Linköping Studies in Science and Technology. Dissertations.
No. 1699
A class of infinite dimensional
stochastic processes with
unbounded diffusion and its
associated Dirichlet forms
John Karlsson
Department of Mathematics
Linköping University, SE–581 83 Linköping, Sweden
Linköping 2015
Linköping Studies in Science and Technology. Dissertations. No. 1699
A class of infinite dimensional stochastic processes with unbounded diffusion and its associated Dirichlet forms
John Karlsson john.karlsson@liu.se www.mai.liu.se Mathematical Statistics Department of Mathematics Linköping University SE–581 83 Linköping Sweden ISBN 978-91-7685-966-7 ISSN 0345-7524
Copyright c 2015 John Karlsson, unless otherwise noted
Abstract
This thesis consists of two papers which focuses on a particular diffusion type Dirichlet form
E(F, G) =Z hADF, DGiHdν,
where A =P∞i=1λihSi, ·iHSi. Here Si, i ∈ N, is the basis in the Cameron-Martin space, H, consisting of the Schauder functions, and ν denotes the Wiener measure.
In Paper I, we let λi, i ∈ N, vary over the space of wiener trajectories in a way that the diffusion operator A is almost everywhere an unbounded operator on the Cameron– Martin space. In addition we put a weight function ϕ on the Wiener measure ν and show that under these changes of the reference measure, the Malliavin derivative and divergence are closable operators with certain closable inverses. It is then shown that under certain conditions on λi, i ∈ N , and these changes of reference measure, the Dirichlet form is quasi-regular. This is done first in the classical Wiener space and then the results are transferred to the Wiener space over a Riemannian manifold.
Paper II focuses on the case when λi, i ∈ N, is a sequence of non-decreasing real numbers. The process X associated to (E, D(E)) is then an infinite dimensional Ornstein-Uhlenbeck process. In this case we show that the distributions of a sequence of certain fi-nite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of the infinite dimensional Ornstein-Uhlenbeck process. We also investigate the quadratic vari-ation for this process, both in the classical sense and in the recent framework of stochastic calculus via regularization. Since the process is Banach space valued, the tensor quadratic variation is an appropriate tool to establish the Itô formula for the infinite dimensional Ornstein-Uhlenbeck process X. Sufficient conditions are presented for the scalar as well as the tensor quadratic variation to exist.
Populärvetenskaplig sammanfattning
En stokastisk process är en matematisk representation av hur ett slumpmässigt system utvecklas under tid. Exempelvis är värdet på en aktie en endimensionell process och po-sitionen på en partikel som rör sig slumpmässigt i rummet är en tredimensionell process. Det är svårare att föreställa sig och analysera processer som tar värden i oändligdimen-sionella rum men det finns olika sätt att behandla problemet matematiskt. Ett sätt är att studera så kallade Dirichletformer. En Dirichletform är ett matematiskt objekt inom om-rådet potentialteori. Genom att använda sig av en sådan framställning får man tillgång till de verktyg som finns inom potentialteorins ämnesområde vilket kan göra det matematiska arbetet enklare.
Det visar sig att endast vissa Dirichletformer har en motsvarande stokastisk process. I det första pappret i den här avhandlingen behandlas en viss typ av Dirichletformer där den så kallade diffusionen ökar för varje dimension. Man kan säga att diffusionen är hastigheten på slumprörelsen. Vi visar hur snabbt diffusionen får öka för att slutprocessen ska vara väldefinierad. Pappret behandlar även fallet då processen lever i ett krökt rum, på en mångfald, som exempel kan man tänka sig ytan av en ballong istället för ytan på ett papper.
I det andra pappret ligger fokus på den associerade oändligdimensionella processen som vi kallar X. Här visas bland annat att man under vissa omständigheter kan approx-imera X med ändligdimensionella processer. Vi beräknar också processens kvadratiska variation som är ett mått på hur mycket processens värde fluktuerar under tid. Genom att visa att en speciell typ av kvadratisk variation existerar kan vi även presentera en Itô formel, ett hjälpsamt verktyg för att bedriva analys på processen.
Acknowledgments
I would like to express my thanks to Linköping University and the department of math-ematics for the opportunity I have had to work there. A special thanks goes to my main supervisor Jörg-Uwe Löbus and my co-supervisor Torkel Erhardsson for all the help and support I have received during this PhD project. Thanks also to the administrative per-sonnel and co-workers at MAI that I have been fortunate to share this department with.
Thanks goes to my fellow PhD students that have been a source of inspiration and good times. A special mention goes to my friend Marcus Kardell that I have now known and had countless discussions with for almost five years.
I would also like to thank my family, other friends, and anyone else I might have forgot to mention, these pages would be full if I were to write down the names of everyone deserving a mention.
Finally, once again I would like to thank my main supervisor Jörg-Uwe Löbus for all the effort and time he put into this work, without him this could never have been done.
Contents
Abstract . . . v
Populärvetenskaplig sammanfattning . . . vii
Acknowledgements . . . ix
Introduction 1 Outline . . . 3
1.1 Motivation for the papers . . . 3
1.2 Paper I specifics . . . 4 1.3 Paper II specifics . . . 5 2 Malliavin calculus . . . 5 2.1 General framework . . . 5 2.2 Hermite polynomials . . . 6 2.3 Wiener Chaos . . . 6
2.4 Multiple stochastic integrals . . . 7
2.5 Derivative operator . . . 8
2.6 Divergence operator . . . 9
3 Dirichlet forms . . . 10
4 Stochastic integration via regularization . . . 12
4.1 One dimensional case . . . 12
4.2 Infinite dimensional case . . . 13
5 The geometric Cameron–Martin formula . . . 15
5.1 Manifolds . . . 15
5.2 Riemannian connection . . . 18
5.3 The orthonormal bundle . . . 20
5.4 Brownian motion on a smooth Riemannian manifold . . . 21
5.5 Analysis on the path space . . . 23
5.6 Directional derivative, gradient, divergence and integration by parts 27 5.7 Tensors and Ricci curvature . . . 29
References . . . 31
Paper I 1 Introduction . . . 37
2 Definitions . . . 40
3 Closability of derivative, divergence, and their inverses . . . 42
4 Closability of the bilinear form . . . 50
xii CONTENTS
5 Quasi-regularity . . . 55
6 Transferring the results to a geometric setting . . . 59
References . . . 67
Paper II 1 Introduction . . . 73
1.1 Main results . . . 75
1.2 Some basic definitions . . . 77
2 Finite dimensional approximation of the infinite dimensional process . . . 78
3 Scalar quadratic variation . . . 85
4 Tensor quadratic variation . . . 93
5 Itô’s formula . . . 99
A Appendix: Some lemmas in extreme value theory . . . 107
1 Outline 3
1
Outline
This thesis consists of two main parts. The first part is an introductory part containing key concepts of the theory used in the second part. The second part contains two papers, A class of infinite dimensional stochastic processes with unbounded diffusionand Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula, written at Linköping university during the years 2012-2015. Both papers are joint work with Jörg-Uwe Löbus. What follows below is a short summary of the different topics in part one.
Malliavin calculus
The purpose of this section is to provide a brief introduction to some notions in Malliavin calculus. Concepts such as the Malliavin derivative are essential for the definition of the bilinear Dirichlet form which is the main object of this study. We also present the Wiener chaos decomposition as we make use of it in Paper I.
Dirichlet forms
This section is devoted to the notion of Dirichlet forms. The section contains the ba-sic definitions needed for understanding the theorem of Ma/Röckner which provides the connection between Dirichlet forms and stochastic processes.
Stochastic integration via regularization
Here we present some main ideas of the recent theory of stochastic integration via reg-ularization. These concepts are used in Paper II in order to obtain a corresponding Itô formula to the stochastic process being studied.
Differential geometry
This is a presentation of basic notions in differential geometry. These concepts are then used to define Brownian motion on a Riemannian manifold. This theory is used in Paper I in the final section which transfers results of the flat case to a geometric setting.
1.1
Motivation for the papers
The theory of Dirichlet forms is a subject showing that certain bilinear forms can serve as a connection between analysis and probability. This connection to probability was established by the fundamental work of Silverstein and Fukushima during the 1970s (see [Meyer, 2009]). In particular, Fukushima showed that if a Dirichlet form on a locally compact state space is regular, it is possible to construct an associated Markov process with right continuous sample paths. By the 1980s, the demand for tools to study Markov processes on infinite dimensional (not locally compact) spaces led to various extensions of Fukushima’s result. In 1992, the general question was settled by the characterization of Ma and Röckner to the extent that a Dirichlet form on a separable metric space is
4 Introduction
associated with a Markov process with decent sample paths if and only if the form is quasi-regular. This result provides a method to construct Markov processes on separable metric spaces.
