Large deviations for stochastic processes

414 Pages·2006·1.999 MB·English

by  Jin Feng, Thomas G. Kurtz

Large Deviations for Stochastic Processes
Jin Feng
Thomas G. Kurtz
Department of Mathematics and Statistics, University of Massa-
chusetts, Amherst, MA 01003-4515
E-mail address: [email protected]
DepartmentsofMathematicsandStatistics, UniversityofWiscon-
sin at Madison, 480 Lincoln Drive, Madison, WI 53706 - 1388
E-mail address: [email protected]
2000 Mathematics Subject Classification. Primary 60F10, 47H20; Secondary
60J05, 60J25, 60J35, 49L25
Key words and phrases. large deviations, exponential tightness, approximation of
nonlinear semigroups, Skorohod topology, Markov processes, random evolutions,
occupation measures, weakly interacting particles, viscosity solutions, comparison
principle, mass transport techniques
Work supported in part by NSF Grants DMS 98-04816 and DMS 99-71571.
Abstract. Generalfunctionallargedeviationresultsforcadlagprocessesun-
der the Skorohod topology are developed. For Markov processes, criteria for
large deviation results are given in terms convergence of infinitesimal gener-
ators. Following an introduction and overview, the material is presented in
threeparts.
• Part1givesnecessaryandsufficientconditionsforexponentialtightness
thatareanalogoustoconditionsfortightnessinthetheoryofweakcon-
vergence. AnalogsoftheProhorovtheoremgivenbyPuhalskii,O’Brien
andVervaat,anddeAcosta,thenimplythelargedeviationprinciple,at
least alongsubsequences. Representations of the ratefunctioninterms
ofratefunctionsforthefinitedimensionaldistributionsareextendedto
include situations in which the rate function may be finite for sample
pathswithdiscontinuities.
• Part 2 focuses on Markov processes in metric spaces. For a sequence
ofsuchprocesses,convergenceofFleming’slogarithmicallytransformed
nonlinearsemigroupsisshowntoimplythelargedeviationprincipleina
manneranalogoustotheuseofconvergenceoflinearsemigroupsinweak
convergence. In particular cases, this convergence can be verified using
thetheoryofnonlinearcontractionsemigroups. Thetheoryofviscosity
solutions of nonlinear equations is used to generalize earlier results on
semigroupconvergence,enablingtheapproachtocoverawidevarietyof
situations. The key requirement is that a comparison principle holds.
Controlmethodsareusedtogiverepresentationsoftheratefunctions.
• Part3discussesmethodsforverifyingthecomparisonprincipleandap-
plies the general theory to obtain a variety of new and known results
onlargedeviationsforMarkovprocesses. ApplicationsincludeFreidlin-
Wentzell theory for nearly deterministic processes, random walks and
Markovchains,Donsker-Varadhantheoryforoccupationmeasures,ran-
domevolutionsandaveragingproblemsforstochasticsystems, rescaled
time-space lattice equations, and weakly interacting particle systems.
ThelatterresultsincludenewcomparisonprinciplesforaclassofHamilton-
JacobiequationsinHilbertspacesandspacesofprobabilitymeasures.
1
1July29,2005
Contents
Preface v
Notation vii
Introduction 1
Chapter 1. Introduction 3
1.1. Basic methodology 4
1.2. The basic setting for Markov processes 6
1.3. Related approaches 8
1.4. Examples 10
1.5. An outline of the major obstacles 25
Chapter 2. An overview 29
2.1. Basic setup 30
2.2. Compact state spaces 30
2.3. General state spaces 33
Part 1. The general theory of large deviations 39
Chapter 3. Large deviations and exponential tightness 41
3.1. Basic definitions and results 41
3.2. Identifying a rate function 50
3.3. Rate functions in product spaces 53
Chapter 4. Large deviations for stochastic processes 57
4.1. Exponential tightness for processes 57
4.2. Large deviations under changes of time-scale 62
4.3. Compactification 64
4.4. Large deviations in the compact uniform topology 65
4.5. Exponential tightness for solutions of martingale problems 67
4.6. Verifying compact containment 71
4.7. Finite dimensional determination of the process rate function 73
Part 2. Large deviations for Markov processes and semigroup
convergence 77
Chapter 5. Large deviations for Markov processes and nonlinear semigroup
convergence 79
5.1. Convergence of sequences of operator semigroups 79
5.2. Applications to large deviations 82
i
ii CONTENTS
Chapter 6. Large deviations and nonlinear semigroup convergence using
viscosity solutions 97
6.1. Viscosity solutions, definition and convergence 98
6.2. Large deviations using viscosity semigroup convergence 106
Chapter 7. Extensions of viscosity solution methods 109
7.1. Viscosity solutions, definition and convergence 109
7.2. Large deviation applications 125
7.3. Convergence using projected operators 129
Chapter 8. The Nisio semigroup and a control representation of the rate
function 133
8.1. Formulation of the control problem 133
8.2. The Nisio semigroup 139
8.3. Control representation of the rate function 140
8.4. Properties of the control semigroup V 141
8.5. Verification of semigroup representation 149
8.6. Verifying the assumptions 153
Part 3. Examples of large deviations and the comparison principle 161
Chapter 9. The comparison principle 163
9.1. General estimates 163
9.2. General conditions in Rd 170
9.3. Bounded smooth domains in Rd with (possibly oblique) reflection 177
9.4. Conditions for infinite dimensional state space 181
Chapter 10. Nearly deterministic processes in Rd 197
10.1. Processes with independent increments 197
10.2. Random walks 204
10.3. Markov processes 205
10.4. Nearly deterministic Markov chains 216
10.5. Diffusion processes with reflecting boundaries 218
Chapter 11. Random evolutions 227
11.1. Discrete time, law of large numbers scaling 228
11.2. Continuous time, law of large numbers scaling 241
11.3. Continuous time, central limit scaling 256
11.4. Discrete time, central limit scaling 263
11.5. Diffusions with periodic coefficients 265
11.6. Systems with small diffusion and averaging 267
Chapter 12. Occupation measures 279
12.1. Occupation measures of a Markov process - Discrete time 280
12.2. Occupation measures of a Markov process - Continuous time 284
Chapter 13. Stochastic equations in infinite dimensions 289
13.1. Stochastic reaction-diffusion equations on a rescaled lattice 289
13.2. Stochastic Cahn-Hilliard equations on rescaled lattice 301
13.3. Weakly interacting stochastic particles 311
CONTENTS iii
Appendix 339
Appendix A. Operators and convergence in function spaces 341
A.1. Semicontinuity 341
A.2. General notions of convergence 342
A.3. Dissipativity of operators 346
Appendix B. Variational constants, rate of growth and spectral theory for
the semigroup of positive linear operators 349
B.1. Relationship to the spectral theory of positive linear operators 350
B.2. Relationship to some variational constants 353
Appendix C. Spectral properties for discrete and continuous Laplacians 363
C.1. The case of d=1 364
C.2. The case of d>1 364
C.3. E =L2(O)∩{ρ:R ρdx=0} 365
C.4. Other useful approximations 366
Appendix D. Results from mass transport theory 367
D.1. Distributional derivatives 367
D.2. Convex functions 371
D.3. The p-Wasserstein metric space 372
D.4. The Monge-Kantorovich problem 374
D.5. Weighted Sobolev spaces H1(Rd) and H−1(Rd) 378
µ µ
D.6. Fisher information and its properties 382
D.7. Mass transport inequalities 390
D.8. Miscellaneous 397
Bibliography 399

