various limit theorems for
stochastic flows. Theseinclude thefollowing:
(a) Approximation theorems of stochastic differential equations and
stochastic flowsdue toBismut, Ikeda -Watanabe, Malliavin, Dow-
elletal.
(b) Limittheorems for drivenprocesses due toPapanicolaou- Stroock-
Varadhan.
(c) Limittheorems forstochastic ordinary differential equations dueto
Khasminskii, Papanicolau -Kohler,Kesten-Papanicolaou etal.
Iintended totreattheselimittheoremsinaunifiedmethod.
IwouldliketothankM.K.Ghoshforhiseffortsincompletingthese
notes. Also I wish to express my gratitude to Professor M.S. Raghu-
nathanandT.I.F.R,forgivingmethisopportunitytovisitIndia. Finally
IwouldliketothankMs. Shanthaforhertyping.
H.Kunita
iii

Contents
0 Introduction 1
1 BrownianFlows 5
1.1 Stochastic FlowswithIndependent Increments . . . . . . 5
1.2 LocalCharacteristics. GeneratorofN-PointMotion . . . 8
1.3 BrownianFlowofHomeomorphisms . . . . . . . . . . . 15
1.4 Stochastic FlowofDiffeomorphisms . . . . . . . . . . . 20
2 StochasticFlowsandStochasticDifferentialEquations 27
2.1 Non-BrownianStochasticFlows . . . . . . . . . . . . . 28
2.2 VectorValuedSemimartingales . . . . . . . . . . . . . . 32
2.3 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . 36
2.4 Construction ofstochastic...... . . . . . . . . . . . . . . . 40
2.5 Representation ofStochasticFlowsbySDES . . . . . . 44
2.6 InverseFlowsandInfinitesimalGenerator . . . . . . . . 50
2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 LimitTheoremsforStochasticFlows 63
3.1 WeakandStrongConvergence of... . . . . . . . . . . . . 64
3.2 Approximation ofStochasticDifferentialEquations . . . 68
3.3 TheMainLimitTheorem . . . . . . . . . . . . . . . . . 74
3.4 Tightnessof(N+M)-PointProcesses . . . . . . . . . . . 76
3.5 WeakConvergence of(N+M)-Pointprocesses . . . . . . 82
3.6 TightnessofSobolevSpace-valued Processes . . . . . . 92
3.7 ProofoftheMainTheorem . . . . . . . . . . . . . . . . 94
v
vi Contents
3.8 ProofoftheApproximation Theorem . . . . . . . . . . 95
3.9 ErgodicCase . . . . . . . . . . . . . . . . . . . . . . . 99
3.10 MixingCase . . . . . . . . . . . . . . . . . . . . . . . . 105
3.11 TightnessandWeakConvergence ofInverse Flows . . . 116
Chapter 0
Introduction
Letusconsider anordinary differential equation onRd 1
dx
= f(x,t), (0.1)
dt
where f(x,t) is continuous in (x,t) and is Lipschitz continuous in x.
Denote by φ (x) the solution of the equation (0.1) starting from x at
s,t
time s, i.e., with initial condition x(s) = x. It is a well known fact that
φ (x)satisfiesthefollowingproperties:
s,t
(a) φ (x)iscontinuous in s,t,x,
s,t
(b) φ (φ (x)) = φ (x)forany s,t,uandany x,
t,u s,t s,u
(c) φ (x) = xforany s,
s,s
(d) themapφ : Rd Rd isahomeomorphism forany s,t.
s,t
→
The map φ with above properties is called a flow of homeomor-
s,t
phisms.
Astochastic flowofhomeomorphisms isanRd valuedrandomfield
φ (x,ω),o s t T, x Rd defined onaprobability space (ΩF,P)
s,t
≤ ≤ ≤ ∈
such that, for almost all ω, it has the above mentioned properties (a)
∼
(d). Inparticular,ifφ ,i = 0,1,...,n 1areindependentforany0
ti,ti+1 − ≤
t < t < ... < t T, itiscalled aBrownian flow. Animportant class
0 1 n
≤
1
2 0.Introduction
ofBrownianflowsisconstructed bysolving Ito’sstochastic differential
equation
τ
dx(t) = F (x,t)dB (t)+F (x,t)dt, (0.2)
k k 0
Xk=1
whereF (x,t),F (x,t),...,F (x,t)arecontinuousin(x,t)andLipschitz
o 1 r
continuous in x, and (B (t), ,B (t)) is a standard Brownian motion.