The papers in the thesis are concerned with Dirichlet forms of type
E(F, G) =Z hDF, ADGiHdν. (1) Here the diffusion operator is given by
A = ∞ X i=1
λihSi, ·iHSi, (2)
Si, i ∈ N are the Schauder functions, H is the Cameron–Martin space C0([0, 1]; Rd) and ν is the Wiener measure. In the paper [Driver and Röckner, 1992] the classical quasi-regular Dirichlet form, i.e. λi = 1, i ∈ N, is studied and and associated with a diffusion process on a compact Riemannian manifold. Similar results but in the case of unbounded diffusion where studied in [Löbus, 2004].
1.2
Paper I specifics
In the first paper we investigate the case where the Wiener measure ν is replaced with the weighted Wiener measure ϕν. Here the weight functions ϕ is of the form
ϕ(γ) = exp 1 Z 0 hb(γs), dγsiRd− 1 2 1 Z 0 |b(γs)|2ds .
This particular choice of weight functions is motivated by the geometric setting in the pa-pers [Wang and Wu, 2008, Wang and Wu, 2009] by F.-Y. Wang and B. Wu. These weight functions are considered both in a geometric framework and in the flat case. We study the form (1) on the set of smooth cylindrical functions of type
F, G ∈ Y = {F (γ) = f (γ(s1), . . . , γ(sk)) : sjis a dyadic point}, and
F, G ∈ Z = {F (γ) = f (γ(s1), . . . , γ(sk)) : sj ∈ [0, 1]}.
Above, γ is a Wiener trajectory. In the paper we formulate necessary and sufficient condi-tions on λ1(γ), λ2(γ), . . . that guarantee closability of (E, Y ) and (E, Z) on L2
(ϕν). The paper then shows locality, Dirichlet property, and quasi-regularity of the closure (E, Z) on L2(ϕν). The final parts of the paper are devoted to transferring the results into a geo-metric setting. In this case we consider Wiener trajectories on a stochastically complete Riemannian manifold, a generalization of the results of [Driver and Röckner, 1992] and [Löbus, 2004] where compact manifolds are studied.
2 Malliavin calculus 5
1.3
Paper II specifics
The second paper focuses on the process associated with the Dirichlet form (1). In Pa-per II the diffusion coefficients λi of the diffusion operator (2) are considered to be real numbers. We show that the associated stochastic process has the representation
Xt:= ∞ X i=1
Gi(λit) · Si, t ≥ 0,
where, for t ≥ 0, the right-hand side converges in C0([0, 1]; Rd). Here Gi, i ∈ N, are independent one-dimensional Ornstein-Uhlenbeck processes, i.e. dGi(t) = −Gi(t)dt + √
2 dWi(t), t ≥ 0, for a sequence of independent one-dimensional Wiener processes Wi, i ∈ N. We show that the distributions of a sequence of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of the infinite dimen-sional Ornstein-Uhlenbeck process, X, associated with the Dirichlet form (E, D(E)). We also investigate the quadratic variation for this process, both in the classical sense and in the recent framework of stochastic calculus via regularization. Since the process is Ba-nach space valued, the tensor quadratic variation is an appropriate tool to establish the Itô formula for the process X. Sufficient conditions are presented for the scalar as well as the tensor quadratic variation to exist.
2
Malliavin calculus
The French mathematician Paul Malliavin developed a calculus that is extending the cal-culus of variations, from functions to stochastic processes. Among other results it makes it possible to calculate the derivative of random variables. The material in this section is based on David Nualart’s monographs [Nualart, 2009] and [Nualart, 2006] and can also be found in [Karlsson, 2013].
2.1
General framework
We let H be a real separable Hilbert space with inner product h·, ·iHand corresponding norm k · kH. Unless otherwise stated, in this presentation we may consider the Hilbert space H to be L2([0, 1]) with the Borel σ-algebraB and Lebesgue measure µ.
Definition 2.1. A stochastic process W = {W (h), h ∈ H} defined in a complete proba-bility space (Ω,F, P) is a Gaussian process on H, if W is a centered Gaussian family of random variables and E[W (g)W (h)] = hg, hiHfor all g, h ∈ H.
This can be illustrated by a simple example. Example .1
Let H = L2
(R+) and Wt:= W (1[0, t]), t ≥ 0. This is standard Brownian motion since E[WsWt] = E[W (1[0,s])W (1[0,t])] = h1[0,s], 1[0,t]iL2 = min(s, t),
6 Introduction
2.2
Hermite polynomials
The Hermite polynomials will play an important role in the upcoming Wiener chaos de-composition of a Gaussian random variable as it turns out they have certain orthogonality properties. Here we define the Hermite polynomials Hnas the coefficients of the Taylor expansion of F (x, t) = exp{tx − t2/2} in powers of t. In other words
F (x, t) := etx−t22 =
∞ X n=0
Hn(x) · tn. (3)
We have the relations
H0(x) = 1, H1(x) = x, H2(x) = 1 2(x 2− 1), Hn(x) = (−1) n n! e x2 2 d n dxn e−x22 and also d dxHn(x) = Hn−1(x), (n + 1)Hn+1(x) = xHn(x) − Hn−1(x) as well as Hn(0) = ( 0 if n odd, (−1)k 2kk! if n even.
2.3
Wiener Chaos
Assume that H (we may think of it as L2([0, 1])) is infinite dimensional and let {e1, e2, . . . } be an ON-basis in H. We let Λ denote the set of all finite multi-indices a = (a1, a2, . . . ), ai ∈ N for i = 1, 2, . . . , i.e. only finitely many ai:s are non-zero. For any a ∈ Λ we define a! := ∞ Y i=1 ai!, |a| := ∞ X i=1 ai, Φa :=√a! ∞ Y i=1 Hai(W (ei)).
Taking the Hermite polynomials of Gaussian variables we obtain an orthogonality relation in the following way. For a two-dimensional Gaussian vector (X, Y ) s.t. E[X] = E[Y ] = 0 and V [X] = V [Y ] = 1, we have that for all m, n ∈ N
E[Hm(X)Hn(Y )] = ( 0 if m 6= n, 1 n!E[XY ] n if m = n.
LetHn denote the closed subspace of L2(Ω,F, P) spanned by {Φa : a ∈ Λ, |a| = n}. It can be shown that L2(Ω,F, P) = L∞
n=0Hn. We call Hn the nth Wiener Chaos. Furthermore we let Jn denote the projection operator onto the nth Wiener Chaos. It follows for F ∈ L2(Ω,F, P) that
F = ∞ X n=0
JnF.
Remark 2.2. The nth Wiener chaos contains polynomials of Gaussian variables of degree n.
While we defined nth Wiener chaos as an expression of ON-basis (ei)∞
i=1, it turns out that the nth Wiener chaos is independent of this choice of ON-basis.
2 Malliavin calculus 7
2.4
Multiple stochastic integrals
We will see that there is another representation for the nth Wiener chaos in the form of iterated Itô integrals. We begin with introducing a type of elementary functions first described by Norbert Wiener.
Definition 2.3. We define Enas the set of all elementary functions of form
f (t1, . . . , tn) = k X i1,...,in=1
ai1...in1Ai1×···×Ain(t1, . . . , tn), (4)
where ai1...in ∈ R and ai1...in = 0 if two indices coincide and A1, . . . , Ak are pairwise
disjoint subsets of [0, 1] with finite Lebesgue measure.
We see that this is the usual simple functions in [0, 1]n with the condition of being 0 on any diagonal described by two equal coordinates. It can be shown that this set has measure 0 and thus the set of elementary functions En is dense in L2([0, 1]n). We can now define a stochastic integral for these elementary functions. For a function of the form (4) we define
In(f (t1, . . . , tn)) = k X i1,...,in=1
ai1,...,inW (Ai1) · . . . · W (Ain).
Remark 2.4. In(f ) can be expressed as an iterated Itô integral
In(f ) = n! 1 Z tn=0 tn Z tn−1=0 . . . t2 Z t1=0 f (t1, . . . , tn) dWt1. . . dWtn.
The following proposition shows some of the properties of this stochastic integral. Proposition 2.5. Inhas the following properties:
(i)In(f ) = In( ˜f ), where ˜f denotes the symmetrization of f , i.e. ˜ f (t1, . . . , tn) = 1 n! X Π f (tΠ(1), . . . , tΠ(n))
whereΠ runs over all permutations of {1, . . . , n}. (ii)
E[Im(f ), In(g)] = (
0 ifm 6= n
n!h ˜f , ˜giL2([0,1]n) ifm = n.
It follows that we get an isometry between L2
s([0, 1]n) := { ˜f : f ∈ L2([0, 1]n)} and
Hn, i.e. √
n!k ˜f kL2([0,1]n)= kI(f )kL2(Ω,F,P).
Since En is dense in L2([0, 1]n) we can extend In to a linear continuous operator from L2([0, 1]n) to L2(Ω,F, P).