Preface
This work began as a research paper intended to show how the convergence
of nonlinear semigroups associated with a sequence of Markov processes implied
the large deviation principle for the sequence. We expected the result to be of
littleutilityforspecificapplications,sinceclassicalconvergenceresultsfornonlinear
semigroupsinvolvehypothesesthatareverydifficulttoverify,atleastusingclassical
methods. We should have recognized at the beginning that the modern theory of
viscosity solutions provides the tools needed to overcome the classical difficulties.
Once we did recognized that convergence of the nonlinear semigroups could be
verified, the method evolved into a unified treatment of large deviation results for
Markov processes, and the research “paper” steadily grew into the current volume.
There are many approaches to large deviations for Markov processes, but this
book focuses on just one. Our general title reflects both the presentation in Part
1 of the theory of large deviations based on the large deviation analogue of the
compactnesstheoryforweakconvergence,materialthatisthefoundationofseveral
of the approaches, and by the generality of the semigroup methods for Markov
processes.
ThegoalofPart2istodevelopanapproachforprovinglargedeviations,inthe
context of metric-space-valued Markov processes, using convergence of generators
in much the same spirit as for weak convergence (e.g. Ethier and Kurtz [36]).
This approach complements the usual method that relies on asymptotic estimates
obtained through Girsanov transformations.
The usefulness of the method is best illustrated through examples, and Part 3
contains a range of concrete examples.
We would like to thank Alex de Acosta, Paul Dupuis, Richard Ellis, Wen-
dell Fleming, Jorge Garcia, Markos Katsoulakis, Jim Kuelbs, Peter Ney, Anatolii
Puhalskii and Takis Souganidis for a number of helpful conversations, and Peter
Ney and Jim Kuelbs for organizing a long-term seminar at the University of Wis-
consin - Madison on large deviations that provided much information and insight.
In particular, the authors’ first introduction to the close relationship between the
theory of large deviations and that of weak convergence came through a series of
lectures that Alex de Acosta presented in that seminar.
v

