Stochastic Analysis With L Evy Noise In The Dual Of A .

Chapter 0. Introductionxiienvironmental pollution.Some other applications are for example to statistical filtering (see Üstünel [104], [106]and D. Ding [27]), chemical kinetics and interacting particles systems (see Bojdecki andGorostiza [9], Gorostiza and Nualart [34], Hitsuda and Mitoma [38], Kallianpur andPérez-Abreu [52], Kallianpur and Mitoma [50], Kallianpur and Xiong [55], and Mitoma[70]).With some few exceptions, much of the works cited above were developed under thehypothesis of the nuclear space being Fréchet or either that its strong dual is also nuclear. Moreover, the stochastic integrals and the noise driving the stochastic differentialequations has been considered either with respect to Wiener processes or Poisson random measures, but to the extent of our knowledge, no theory has been developed withrespect to the general Lévy process case. This is the main motivation for the development of our theory in this thesis. We are also interested in developing this theoryunder the weakest possible assumptions on the nuclear space and its strong dual.In general terms, our contribution to theory of stochastic analysis on nuclear spaces canbe divided into four main aspects. First, we will show some extensions of the regularization theorem of Itô and Nawata [44] to the case of cylindrical processes on the dualof a nuclear space. These theorems will be a corner stone for our theory of stochasticanalysis. Our second main contribution is the proof of a Lévy-Itô decomposition forLévy process taking values in the dual of a complete, barrelled, nuclear space. Thethird is the introduction of a new theory of stochastic integration with respect to someclasses of cylindrical martingale-valued measures on the dual of a nuclear space. Thistheory allow us to introduce stochastic integrals with respect to Lévy process by meansof the Lévy-Itô decomposition. Finally, our last main contribution is the application ofour theory of stochastic integration to model stochastic evolution equations on the dualof a nuclear space. Contrary to what can be found in the literature, we will considersemi-linear equations driven by multiplicative noise.This thesis is organized as follows:Chapter 1 is devoted to the introduction to the main tools that we will need on thesubsequent chapters. First we review the basic properties of classes of locally convexspaces encountered on this thesis. We focus our attention on those concepts related tonuclear spaces and their strong duals. Later, we review basic concepts of cylindricaland stochastic processes on the dual of a nuclear space. Then, we proceed to proveour extensions of the regularization theorem. We finalize this chapter by studyingmartingales.In Chapter 2 we study basic properties of Lévy processes in the dual of a nuclear space.The proof of the Lévy-Itô decomposition is going to take most of our effort in thischapter. As a corollary we will proof the Lévy-Khintchine formula for the characteristicfunction of any Lévy processes.The aim of Chapter 3 is to develop the theory of stochastic integration. First, weintroduce the class of cylindrical martingale-valued measures that will be the integratorsof our integrals. Then, we develop the stochastic integration theory in two steps. Thefirst is a theory of weak stochastic integration with respect to cylindrical martingalevalued measures. For the second stage, we use the regularization theorems of Chapter 1and the weak stochastic integral to introduce a theory of strong stochastic integration,i.e. we define stochastic integrals for some families of operator-valued processes withrespect to the cylindrical martingale-valued measures. Applications to define stochastic

xiiiintegrals with respect to Lévy process will be given.Finally, in Chapter 4 we apply our theory of stochastic integration to study stochasticevolution equations driven by cylindrical martingale-valued measures. We start byintroducing some notions of deterministic integration for random integrands. Then,we introduce the class of stochastic evolution equations that we are going to considerin this thesis. In particular we will focus on the study of equivalence between weakand mild solutions, and we consider conditions on the coefficients for the existence anduniqueness of these types of solutions. We finalize this chapter with an example of anapplication to stochastic evolution equations driven by Lévy noise.