8 Introduction
Remark 2.6. The image of L2
s([0, 1]n) under Inis the nth Wiener chaosHnsince In(f ) is a polynomial of degree n in W (A1), . . . , W (An) when f has the form (4), and Propo-sition 2.5(ii) shows that stochastic integrals of different order are orthogonal. The claim then follows from induction.
Since we can express a random variable in its chaos expansion we note that for F ∈ L2(Ω,F, P), there exists fn∈ L2 s([0, 1]n), n ∈ N such that F = E[F ] + ∞ X n=1 In(fn).
2.5
Derivative operator
To define the calculus on the Wiener space a natural way of proceeding is to first look at a simple class of functions. Let Z denote the set of smooth cylindrical functions
Z = {F : F = f (W (h1), . . . , W (hn)), h1, . . . , hn∈ H, f ∈ Cp∞} where Cp∞denotes smooth functions with polynomial growth.
Definition 2.7. The gradient of a smooth cylindrical function is defined by: DF := n X n=1 ∂f ∂xi(W (h1), . . . , W (hn)) · hi. Remark 2.8. We have that DW (h) = h.
Clearly we have the product rule D(F G) = DF · G + F · DG and we also get an integration by parts formula as will be shown shortly. We also define the directional derivative in the following manner.
Definition 2.9. The directional derivative Dhis defined by DhF := hDF, hi h ∈ H.
Proposition 2.10. Let F ∈ D1,2and assume thatF has representation
F = E[F ] + ∞ X n=1 In(fn) wherefn∈ L2 s([0, 1]n). Then DtF = ∞ X n=1 nIn−1(fn(·, t)), t ∈ [0, 1].
Proof: First let F = In(fn) where fn(t1, . . . , tn) =
k X i1,...,in
2 Malliavin calculus 9 Then F = In(fn) = k X i1,...,in=1
ai1...inW (Ai1) · . . . · W (Ain)
and F ∈ Z. We get DtF = n X j=1 k X i1,...,in=1
ai1...inW (Ai1) · . . . · 1Aij(t) · . . . · W (Ain) = nIn−1(f (·, t))
by symmetry. The claim follows from the closedness of D1,2.
2.6
Divergence operator
We introduce the divergence operator δ. It turns out that δ is the adjoint operator to D. Let Dom δ be the set of all u ∈ L2(Ω,F, P; H) such that there exists c(u) > 0 with
|E[hDF, uiH]| ≤ ckF kL2 (5)
for all F ∈ D1,2. Then E[hDF, uiH] is a bounded linear functional from D1,2
to R. Now since D1,2is dense in L2(Ω,F, P) Riesz representation theorem states that there exists a unique representing element δ(u), bounded in L2(Ω,F, P). That is
E[hDF, uiH] = E[F δ(u)], ∀ F ∈ D1,2. (6)
Just like for the derivative there is a connection between the Wiener chaos expansion and the divergence operator. Recall that for u ∈ L2([0, 1] × Ω), u has a Wiener chaos expansion of form u(t) = ∞ X n=0 In(fn(·, t)),
where fn∈ L2([0, 1]n) and fnis symmetric in the n first variables.
Proposition 2.11. We have u ∈ Dom δ if and only if the series ∞
X n=0
In+1( ˜fn)
converges inL2(Ω,F, P). In this case we have δ(u) =
∞ X n=0
10 Introduction
Proof: Suppose G = In(g) where g ∈ L2
s([0, 1]n). Now using Proposition 2.10 we get E[hu, DGiH] = E[hu, nIn−1(g(·, t))iH] = E[hIn−1(fn−1(·, t)), nIn−1(g(·, t))iH]
= Z [0,1] E[In−1(fn−1(·, t))nIn−1(g(·, t))] dt = n(n − 1)! Z [0,1] hfn−1, giL2([0,1]n−1)dt = n!hfn−1, giL2([0,1]n= n!h ˜fn−1, giL2([0,1]n) = E[In( ˜fn−1)In(g)] = E[In( ˜fn−1)G]
where the third and last lines come from the isometry between L2s([0, 1])nandHn. One can see from this that δ is an integral. δ(u) is called the Skorokhod stochastic integral of the process u and if the process is adapted then the Skorokhod integral will coincide with the Itô integral, i.e.
δ(u) = 1 Z 0 utdWt.
3
Dirichlet forms
The theory of Dirichlet forms was first introduced in the works by Beurling and Deny in 1958 and 1959. Dirichlet form theory can be used in the area of Markov processes giving a different approach to problems. The usual tools for studying diffusion theory are methods from partial differential equations, whereas Dirichlet forms are connected to potential theory and energy methods. This section is based on the book [Ma and Röckner, 1992] which contains a much more through treatment of the subject.
Let H be a Hilbert space with corresponding norm k · kHand inner product h·, ·iH. Definition 3.1. A Dirichlet form is a densely defined positive symmetric bilinear form on L2(E, µ) such that
(i) D(E) is a real Hilbert space with inner product hu, viE:= h, i2E+E(u, v) 2
, (ii) For every u ∈ D(E) it holds u+∧ 1 ∈ D(E) and E(u+∧ 1, u+∧ 1) ≤E(u, u).
Not all Dirichlet forms have an associated process. What follows are the different notions needed for the formulation of Ma–Röckner’s theorem.
Definition 3.2. Let H = L2(E, m). A bilinear formE on E is regular if D(E) ∩ C0(E) is dense in D(E) w.r.t. E1-norm and dense in C0(E) w.r.t. the uniform norm. Here C0(E) denotes all continuous functions on E with compact support.
3 Dirichlet forms 11
Definition 3.3. We say that a function is cadlag if it is right continuous with left limits, i.e.
(i) f (x−) := lim
t%xf (t) exists, (ii) f (x+) := lim
t&xf (t) exists and is equal to f (x). The corresponding term for left continuous functions is caglad.
Definition 3.4. Two stochastic processes X and Y with a common index set T are called versions of one another if
t ∈ T, P ({ω : X(t, ω) = Y (t, ω)}) = 1. Such processes are also said to be stochastically equivalent.
If in addition the process Xtis left or right continuous then for a version Ytwe have Xt= Ytalmost surely.
Definition 3.5. (i) An increasing sequence (Fk)k∈Nof closed subsets of E is called an E-nest if ∪k≥1D(E)Fkis dense in D(E) w.r.t. k·k˜E1, where D(E)Fkdenotes {u ∈ D(E) :
u = 0 m-a.e. on E \ Fk}.
(ii) A subset N ⊂E is called E-exceptional if N ⊂ ∩k≥0Fkc for someE-nest (Fk)k∈N. (iii) We say that a property holds E-quasi-everywhere if it holds everywhere outside someE-exceptional set.
We can relate the definition of quasi-continuity to a similar notion on theE-nest. Definition 3.6. AnE-quasi-everywhere defined function f is called E-quasi-continuous if there exists anE-nest (Fk)k∈Nsuch that f is continuous on (Fk)k∈N.
Definition 3.7. A Dirichlet form (E, D(E)) on L2
(E, m) is called quasi-regular if: (i) There exists anE-nest (Fk)k∈Nconsisting of compact sets,
(ii) There exists an ˜E1/21 -dense subset of D(E) whose elements have E-quasi-continuous m-versions,
(iii) There exist un ∈ D(E), n ∈ N, having E-quasi-continuous m-versions ˜un, n ∈ N, and an E-exeptional set N ⊂ E such that {˜un: n ∈ N} separates the points of E \ N .
Definition 3.8. A process M with state space E is called µ-tight if there exists an in-creasing sequence (Kn)n∈Nof compact sets in E such that
Pµ( lim
n→∞inf{t > 0 : Mt∈ E \ Kn} < ∞) = 0 where inf ∅ := ∞.
We note that it simply means that for every ε > 0 there exists a compact set K such that Pµ(Mt∈ K) > 1 − ε.
12 Introduction
Definition 3.9. A cadlag Markov process M with state space E and transition semigroup (pt)t>0is said to be properly associated with (E, D(E)) and its corresponding semigroup (Tt)t>0 if ptf is anE-quasi continuous µ-version of Ttf for all t > 0 and all bounded f ∈ L2(E; µ).
The following theorem is the main purpose of this section. It provides the connection between Dirichlet forms and process theory.
Theorem 3.10. Let (E, D(E)) be a quasi-regular Dirichlet form on L2(E, µ). Then there exists a pair (M, ˆM ) of µ-tight special standard processes which is properly associated with (E, D(E)).
Proof: The proof is omitted but can be found in [Ma and Röckner, 1992].
Remark 3.11. It is known that if E is locally compact, i.e. finite dimensional, then there exists an associated process to every regular Dirichlet form on L2(E, µ). See [Fukushima et al., 2011].