Notation
(1) (E,r). A complete, separable metric space.
(2) B(E). The σ-algebra of all Borel subsets of E.
(3) A⊂M(E)×M(E). An operator identified with its graph as a subset in
M(E)×M(E).
(4) B(E). The space of bounded, Borel measurable functions. Endowedwith
the norm kfk=sup |f(x)|, (B(E),k·k) is a Banach space.
x∈E
(5) B (E). The space of locally bounded, Borel measurable functions, that
loc
is, functions in M(E) that are bounded on each compact.
(6) B (x)={y ∈E :r(x,y)<(cid:15)}. The ball of radius (cid:15)>0 and center x∈E.
(cid:15)
(7) buc-convergence, buc-approximable, buc-closure, closed and dense, and
buc-lim, See Definition A.6.
(8) C(E). The space of continuous functions on E.
(9) C (E)=C(E)∩B(E).
b
(10) C(E,R). The collection of functions that are continuous as mappings
from E into R with the natural topology on R.
(11) C (E). ForE locallycompact,thefunctionsthatarecontinuousandhave
c
compact support.
(12) Ck(O), for O ⊂ Rd open and k = 1,2,...,∞. The space of functions
whose derivatives up to kth order are continuous in O.
(13) Ck(O)=Ck(O)∩C (O).
c c
(14) Ck,α(O), for O ⊂ Rd open, k = 1,2,..., and α ∈ (0,1]. The space of
functions f ∈Ck(O) satisfying
(cid:12)∂βf(cid:12) h∂βfi
kfk = sup sup(cid:12) (cid:12)+ sup <∞,
k,α 0≤β≤k O (cid:12)∂xβ(cid:12) |β|=k ∂xβ α
where
h i |f(x)−f(y)|
f =sup , α∈(0,1].
α x6=y |x−y|α
(15) Ck,α(O). The space of functions f ∈ C(O) such that f| ∈ Ck,α(D) for
loc D
every bounded open subset D ⊂O.
(16) C [0,∞). The space of E-valued, continuous functions on [0,∞).
E
(17) Cb(U),forU locallycompact. Thespaceofcontinuousfunctionsvanishing
at infinity.
(18) D(A)={f :∃(f,g)∈A}. The domain of an operator A.
(19) D+(A)={f ∈D(A),f >0}.
(20) D++(A)={f ∈D(A),inf f(y)>0}.
y∈E
(21) D [0,∞). ThespaceofE-valued,cadlag(rightcontinuouswithleftlimit)
E
functions on [0,∞) with the Skorohod topology, unless another topology
is specified. (See Ethier and Kurtz [36], Chapter 3).
vii
viii NOTATION
(22) D(O), for O ⊂Rd open. The space C∞(O) with the topology giving the
c
space of Schwartz test functions. (See D.1).
(23) D0(O). The space of continuous linear functionals on D(O), that is, the
space of Schwartz distributions.
(24) g∗ (respectively g ). The upper semicontinuous (resp. lower semicontinu-
∗
ous)regularizationofafunctiong onametricspace(E,r). Thedefinition
is given by (6.2) (resp. (6.3)).
(25) limsup G and liminf G for a sequence of sets G . Defini-
n→∞ n n→∞ n n
tion 2.4 in Section 2.3.
(26) Hk(Rd), for ρ∈P(Rd). A weighted Sobolev space. See Appendix D.5.
ρ
(27) K(E)⊂C (E). Thecollectionofnonnegative,bounded,continuousfunc-
b
tions.
(28) K (E) ⊂ K(E). The collection of strictly positive, bounded, continuous
0
functions.
(29) K (E)⊂K (E). The collection of bounded continuous functions satisfy-
1 0
ing inf f(x)>0.
x∈E
(30) M(E). The R-valued, Borel measurable functions on E.
(31) Mu(E). The space of f ∈M(E) that are bounded above.
(32) Ml(E). The space of f ∈M(E) that are bounded below.
(33) M(E,R). The space of Borel measurable functions with values in R and
f(x)∈R for at least one x∈E.
(34) M [0,∞). The space of E-valued measurable functions on [0,∞).
E
(35) Md×d. The space of d×d matrices.
(36) Mu(E,R)⊂M(E,R)(respectively,Cu(E,R)⊂C(E,R)). Thecollection
of Borel measurable (respectively continuous) functions that are bounded
above (that is, f ∈Mu(E,R) implies sup f(x)<∞).
x∈E
(37) Ml(E,R)⊂M(E,R) (respectively, Cl(E,R)⊂C(E,R)). The collection
of Borel measurable (respectively continuous) functions that are bounded
below.
(38) M(E). The space of (positive) Borel measures on E.
(39) M (E). The space of finite (positive) Borel measures on E.
f
(40) M (U),U ametricspace. Thecollectionofµ∈M(U×[0,∞))satisfying
m
µ(U ×[0,t])=t for all t≥0.
(41) MT(U) (T > 0). The collection of µ ∈ M(U ×[0,T]) satisfying µ(U ×
m
[0,t])=t for all 0≤t≤T.
(42) P(E)⊂M (E). The space of probability measures on E.
f
(43) R=[−∞,∞].
(44) R(A)={g :∃(f,g)∈A}. The range of an operator A.
(45) T#ρ=γ. γ ∈P(E) is the push-forward (Definition D.1) of ρ∈P(E) by
the map T.