1 r
···
Let φ (x,ω) be the solution of the equation under the initial condition
s,t
2 x(s) = x. Then, taking a suitable modification, it defines a Brown-
ian flow. This fact has been established by many authors, e.g. Elwor-
thy [7], Malliavin [8], Bismut [3], Ikeda - Watanabe [13], Kunita [18]
. However, not all stochastic flows can be constructed by the above
method. In fact, we need an infinite number of independent Brownian
motions B (t),B (t),... and functions F (x,t),F (x,t),..., or equiva-
1 2 O 1
lentlyaBrownianwithvaluesinC =C(Rd;Rd),i.e.,aC-Brownianmo-
tion. Here, acontinuous Rd valuedrandom field X(s,t),x Rd,0 t
∈ ≤ ≤
T,iscalledaC-BrownianmotionifX(x,t ) X(x,t),i = 0,1,...,n 1
i+1 i
− −
are independent for any 0 t < t ... < t T. Then under a mild
0 1 n
≤ ≤
conditionanyBrownianflowcanbeobtainedbyastochasticdifferential
equation oftheform
DC(t)= X(x(t),dt). (0.3)
ThisfactisduetoLeJan[23]andBaxendale[1]. SeealsoFujiwara
-Kunita[8].
The first part of these lectures will be devoted to the study of the
basic properties of stochastic flowsincluding the above facts. InChap-
ter 1 we shall characterize the Brownian flown by means of its local
characteristics:
1
b(x,t) = lim (E[φ (x)] x) (0.4)
t,t+h
h −
1
a(x,y,t) = lim (E[φ (x) x)(φ (y) y) ],
t,t+h t,t+h ∗
h − −
where is the transpose of the column vector (.).b(x,t) and a(x,y,t)
∗
will be referred to as infinitesimal mean and infinitesimal covariance
respectively. It will be shown that these two objects determine the law
oftheBrownianflow.
3
In Chapter 2, we shall consider the stochastic differential equation
basedonC-valued Brownianmotionormoregenerally C-valued con-
tinuoussemimartingale X(x,t),givenintheform(0.3). Itwillbeshown
thattheequationhasauniquesolutionifitslocalcharacteristicsareLip-
schitz continuous and that the solution defines a stochastic flow. Con- 3
versely, under some conditions the flow can be expressed as a solution
ofasuitable stochastic differential equation.
The second part of these notes will be devoted to various limit the-
orems concerning stochastic flows. We shall consider three types of
limit theorems. The first is the approximation theorem for a stochas-
tic differential equation given by (0.2). Let vε(t) = (vε(t),...,vε(t)),
1 r
ε > 0 be a piecewise smooth r dimensional process such that Bε(t) =
tvε(r)dr,ε > 0 converges to a Brownian motion B(t) as ε 0. Con-
0 →
Rsiderthestochastic ordinary differential equation
r
dx
= F (x,t)vε(t)+F (x,t). (0.5)
dt k k o
Xk=1
Let φε (x) be the solution of (0.5) starting from x at time s. The
s,t
question is whether the family of stochastic flows φε,ε > 0 converges
weaklyorstronglytoaBrownianflowdeterminedbythestochasticdif-
ferential equation (0.2) ( with some correction term). The problem has
been considered by many authors in several special cases of vε(t), e.g.,
Bismut [3], Dowell [6], Ikeda - Watanabe [14], Kunita [15], Malliavin
[25].
Thesecond limittheorem weshall beconcerned withisthatbyPa-
panicolaou Stroock-Varadhan [29]. Consider the following system of
stochastic differential equations
p
1
dxε(t) = F(xε(t),zε(t))dt+G(xε(t),zε(t))dt+ σ (xε(t),zε(t))dβ (t)
j j
ε
Xj=1
q
1 1
dz (t) = F˜(xε(t),zε(t))dt+ σ (xε(t),zε(t))dβ˜ (t),
∈ ε2 ε j j
Xj=1
where (β (t),...,β (t)) and (β˜ (t),...,β˜ (t)) are independent Brown-
1 p 1 q
ian motions, F,G,F˜,σ,σ˜ are bounded smooth functions with bounded
4 0.Introduction
derivatives. The processes zε(t),ε > 0(called the driving processes) do
4 not converge. However, under suitable conditions on the coefficients,
the processes xε(t),ε > 0(the driven processes) canconverge weakly to
adiffusionprocess.
The third one is concerned with the stochastic ordinary differential
equation
dx
= εf(x,t,ω)+ε2g(x,t,ω),
dt
where f,g are random velocity fields satisfying suitable strong mixing
conditions and E[f] = 0. The stochastic flows φε determined by the
s,t
above equation converge to the trivial flow φ0 (x) x. Khasminskii
s,t ≡
[16], Papanicolaou - Kohlet [28], Borodin [5], Kesten- Papanicolaou
[15]etalhaveshownthatafterchangingthescaleoftime,theprocesses
ψε = φε ,ε> 0converge weaklytoadiffusionprocess.
t 0,t/ε2
In Chapter 3 we shall present a general limit theorem so that the
abovementioned casescanbehandledtogether.