4
Stochastic integration via regularization
This section contains some notions in the area of stochastic integration via regulariza-tion. Stochastic integration via regularization is a recently developed theory using tech-niques from integration. The different objects in this theory are presented first for finite dimensional stochastic processes, before considering the corresponding objects in the in-finite dimensional case. This section is based on material in [Russo and Vallois, 2007], [Di Girolami and Russo, 2014] and [Di Girolami et al., 2014].
4.1
One dimensional case
In this section let X(t), t ≥ 0, be a real-valued continuous process and Y (t), t ≥ 0, be a real-valued locally integrable process. Below we introduce the key concept of uniform convergence in probability, abbreviated ucp.
Definition 4.1. A sequence of real-valued processes (Xtδ)t∈[0,T ]is said to converge to (Xt)t∈[0,T ]in the ucp sense as δ → 0, if for all ε > 0
lim δ→0P t∈[0,T ]sup |X δ t − Xt| > ε ! = 0.
We may now introduce the following integral and covariation.
Definition 4.2. Provided that the following limits exist in the ucp sense we define the forward integral Z Y d−X := lim δ→0 t Z 0 YsXs+δ− Xs δ ds
4 Stochastic integration via regularization 13
and the covariation
[X, Y ]t:= lim δ→0 t Z 0 (Xs+δ− Xs)(Ys+δ− Ys) δ ds.
We also denote [X]t:= [X, X]tand call it the quadratic variation of X.
Remark 4.3. In e.g. the case of Y being continuous with bounded variation then the forward integral coincides with the usual Itô integral, see [Russo and Vallois, 2007].
In the one dimensional case we have the following Itô formula from [Russo and Vallois, 2007] Proposition 12.
Proposition 4.4. Suppose that [X]t,t ≥ 0 exists and f ∈ C2
(R). ThenR f0(Xs)d−Xs exists and f (Xt) = f (X0) + t Z 0 f0(Xs) d−Xs+ 1 2 t Z 0 f00(Xs) d[X]s.
4.2
Infinite dimensional case
We now consider the corresponding notions in the infinite dimensional case. We now consider X to be a process taking values in a Banach space B. Ucp convergence is defined in a similar way as above.
Definition 4.5. A sequence of B valued processes (Xδ
t)t∈[0,T ]is said to converge to (Xt)t∈[0,T ]in the ucp sense as δ → 0, if for all ε > 0
lim δ→0P t∈[0,T ]sup kX δ t − XtkB > ε ! = 0.
In [Di Girolami and Russo, 2014] Definition 5.1 the authors introduce the following definition.
Definition 4.6. Let X be a B valued process and Y be a B∗valued process such that X is continuous andRT
0 kYskB∗ds < ∞ a.s.. Provided that the following limit exist in probability for every t ∈ [0, T ] we define the forward integral
t Z 0 B∗hYs, d−XsiB := lim δ→0 t Z 0 B∗ Ys, Xs+δ− Xs δ B ds if the process t Z 0 B∗hYs, d−XsiB t∈[0,T ] admits a continuous version.
14 Introduction
In e.g. the case of B = L1([0, 1]), the duality pairing
B∗hg∗, f iB has the integral
representation B∗hg∗, f iB= 1 Z 0 f · g dx
for f ∈ B = L1([0, 1]), g∗∈ B∗and some representing element g ∈ L∞([0, 1]) ∼= B∗. In the case of F ∈ B⊗πb B ∼= L1([0, 1]2; Rd
2
), G∗ ∈ (B⊗πB)b ∗and some representing element G ∈ L∞([0, 1]2
; Rd2) ∼= (B b
⊗πB)∗the pairing duality becomes
(Bb⊗πB)∗hG
∗, F i
(B⊗bπB)∗∗ =
Z
F (u, v) qG(u, v) dudv,
where the symbol q to denotes the scalar product in Rd2.
The paper [Di Girolami and Russo, 2014] also defines the following notions of co-variation.
Definition 4.7. Let X and Y be a B1respectively B2valued processes. We define the scalar covariation as the ucp limit
[X, Y ]t:= lim δ→0 t Z 0 kXs+δ− XskB1kYs+δ− YskB2 δ ds,
provided it exists. We also denote [X]t:= [X, X]tand call it the scalar quadratic variation of X.
Definition 4.8. Let X and Y be B1 respectively B2 valued processes. We define the tensor covariation as the ucp limit
[X, Y ]⊗t := lim δ→0 t Z 0 (Xs+δ− Xs) ⊗ (Ys+δ− Ys) δ ds,
provided it exists. We also denote [X]⊗t := [X, X]⊗t and call it the tensor quadratic variation of X.
It is worth mentioning that quadratic variation in the sense above is essentially only suitable for semimartingale processes, see [Di Girolami et al., 2014]. However in the sit-uation where a process admits having both scalar and tensor quadratic variation we get the Itô formula in the form of [Di Girolami and Russo, 2014] Theorem 5.2.
Proposition 4.9. Suppose that X is a B valued continuous process admitting a scalar quadratic variation and a tensor quadratic variation. Furthermore letF be a mapping F : [0, T ] × B → R such that F is one time continuously Fréchet differentiable and two times continuously Fréchet differentiable in the second argument. That is, denoting the Fréchet derivative with respect to the second variable byD, for every t ∈ [0, T ] we
5 The geometric Cameron–Martin formula 15
haveDF (t, ·) : B → B∗andD2F (t, ·) : B → (B b
⊗πB)∗continuously. Under these conditions for everyt ∈ [0, T ] the forward integral
t Z 0 B∗hDF (s, Xs), d−XsiB exists and F (t, Xt) = F (0, X0) + t Z 0 ∂tF (s, Xs) ds + t Z 0 B∗hDF (s, Xs), d−XsiB +1 2 t Z 0 (B⊗bπB)∗D 2F (s, Xs), d[X]s (B⊗bπB)∗∗.
Remark 4.10. The above proposition makes use of the existence of scalar quadratic vari-ation and a tensor quadratic varivari-ation. The statement then follows as a direct conse-quence of [Di Girolami et al., 2014] Remark 5.7 applied to the mentioned Theorem 5.2 of [Di Girolami and Russo, 2014].
5
The geometric Cameron–Martin formula
This chapter serves to introduce the concepts needed for construction of Brownian mo-tion on a manifold M . The main part of secmo-tion is based on the series of lecture notes [Löbus, 1995]. Much of the material can also be found in [Rogers and Williams, 2000]. We first recall some basic definitions to introduce the notation used.
5.1
Manifolds
Definition 5.1. A subset M ⊂ RN is called a d-dimensional C∞
-manifold in RN if for each x ∈ M there is an open (in the relative topology) set G ⊂ M with x ∈ G and there exist F ∈ C∞(Rd −→ RN −d) such that (if necessary, after a permutation of coordinates)
G = {(y, F (y)) : y ∈ A}, for an open set A ⊂ Rd.
Definition 5.2. We say that y(t), t ∈ [0, 1] is a smooth curve in Rdif
y(t) = d X i=1 yi(t)ei, t ∈ [0, 1], where ei = (0, . . . , 0, 1 i, 0, . . . , 0), y i∈ C∞([0, 1]), i = 1, . . . , d, such that d X i=1 ( ˙yi(t))26= 0, t ∈ [0, 1].
16 Introduction
With the notation used in the previous definition, (x(t), t ∈ [0, 1]) defined as x(t) :=y(t), F y(t)
, t ∈ [0, 1], is called a smooth curve on M .
We recall the definition of tangents and tangent space.
Definition 5.3. We say that v(t) = ˙y(t) is a tangent to (y(s), s ∈ [0, 1]) (at t). Let Id denote the d × d identity matrix and J (y) denote the Jacobian matrix
Jir(y) i,r=1,...,d= ∂Fr ∂yi i,r=1,...,d. We use the notation (Id, J )T for the matrix
1 · · · 0 . .. 0 · · · 1 ∂F1 ∂y1 · · · ∂F1 ∂yd .. . ... ∂Fn ∂y1 · · · ∂Fn ∂yd. We call
u(t) := ˙x(t) =Id, J y(t)T
v(t) (7)
a tangent to (x(s), s ∈ [0, 1]) at t.
Definition 5.4. For a fixed y ∈ Rd and the corresponding point x = (y, F (y)) ∈ M . We define Tx:= {x + t : x ∈ M, t = (Id, J (y))Tv for some v ∈ Rd} which we identify with all the tangents to M at the point x and call it the tangentspace to M at x.
We may induce a metric on the tangent space from the Euclidean metric in RN. Let u1, u2 ∈ Txwhere x = (y, F (y)), y ∈ Rd, and v1, v2∈ Rdwhere u1= (Id, J (y))Tv1, u2= (Id, J (y))Tv2. Then
hu1, u2iRN = vT1(Id, J (y))(Id, J (y))Tv2. (8)
We introduce the notation
g(y) := (Id, J (y))(Id, J (y))T = Id+J (y)J (y)T (9) and write (8) as
hu1, u2iTx= v
T
1g(y)v2=: ghv1, v2i.