Notation and Useful FactsIn this thesis N, Z Q, R and C denote the sets of natural, integers, rational, realand complex numbers respectively. Denote R [0, ). For any n N, Rn is then-dimensional Euclidean space.For a, b R, we will use a b : max {a, b} and a b : min {a, b}. If I is a countableset, we denote by δij the Kronecker delta for i, j I , i.e. δij 0 if i 6 j andδij 1 if i j .For any two sets A and B , we denote by A B , A B and A \ B the union, theintersection and the complement of B in A respectively. When we are considering asubset U of a given set E , we write U c E \ U . If A is a finite set, we denote by #Athe number of elements of A.If A is a collection of subsets of a set S , we denote by σ(A) the σ -algebra generatedby A. If (Ω, F) and (S, S) are measurable spaces andY : Ω S is a F/S measurable 1map, the σ -algebra generated by Y is σ(Y ) Y (B) : B S .We denote by 1A (·) the indicator function of the set A, defined byx A and 1A (x) 0 for x / A.1A (x) 1 forLet T1 , T2 be any two topologies on a set X . If T1 is contained in T2 (i.e. if anyelement of T1 is also an element of T2 ), we denote this fact by: T1 T2 . In this casewe say that T1 is coarser than T2 , and that T2 is finer than T1 .If U is a subset of a topological space (X, τ ) we denote by U its closure and by Ů itsinterior.For a topological space (X, τ ), we denote by B(X) the Borel σ -algebra of X . It isthe smallest σ -algebra of subsets of X which contains all the open sets. R and Rdwill be always assumed to be equipped with their Borel σ -algebra. A measure µ on(X, B(X)) is called a Borel measure.The Dirac measure on X for a given x X will be denoted by δx and is defined byδx (A) 1A (x), for any A X .For two Borel measures µ and ν on a topologicalvector space X , denote by µ ν theirRconvolution. Recall that µ ν(A) X X 1A (x y) µ(dx)ν(dy), for any A B(X).Denote ν n ν · · · ν (n-times) and we use the convention ν 0 δ0 .Let (S, Σ, µ) be a measure space. For 1 p , Lp (S, Σ, µ) is the usual spaceof (equivalence classes of) real-valued measurable functions that agree almost every 1Rwhere with respect to µ and for which f p : S f (x) p µ(dx) p for allf Lp (S, Σ, µ). It is a Banach space with respect to the norm · p and for p 2 it is 1Ra Hilbert space with respect to the inner product hf , gi2 : S f (x)g(x)µ(dx) 2 for all f, g L2 (S, Σ, µ).Let (Ω, F , P) be a probability space and (S, Σ) be a measurable space. A F /Σxv

Chapter 0. Notation and Useful Factsxvimeasurable map X : Ω S will be called a S -valued random variable. In thisthesis we will only consider Borel random variables, i.e. S will be a topologicalspace and Σ B(S).Let J be R or [0, T ] for T 0. A S -valued process is a collection X {Xt }t J ofS -valued random variables. We say that X is continuous (respectively càdlàg) if forP-a.e. ω Ω, the sample paths t 7 Xt (w) S of X are continuous (respectivelyright-continuous with left limits).Let X be a real-valued random variable. If X is integrable, i.e. X L1 (Ω, F , P), wedefine its expectation to beZX(ω)P(dω).EX ΩWe say that the random variable is p-integrable (1 p ) if X Lp (Ω, F , P). Inthis case, the p-moment of X is EX p .Unless otherwise stated, throughout this document we will only consider vector spacesover a field K, which will always be R or C. Usually we denote a vector space by E .If S is a subset of E , span{S} denotes the linear span of S .If A and B are subsets of E , let A B : {x y : x A, y B}, λA : {λx : x A}where λ R (or λ C) , and A y : A {y} for y E .Let A and B be subsets of E . We say that A absorbs B if there exist some η0 Ksuch that B ηA whenever η η0 . A subset U of E is called absorbing if Uabsorbs every finite subset of E . A subset C of E is balanced if αC C wheneverα K, α 1. A subset D of E is said to be convex if x, y D implies thatλx (1 λ)y D for all 0 λ 1.

Chapter 1Probabilities on the Dual of aNuclear SpacesThe main purpose of this chapter is to introduce the main concepts of probability onnuclear spaces that we will need on this thesis. This chapter is divided into two mainsections. In the first, we review some concepts of locally convex spaces, linear operatorsand of nuclear spaces that will be used throughout this thesis. In the second section westart by reviewing basic properties of cylindrical and stochastic processes in the dualof a nuclear space. Then, we show some new results on the existence of continuousand càdlàg versions for cylindrical and stochastic process taking values in the dual of anuclear space. Finally, we apply these results to the study of martingales taking valuesin the dual of a nuclear space.§ 1.1Review of Locally Convex SpacesIn this section we give a brief presentation of those concepts on locally convex spaceswhich are used on this thesis. For a more detailed treatment the reader is referred toSchaefer [93], Trèves [99], Jarchow [48] or Narici and Beckenstein [77].1.1.1Semi-normsLet E be a vector space over a field K, which will always be R or C. A non-negativereal-valued function p on E having the properties:p(x y) p(x) p(y),p(αx) α p(x), x, y E, α K,is called a semi-norm on E and a norm if it additionally satisfies: x 6 0, impliesp(x) 0.Let p be a semi-norm on E . The set Bp (r) {x E : p(x) r} for r 0, is calledthe closed ball of radius r of p. In the case r 1 we

theory of tensor products of locally convex spaces. However, it is hardly an exaggeration to say that much of the true power behind the theory of nuclear spaces was better understood thanks to the characterization of nuclear spaces in terms of summable and absolute summable families of