Remark 5.5. The length of a smooth curve (x(t), t ∈ [0, 1]) on M is calculated by 1
Z
0
˙
y(t)Tg(y(t)) ˙y(t) 1/2
dt,
where x(t) =y(t), F y(t)
5 The geometric Cameron–Martin formula 17
We now recall the notion of parallel transport along a curve.
Definition 5.6. For a smooth curve (y(s), s ∈ [0, 1]) in Rdand the corresponding smooth curve (x(s), s ∈ [0, 1]) on M , i.e. x(s) =y(s), F y(s)
, s ∈ [0, 1], let t1, t2 ∈ [0, 1] and u(t1) ∈ Tx(t1). Assume 0 ≤ t1≤ t2≤ 1 and let v(t) ∈ Tx(t)such that w(t) ∈ R
d, t ∈ [t1, t2] and
v(t) =Id, J y(t) T
w(t), t ∈ [t1, t2].
Let ˙v(t) ∈ RN denote the rate of change of v(t) and let Px(t)(·) denote the orthogonal projection onto the tangent space Tx(t). We say that v(t) moves parallel in t ∈ [t1, t2], along x(s), if for all t ∈ [t1, t2] we have Px(t)˙v(t) = 0.
Remark 5.7. In this case we have Px(t)(·) = Id, J y(t) T g−1 y(t) Id, J y(t) (·), since for all w ∈ Tx(t)and all z ∈ RN we have for w =Id, J y(t)
T q D Px(t)z, w E RN
=D(Id, J )Tg−1(Id, J )z, (Id, J )TqE RN =g−1(Id, J )z T (gq) =Dz, (Id, J )Tq E RN = hz, wiRN.
It follows that v(t) moves parallel if 0 = Px(t)˙v(t) = Id, J y(t) T g−1 y(t) Id, J y(t) ˙v(t), t ∈ [t1, t2]. (10) Definition 5.8. Let ˜v(t), t ∈ [0, 1] be the unique solution to the ODE (10) with boundary conditions ˜v(t1) = u(t1) ∈ Tx(t1). We say that u(t2) := ˜v(t2) ∈ Tx(t2)is the parallel
transport of the vector u(t1) along (x(s), s ∈ [0, 1]).
For notational clarity the rest of this section uses the Einstein summation convention. Remark 5.9. Take v(t) and w(t), t ∈ [t1, t2], as above and denote (gij)i,j=1,...,d:= g−1. We say that w(t) is the induced parallel transport along (y(s), s ∈ [0, 1]). That is w(t) solves the equation
0 = g−1 y(t)
Id, J (y(t)[(Id, J (y(t))Tw(t)]q = ˙w(t) + g−1 y(t) Id, J (y(t)[(Id, J (y(t))] q w(t) = ˙w(t) + gilX r JlrDjJkr y(t) ˙yj(t)wk(t) i=1,...,d . Using the notation
Γjki := gilX r
JlrDjJkr= gilX r
JlrDjDkFr, i, j, k ∈ {1, . . . , d}, (11)
the above equation turns into
0 = ˙w(t) +nΓjki y(t) ˙yj(t)wk(t)o i=1,...,d
18 Introduction
5.2
Riemannian connection
In this section we use a different definition of Γ and instead consider it in the following way. Definition 5.10. A function Γ = Γjki (y) i,j,k=1,...,d y ∈ R d,
Γ ∈ C∞(Rd× {1, . . . , d}3 → R) is called a connection i.e. Γ is a collection of d3 smooth functions on Rd. Let M = {(y, F (y)) : y ∈ Rd} be a C∞-manifold of RN and g(y) := (Id, J (y))(Id, J (y))T, y ∈ Rd. The pair (Rd, g) is called a Riemannian space and the pair (M, g) is called a Riemannian manifold.
Definition 5.11. Let Γ be a connection on (Rd, g) and let (w(s), s ∈ [0, 1]) ∈ C∞([0, 1] → Rd) be a vector field along a smooth curve (y(s), s ∈ [0, 1]). If equation (12) holds i.e.
0 = ˙w(t) +nΓjki y(t) ˙yj(t)wk(t)o i=1,...,d
t ∈ [t1, t2], (13)
we say that w(t) is parallel transported along (y(s), s ∈ [0, 1]) under Γ , for t ∈ [t1, t2]. Remark 5.12. The objects Γi
jk, i, j, k ∈ {1, . . . , d} are called Christoffel symbols. Definition 5.13. A connection Γ on a Riemannian space (Rd, g) is called Riemannian if the following holds.
(i) For all i ∈ {1, . . . , d} we have Γjki = Γ i
kj, j, k ∈ {1, . . . , d}.
(ii) For vector fields (w(s), s ∈ [0, 1]), (v(s), s ∈ [0, 1]) parallel transported along a smooth curve (y(s), s ∈ [0, 1]), it holds that
ghw(t1), v(t1)i = ghw(t2), v(t2)i, t1, t2∈ [0, 1].
If the parallel transport under Γ preserves inner products in the above sense we say that Γ is compatible with g.
The following theorem explains why we defined Γ by (11).
Theorem 5.14. There is exactly one Riemannian connection on (Rd, g). This connection is given by
Γjki = 1 2g
il Djglk+ Dkglj− Dlgjk, i, j, k ∈ {1, . . . , d}.
Proof: Step 1: (Existence) Let M = {(y, F (y)) : y ∈ Rd} be a d-dimensional C∞ -manifold in RN, J the corresponding Jacobian matrix
J = Jir(y) i=1,...,d; r=1,...,n= ∂Fr ∂yi i=1,...,d; r=1,...,n
5 The geometric Cameron–Martin formula 19
and g(y) = Id, J (y) Id, J (y) T , y ∈ Rd. Set Γjki = gilX r JlrDjDkFr, i, j, k ∈ {1, . . . , d},
i.e. (11). Under this assumption the symmetry condition (i) is satisfied. Let (w(s), s ∈ [0, 1]) and (v(s), s ∈ [0, 1]) be vector fields along a smooth curve (y(s), s ∈ [0, 1]) in Rd, which are parallel transported along this curve under Γ . Furthermore define
q(s) := Id, J (s)T w(s), p(s) := Id, J (s)T u(s), s ∈ [0, 1]. We have ghw(s), v(s)i = q(s)Tp(s), and ghw(s), v(s)i q = ˙q(s)Tp(s) + q(s) ˙p(s)T, s ∈ [0, 1].
Due to (12) and (10), q(s) ∈ Tx(s)as well as p(s) ∈ Tx(s)are parallel transported along x(s) = y(s), F (y(s)) then per definition ˙q(s) ⊥ Tx(s)and ˙p(s) ⊥ Tx(s). It follows that
ghw(s), v(s)i q = 0, i.e. the isometry condition (ii) is satisfied.
Step 2: (Uniqueness)Let Γ be a connection and (y(s), s ∈ [0, 1]), (v(s), s ∈ [0, 1]) and (w(s), s ∈ [0, 1]) be as above. Then from (ii) it holds
wi(s)gij(s)vj(s) q = 0. Using (12) i.e. ˙w(s) = −{Γi
jk(y(s)) ˙yj(s)wk(s)}i=1,...,das well as a similar relation for ˙v(s), s ∈ [0, 1] we get 0 = −Γkli yw˙ lgijvj+ wiy˙kDkgijvj− wigijΓkliy˙ kvl =− Γi klgij+ Dkglj− gliΓ i kj wly˙kvj. Thus it holds Γkligij+ Γkji gli= Dkglj. We swap cyclically k → l → j → k to obtain
− Γi
ljgik+ Γlkigji = −Dlgjk, and
Γjki gil+ Γjligki= Djgkl.
Adding these last three equations together, using the symmetry (i) and the symmetry of (gij) we get
2Γjki gil= Djgkl+ Dkglj− Dlgjk. The statement follows.
Remark 5.15. The matrix g defined by (9) uniquely defines Γjki in Theorem 5.14. Thus g gives a complete description of the geometry of the smooth Riemannian manifold (M, g).
20 Introduction
5.3
The orthonormal bundle
Having defined parallel transport it is natural to define the concept of moving bases. Definition 5.16. Let y(s), s ∈ [0, 1] be a smooth curve in Rd and (x(s), s ∈ [0, 1]) a smooth curve on M = {(y, F (y)) : y ∈ Rd} with x(s) = y(s), F (y(s)), s ∈ [0, 1]. Let E = {e1, . . . , ed} be an orthonormal basis (or short ON-basis) in Tx(0) and let f1, . . . , fd ∈ Rdwhere el= Id, J (y(0))T
fl, l = 1, . . . , d. Also let (fl(s), s ∈ [0, 1]) be vector fields along (y(s), s ∈ [0, 1]) satisfying the differential equation
( ˙fs(s) =Γi jk(y(s)) ˙y j(s)fk l(s) i=1,...,d s ∈ [0, 1], f (0) = fl, l = 1, . . . , d. (14) We define
el(s) := Id, J (y(s))T
fl(s), s ∈ [0, 1], l = 1, . . . , d.
We have that {e1(s), . . . , ed(s)} is an orthonormal basis in Tx(s), s ∈ [0, 1] and say that {e1(s), . . . , ed(s)} is a moving basis along x(s).
These final notions describe how curves behave on a manifold. Definition 5.17. The family
{(x, e1, . . . , ed) : x ∈ M, {e1, . . . , ed} is an ON-basis on Tx}, is called the orthonormal bundle and is denoted O(M ).
Definition 5.18. We say that a curve (u(s), s ∈ [0, 1]) in O(M ) is horizontal if u(s) = (x(s), E(s), s ∈ [0, 1]), where E(s) is a moving basis along (x(s), s ∈ [0, 1]). The curve (u(s), s ∈ [0, 1]), is called the horizontal lift of (x(s), s ∈ [0, 1]).
Proposition 5.19. Let (w(s), s ∈ [0, 1]) be a smooth curve in Rd. Then there is a hori-zontal curveu(s) on O(M ) satisfying
˙ w(s) = r1(s), . . . , rd(s)T , ˙ x(s) = E(s)r(s), x(s) = y(s), F (y(s)),
s ∈ [0, 1]. For given initial condition u0= (x0, e1, . . . , ed) ∈ O(M ) the horizontal curve u(s) = (x(s), E(s)), s ∈ [0, 1], with u(0) = u0is the unique solution to the system of differential equations ( ˙fl(s) +Γi jk(y(s)) ˙y j(s)fk l(s) i=1,...,d= 0, l = 1, . . . , d, ˙ y(s) = rl(s)fl(s), s ∈ [0, 1]. (15) We call the mapw(s) → u(s) the development map.
Remark 5.20. As a motivation for (15) consider a curve (w(s), s ∈ [0, 1]), in Rd. Now let r(s) be the velocity along w(s) i.e. r(s) = ˙w(s). Solving (15) gives y(s), a curve in Rd, and f
l(s), l = 1, . . . , d, basis vectors in Rd. These objects provide an equivalent description of the curve x(s) on M , with basis vectors el(s), l = 1, . . . , d, on Tx(s).
5 The geometric Cameron–Martin formula 21
5.4
Brownian motion on a smooth Riemannian manifold
In order to construct Brownian motion on smooth Riemannian manifolds we first remind of the following relation between the Itô and Stratonovich integral. Below the multiplica-tions are made in the sense of scalar products in Rm.
Definition 5.21. Let X, Y be continuous semimartingales in Rm. The Stratonovich in-tegralR Y ∂X is defined by St:= t Z 0 Ys∂Xs:= t Z 0 YsdXs+ 1 2[Y, X]t,
where t ∈ T (= [0, 1], [0, ∞), . . .),R Y dX is the usual Itô integral and [·, ·]tdenotes the covariation. We may write
∂S = Y ∂X = dS = Y dX +1
2dhY, Xi = Y dX + 1 2dY dX using differential notation.
Remark 5.22. The Stratonovich calculus is compatible with the classical non-stochastic differential calculus. For example for a smooth function, F ∈ C∞(Rm), we have
(i) ∂(F (X)) = F0(X)∂X, (ii) ∂(XY ) = Y ∂X + X∂Y. We consider the system of differential equations (15) ( ˙ y(s) = fm(s) ˙Wm(s), ˙ fl(s) = − Γjki (y(s))fmj(s)flk(s) i=1,...,dW˙ m(s), l = 1, . . . , d, s ∈ [0, 1], with initial conditions
(y(0), f1(0), . . . , fd(0)) = (y0, f10, . . . , fd0) (16a) x0:= (y, F (y0)), el0:= Id, J (y0)
T
fl0, l = 1, . . . , d, (16b)
such that
(x0, e10, . . . , ed0) ∈ O(M ). (16c) For m = 1, . . . , d use the notation
ϕm= ϕm y(s), f1(s), . . . , fd(s) := − Γjki (y(s))f j m(s)f k l(s) i,l=1,...,d. Note that ϕm, m = 1, . . . , d, is a d2-dimensional vector whose entries corresponds to the d different d-dimensional parallel transports of the basis vectors fi, i = 1, . . . , d. Let
Φ y(s), f1(s), . . . , fd(s) := | | f1 · · · fd | | −− −− −− | | ϕ1 · · · ϕd | | | {z } d
22 Introduction
With this notation (15) takes the form ˙
y(s), ˙f1(s), . . . , ˙fd(s)T = Φ y(s), f1(s), . . . , fd(s)W (s), s ∈ [0, 1].˙
The main idea for constructing Brownian motion on a manifold is to first get a description on how to transfer smooth curves to a manifold via differential calculus. We then wish to replace these smooth curves with the jagged curves of Brownian motion. This is done by replacing the usual differentials with Stratonovich ones. Work has to be done to ensure that this procedure yields a well defined result. We consider the Stratonovich SDE
(
(∂y, ∂f1, . . . , ∂fd) = Φ(y, f1, . . . , fd)∂γ,
deterministic initial condition (16), (17) where (γs,Fγ
s, s ∈ [0, 1]) is a d-dimensional Brownian motion on a probability space (Ω,F, P) equipped with the filtration (Fs)s∈[0,1]. Let
Ψ := i X q=1 DiΦjq i,j=1,...,d(d+1) .
Since there exist K > 0 such that for all ξ, η ∈ Rd(d+1)
kΦ(ξ) − Φ(η)k + kΨ (ξ) − Ψ (η)k ≤ Kkξ − ηk and
kΦ(ξ)k2+ kΨ (ξ)k2≤ K(1 + kξk2) we may formulate the following theorem.
Theorem 5.23. The Itô SDE (
d(y, f1, . . . , fd) = Φ(y, f1, . . . , fd)dγ +12Ψ (y, f1, . . . , fd)dt,
(16), (18)
has a unique strong solution. This implies that the Stratonovich SDE(17) has a unique strong solution namely the solution to the Itô SDE(18).
Definition 5.24. (a) Let y(s), f1(s), . . . , fd(s), s ∈ [0, 1], (Fs)s∈[0,1], Pbe a so-lution to (17). Furthermore let
ˆ γ(s) := y(s), F (y(s)), el(s) := Id, J y(s) T fl(s), l = 1, . . . , d, s ∈ [0, 1].
The process (ˆγ, E) = ˆγ(s), e1(s), . . . , ed(s), s ∈ [0, 1], (Fs)s∈[0,1], P is called hori-zontal Brownian motion.
(b) We call ˆγ := (ˆγ(s), s ∈ [0, 1]), (Fs)s∈[0,1], P Brownian motion on M . (c) Let 0 ≤ s1, s2≤ 1 and v = viei(s1) ∈ T
ˆ γ(s1). The quantity Tsˆγ 1→s2v := v Te i(s2),
5 The geometric Cameron–Martin formula 23
is called stochastic parallel transport from Tˆγ(s1)to Tγ(sˆ 2)along ˆγ.
(d) Let ˆγ0∈ M . The set of curves Pˆγ0(M ) := {ˆγ ∈ C([0, 1]; M ), ˆγ(0) = ˆγ0} is called
the path space over M . (e) The mapping ˆγ := I(γ)
C([0, 1]; Rd) 3 γ(·, ω) → ˆγ(·, ω) ∈ Pˆγ(0)(M ), P-a.e. ω ∈ Ω
is called the Itô map.
(f) Let ν denote the Wiener measure on {ϕ ∈ C([0, 1]; Rd) : ϕ(0) = 0}. The image measure ˆν := I∗ν under I, is called Wiener measure on Pˆγ0(M ).
(g) The mapping γ → (ˆγ, E) is called the stochastic development map and the mapping H : ˆγ → (ˆγ, E) is called stochastic horizontal lift.
The stochastic parallel transport has the following properties. Proposition 5.25. Let 0 ≤ s1, s2, s3≤ 1, v1= vi 1ei(s1) ∈ Tγ(sˆ 1), andv2= v i 2ei(s1) ∈ Tγ(sˆ 1). Then P-a.s. Tsγˆ 2→s3T ˆ γ s1→s2v1= T ˆ γ s1→s3v1, (19) and hv1, v2iTγ(s1)ˆ = hTsγˆ 1→s2v1, T ˆ γ s1→s2v2iTˆγ(s2). (20)
Proof: Step 1: From Definition 5.24, (19) is clear for fixed 0 ≤ s1, s2, s3≤ 1, P-a.s. The P-a.s. continuity of e1, . . . , edimplies P-a.s. continuity in the left hand side of (19) in s1, s2, s3as well as for the right hand side of (19) for s1, s3. Therefore (19) holds P-a.s. for all 0 ≤ s1, s2, s3≤ 1 and v1∈ Tγ(sˆ 1).
Step 2: Since e1, . . . , ed is P-a.s. continuous it follows that v1iei, vi2ei is continuous as well. It is therefore sufficient to prove (20) for fixed 0 ≤ s1, s2≤ 1. We note
hv1, v2iTˆ γ(s1) = X i,j v1iv2jhei(s1), ej(s1)iTˆ γ(s1) = X i v1iv2i, and hTγˆ s1→s2v1, T ˆ γ s1→s2v2iTˆγ(s2)= hv i 1ei(s2), v j 2ej(s2)iTˆγ(s2)= d X i=1 vi1vi2,
i.e. the left and right hand sides of (20) coincide.
5.5
Analysis on the path space
This section contains a brief overview of the calculus on the path space of Brownian motion on a Riemannian manifold. Detailed calculations and more results can be found in [Driver and Röckner, 1992], [Hsu, 1995], [Hsu, 2002]. Let (Ω,F, P) be a probability space and γ(ω) be a d-dimensional Brownian motion on (Ω,F, P) with filtration Fs, s ∈ [0, 1] and corresponding measure ν. In addition letO(d) denote the space of d × d orthogonal matrices.
24 Introduction
Proposition 5.26. Let A = A(s), s ∈ [0, 1], (Fs)s∈[0,1], Pbe anO(d)-valued pro-cess and α = α(s), s ∈ [0, 1], (Fs)s∈[0,1], P be an Rd-valued process such that E R |α|2ds < ∞, and E R kAk2ds < ∞. Introduce
W (s) = s Z 0 A(r) dγ(r) + s Z 0 α(r) dr, s ∈ [0, 1].
We letν := W˜ ∗ν denote the image measure under the map γ 7→ W . We have
d˜ν dν(γ) = exp Z1 0 α(r)A(r) dγ(r) −1 2 1 Z 0 |α(r)|2dr .
Proof: The process γ0(s) =Rs
0 A(r) dγ(r), s ∈ [0, 1] is a Brownian motion with respect to (Fs)s∈[0,1], since γ0is a continuous local martingale with quadratic variation
· Z 0 A dγ, · Z 0 A dγ s = s Z 0 ATA dγ, γr= Id·s, s ∈ [0, 1].
That is ν γ0(ω) : ω ∈ Ω = 1 and W = γ0+R α dr. The statement then follows from the Girsanov formula.
We use the notation
H :=h ∈ C0 [0, 1]; Rd : h absolutely continuous Z
|h0(s)|2ds < ∞
to denote the d-dimensional Cameron–Martin space.
Theorem 5.27. Let h ∈ H. There exist families (A(t, ·))t∈R,(α(t, ·))t∈Rof processes such that the following holds.
(i) The conditions of Proposition 5.26 are satisfied for every t ∈ R. (ii) It holds that α(·, s) ∈ C1
(R; Rd), A(·, s) ∈ C1
(R; O(d)), s ∈ [0, 1], P-a.s. as well as Rd3 α(·, 0) = 0 and O(d) 3 A(·, 0) = 0.
(iii) Let W (t, s) = s Z 0 A(t, r) dγ(r) + s Z 0 α(t, r) dr, and define ξ(t, s) := I(W (t, s)), u(t, s) := H(ξ(t, s)), t ∈ R, s ∈ [0, 1],
5 The geometric Cameron–Martin formula 25
whereI and H are the Itô mapping respectively horizontal lift from Definition 5.24. Thenξ has the representation
ξ(t, s) = ˆγ(s) + t Z
0
u(r, s)h(s) dr, (21)
and is a solution to thegeometric flow equation (d
dtξ(t, s) = u(t, s)h(s) = H(ξ(t, s))h(s), ξ(0, s) = ˆγ(s), s ∈ [0, 1], t ∈ R. is satisfied with the solution
Hereˆγ is a Brownian motion on M .
In(21) u(r, s) is to be understood by the mapping Rd → Tξ(r,s),u(r, s)k := kie
i W (r, s) whereei W (r, s) denotes the walking basis vector eialongW (r, s).
Proof: The proof is omitted but can be found in the papers [Hsu, 1995] and [Hsu, 2002].
We denote
νt:= W (t, ·)∗P, t ∈ R,
where ν0 = ν. Furthermore we let γ0(t) = (γ0(t, s), s ∈ [0, 1]), (Fs)s∈[0,1], Pgiven by γ0(t, s) := s Z 0 A(t, r) dγ(r), t ∈ R.
According to the proof of Proposition 5.26 γ0(t, ·) is a Brownian motion with respect to the filtrationFs, s ∈ [0, 1]. The map γ(ω) → γ0(t)(ω) induces an injective map b : Ω → Ω.
The flow equation is used to calculate the directional derivative on the manifold, see (25) below.
Theorem 5.28. (a) (Quasi invariance) The measures νt,t ∈ R are equivalent and we have for P-a.e. ω ∈ Ω
dνt dν(γ(ω)) = exp Z α(t, r)A(t, r) dγ(r) − 1 2 Z |α(t, r)|2dr (ω). (22)
(b) Almost surely the Radon–Nikodym derivative dνt
dν(γ) belongs to C 1 (R) and d dt t=0 dνt dν(γ) = Z d dt t=0 α(t, r) dγ(r), t ∈ R. (23)
26 Introduction
Proof: (a) Let
˜ νt:= γ(s)(·) + s Z 0 α(t, r)(b−1·) dr ∗ ν .
The Girsanov formula yields d˜νt dν (γ(ω)) = exp Z1 0 α(t, r)(b−1(ω)) dγ(r)(ω) − 1 2 1 Z 0 |α(t, r)(b−1(ω))|2dr .
On the other hand we have νt = b∗˜νt, t ∈ R. Taking into account that γ and γ0 are Brownian motions it follows ν(dγ(ω)) = ν(dγ0(t)(ω)) and
dνt dν γ(ω) = νt dγ(ω) ν dγ(ω) = ˜ νt dγ0(t)(ω) ν dγ(ω) = ˜ ν dγ0(t)(ω) ν dγ0(t)(ω) = d˜νt dν γ b(ω) = exp Z1 0 α(t, r)(ω) dγ0(r)(ω) − 1 2 Z |α(t, r)(ω)|2dr = exp Z1 0 α(t, r)(ω)A(t, r)(ω) dγ(ω) −1 2 Z |α(t, r)(ω)|2dr .
(b) Using the stochastic version of dominated convergence theorem, see [Protter, 2005], and taking under consideration α(0) = 0, A(0) = Id, we obtain (23) by differentiating (22).
Corollary 5.29. (a) (Quasi invariance) The measures ˆνt := I∗νt,t ∈ R are equivalent and we have for P-a.e. ω ∈ Ω
dˆνt
dˆν ˆγ(ω) = dνt
dν γ(ω). (b) Almost surely the Radon–Nikodym derivativedˆνt
dˆν(ˆγ) ∈ C 1 (R) and we have d dt t=0 dˆνt dˆν(ˆγ) = d dt t=0 dνt dν(γ).
Definition 5.30. A vector field on an open set N ⊂ M is a smooth map S : N → TN = S
x∈NTxwhere S(x) ∈ Tx, x ∈ N . Let N1be an open subset of M and let N ⊂ N1. If S is a vector field on N1 then S|N is a vector field on N . In particular if N is a smooth curve, N := (x(s), s ∈ [0, 1]) on M , we say that S is a vector field along (x(s), s ∈ [0, 1]).
Remark 5.31. Let S be a vector field on N ⊂ M . If x = F (y), y ∈ Rd, and S(x) = (I, J (y))V (y), we can identify S with the first order differential operator vi ∂
5 The geometric Cameron–Martin formula 27
5.6
Directional derivative, gradient, divergence and integration
by parts
In this section we introduce the geometrical objects corresponding to the usual differential notions. This is done by means of specifying the objects on a class of smooth functions. Definition 5.32. Let ˆγ0∈ M . We define
ˆ
Z :=nΦ(ˆγ) = ϕ ˆγ(s1), . . . , ˆγ(sk), ˆγ ∈ Pˆγ0(M ) :
0 < s1< . . . < sk = 1, ϕ ∈ Cb∞(Mk), k ∈ No,
to be the set of smooth cylindrical functions on M .
For f ∈ Cb∞(M ) we define the gradient on the tangent space ˜
∇f := (Id, J )Tg−1∇f ◦ F,
here recall that F denotes the function describing the manifold embedding in RN. With this definition and for ψ ∈ C1(R; M ) it follows that
d dt(f ◦ ψ(t)) = d dt f ◦ F ◦ F −1◦ ψ(t) =D∇f ◦ F, d dtF −1ψ(t)E Rd = gDg−1∇f ◦ F, d dtF −1ψ(t)E=D ˜∇f (x), d dtψ(t) E TxM . Definition 5.33. Let Φ ∈ ˆZ where Φ(ˆγ) = ϕ(ˆγ(s1), . . . , ˆγ(sk)). We define
ˆ DsΦ(ˆγ) := k X i=1 χ[0,si](s) n Tsˆγi→s( ˜∇iϕ)(ˆγ(s1), . . . , ˆγ(sk))o, s ∈ [0, 1],
to be the gradient of Φ. Here ˜∇idenotes the operator ˜∇ corresponding to the ith variable x(si).
Remark 5.34. We mention that the difference between ˆDsin Definition 5.33 and Section 6 in Paper I, is a parallel transport from Tγ(s)to Tγ(0).
Recall that H denotes the d-dimensional Cameron–Martin space.
Definition 5.35. Let (ˆγ, E) := (ˆγ(s), e1(s), . . . , ed(s)), s ∈ [0, 1], (Fs)s∈[0,1], P
be the process constructed from the solution of the system of equations (15) for fixed initial conditions (ˆγ(0), e1(0), . . . , ed(0)) ∈ O(M ). For h ∈ H and Φ ∈ ˆZ we define the directional derivative in the direction of h as
ˆ DhΦ(ˆγ) := 1 Z 0 D ˆDsΦ(ˆγ), d dsh j sej(s) E Tγ(s)ˆ ds.
28 Introduction We obtain ˆ DhΦ(ˆγ) = 1 Z 0 D ˆDsΦ(ˆγ), d dsh j sej(s) E Tγ(s)ˆ ds = k X i=1 1 Z 0 χ[0,si](s) · d dsh j s D Tsˆγi→s∇iϕ, ej(s)˜ E Tˆγ(s) ds = k X i=1 1 Z 0 χ[0,si](s) · d dsh j s D Tsˆγ i→0 ˜ ∇iϕ, ej(0)E Tγ0ˆ ds = k X i=1 hjsi·DTsγˆ i→0 ˜ ∇iϕ, ej(0)E Tˆγ0 = k X i=1 hjsi· D ˜∇iϕ, ej(si)E Tˆγ(si) (24)
as another representation of ˆDhΦ(ˆγ). Finally for the solution ξhof (21) we have P-a.s.
d dt t=0 Φ ◦ ξh(t, ·) = k X i=1 ˜ ∇ϕ, d dt t=0 ξh(t, si) Tγ(si)ˆ = k X i=1 D ˜∇ϕ, u(0, si)h(si)E Tˆγ(si) = ˆDhΦ(ˆγ). (25)
Remark 5.36. The limit
lim t→0 1 t Φ ◦ ξh− Φ(ˆγ), exist in L2= L2(Pˆ γ0(M ), ˆν). Therefore, lim t→0 1 t Φ ◦ ξh− Φ(ˆγ)= ˆDhΦ(ˆγ) in L2. (26)
Let ˆν denote the Wiener measure on Pˆγ0(M ), ˆγ0∈ M . We study ˆγ → ξ
h(t) trajecto-rywise as a mapping Cˆγ0([0, 1]; M ) → Cˆγ0([0, 1]; M ). For Φ as before and Ψ ∈ ˆZ such
that Ψ (ˆγ) = ψ(ˆγ(s1), . . . , ˆγ(sk)), ˆγ ∈ Pˆγ0(M ), ψ ∈ C ∞ b (M ) we have h ˆDhΦ, Ψ iL2 = d dt t=0Φ ◦ ξ h, Ψ L2 = lim t→0 1 t h hΦ ◦ ξh(t), Ψ i L2− hΦ, Ψ iL2 i = lim t→0 1 t Z Φ ◦ ξh(t)(ˆγ)Ψ (ˆγ) ˆν(dˆγ) − hΦ, Ψ iL2 = lim t→0 1 t Z ΦΨ ◦ ξh(−t)(ˆγ) ˆν dξh(−t)(ˆγ) − hΦ, Ψ iL2 = lim t→0 1 t Z ΦΨ ◦ ξh(−t)(ˆγ)ˆν dξ h(−t)(ˆγ) ˆ ν(dˆγ) ν(dˆˆ γ) − hΦ, Ψ iL2 . (27)
5 The geometric Cameron–Martin formula 29
We use the abbreviations dξh(−t) ∗νˆ dˆν = dˆν−t dˆν and z h:= d dt t=0 dˆν−t dˆν . From (26) and (27) we obtain the integration by parts formula ˆ DhΦ, Ψ L2 = Φ, d dt t=0 Ψ ◦ξh(−t)(ˆγ)ν dξˆ h(−t)(ˆγ) ˆ ν(dˆγ) L2 =Φ, − ˆDhΨ +zhΨ L2. Let g ∈ L2 ([0, 1] → Rd) and h :=R q
0 gsds. We note that h ∈ H. Furthermore let Φ and Ψ be cylindrical functions. We have
Z 1 Z 0 D DsΦ, Ψ gjej(s)E T·(s) ds dˆν = Z 1 Z 0 D DsΦ, gjej(s)E T·(s) ds Ψ dˆν =ˆ DhΦ, Ψ L2 =Φ, − ˆDhΨ + zhΨ L2 = Φ, − k X i=1
hj(si)D ˜∇iψ, ej(si)E T·(si)
+ zhΨ
L2
,
where for the second last equality we have used the above integration by parts formula. Definition 5.37. We define ˆ δ(Ψ gjej) := − k X i=1 d X j=1
hj(si)D ˜∇iψ, ej(si)E
T·(si)+ z h
Ψ
and call it the divergence of Ψ gjej.
Remark 5.38. The divergence is the adjoint operator to the gradient. This is true also for the non-geometric Malliavin gradient and divergence. That is one of the most important relations in Malliavin calculus.
5.7
Tensors and Ricci curvature
This presentation of the theory is based on the book [Kühnel, 2002]. Definition 5.39. An (r, s)-tensor, A, at a point x ∈ M , is a multilinear map
(Tx)∗× . . . × (Tx)∗ | {z } r × (Tx) × . . . × (Tx) | {z } s → R. Here (Tx)∗denotes the dual space to the tangent space at x.
Let us a consider a vector field S on M . Let x = F (y), y = (y1, . . . , yd) ∈ Rd, and S(x) = (I; J (y))V (y). We can then identify S with the first order differential operator
S(x) = d X i=1 vi(y) ∂ ∂yi x .
30 Introduction
Using the Christoffel symbols of (11) we may introduce the Riemann curvature tensor R in coordinate form as the object
Rijks :=∂Γ s ik ∂xj −∂Γ s ij ∂xk + ΓikrΓrjs − Γr ijΓ s rk. (28) We define R ∂ ∂xj , ∂ ∂xk ∂ ∂xi = d X s=1 Rsijk ∂ ∂xs .
Thus for vector fields, X = (ξ1, . . . , ξd), Y = (η1, . . . , ηd), and Z = (ζ1, . . . , ζd) on M , we get the curvature tensor
R(X, Y )Z = R d X i=1 ξi ∂ ∂yi , d X j=1 ηj ∂ ∂yj d X k=1 ζk ∂ ∂yk = d X i,j,k=1 ξiηjζkR ∂ ∂yi, ∂ ∂yj ∂ ∂yk = d X i,j,k,s=1 ξiηjζkRskij ∂ ∂xs.
Note that the Riemann curvature tensor is a (1, 3)-tensor. Using the Riemann curvature tensor we now define the Ricci tensor.
Definition 5.40. The Ricci tensor is a (0, 2)-tensor given by the trace of the curvature tensor. That is for an ON-basis (E1, . . . , Ed)
Ric(Y, Z)(x) := Ric(X 7→ R(X, Y )Z) = d X i=1
hR(Ei(x), Y (x))Z(x), Ei(x)iTx.
Remark 5.41. We recall that ∂y∂
1, . . . ,
∂
∂yd is a basis for Tx at x. In the same way
dy1, . . . , dydis a basis for the dual space (Tx)∗with
dyi x ∂ ∂yj x = δij= ( 1, i = j 0, i 6= j. In the same way
dyi x⊗ dyj x ∂ ∂yk x⊗ ∂ ∂yl x = δikδjl= ( 1, i = k, j = l 0, otherwise.
Remark 5.42. Taking into account (29) and Remark 5.41 we note that the Ricci tensor has representation
5 The geometric Cameron–Martin formula 31
It follows that the Ricci tensor is given by the matrix described by
Rjk= d X i=1
Riijk. (29)
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