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This volume presents the lecture notes from two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second Barcelona Summer School on Stochastic Analysis.

René Schilling’s notes are an expanded version of his course on Lévy and Lévy-type processes, the purpose of which is two-fold: on the one hand, the course presents in detail selected properties of the Lévy processes, mainly as Markov processes, and their different constructions, eventually leading to the celebrated Lévy-Itô decomposition. On the other, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, making it possible to study the properties of Feller processes as space inhomogeneous processes that locally behave like Lévy processes. The presentation is self-contained, and includes dedicated chapters that review Markov processes, operator semigroups, random measures, etc.

In turn, Davar Khoshnevisan’s course investigates selected problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and to deduce, as an application, comparison-type results. The framework in which these problems are addressed goes beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as the existence and uniqueness of a solution for the corresponding equation, become relevant in their own right. These aspects are also developed and supplemented by a wealth of illustrative examples.



An Introduction to Lévy and Feller Processes


Chapter 1. Orientation

Stochastic processes with stationary and independent increments are classical examples of Markov processes. Their importance both in theory and for applications justifies to study these processes and their history.
René Schilling

Chapter 2. Lévy Processes

Throughout this chapter, \( \left( {\Omega, \,\mathcal{A},\,\mathbb{P}} \right) \) is a fixed probability space, \( t_0 = 0 \leqslant t_1 \leqslant \ldots \leqslant t_n \,and\,0 \leqslant s < t \) are positive real numbers, and \( \xi _k, \,\eta _k, \,k = 1, \ldots, \,n \), denote vectors from \( \mathbb{R}^d \); we write \( \xi \cdot \eta \) for the Euclidean scalar product.
René Schilling

Chapter 3. Examples

We begin with a useful alternative characterisation of Lévy processes.
Theorem 3.1. Let \( X = \left( {X_t } \right)_{t \geqslant 0} \) be a stochastic process with values in \( \mathbb{R}^d, \,X_0 = 1\,a.s.,\,and\,\mathcal{F}_t = \mathcal{F}_t^X = \sigma \left( {X_r, \,r \leqslant t} \right) \). The process X is a Lévy process if, and only if, there exists an exponent \( \psi :\,\mathbb{R}^d \to \mathbb{C} \) such that
$$ E\left( {e^{i\xi \cdot \left( {X_t - X_s } \right)} \left| {F_s } \right.} \right) = e^{ - \left( {t - s} \right)\psi \left( \xi \right)} \,for\,all\,s < t,\,\xi \in \mathbb{R}^d $$
René Schilling

Chapter 4. On the Markov Property

Let \( \left( {\Omega, \,\mathcal{A},\,\mathbb{P}} \right) \) be a probability space with some filtration \( \left( {\mathcal{F}_t } \right)_{t \geqslant 0} \) and a d-dimensional adapted stochastic process \( X = \left( {X_t } \right)_{t \geqslant 0} \), i.e., each Xt is \( \mathcal{F}_t \) measurable.We write \( \mathcal{B}\left( {\mathbb{R}^d } \right) \) for the Borel sets and set \( F_\infty : = \sigma \left( { \cup _{t \geqslant 0} F_t} \right) \).
René Schilling

Chapter 5. A Digression: Semigroups

We have seen that the Markov kernel \(p_{t}(x,B)\) of a Lévy or Markov process induces a semigroup of linear operators \((P_{t})_{t}\geq0\). In this chapter we collect a few tools from functional analysis for the study of operator semigroups.
René Schilling

Chapter 6. The Generator of a Lévy Process

We want to study the structure of the generator of (the semigroup corresponding to) a Lévy process \(X\;=\;{(X_{t})}{t}\geqslant {0}\). This will also lead to a proof of the Lévy– Khintchine formula.
René Schilling

Chapter 7. Construction of Lévy Processes

Our starting point is now the Lévy–Khintchine formula for the characteristic exponent ψ of a Lévy process
$$\psi(\xi)\;=\;- il\cdot\xi\;+\;\frac{1}{2}\xi\cdot Q\xi\;+\;\int_{y\neq 0}[1-e^{iy\cdot\xi}\;+\;i\xi\cdot y1\!\!\!\!1_{(0,1)}(|y|)]v(dy)$$
where (l,Q, ν) is a Lévy triplet in the sense of Definition 6.10; a proof of (7.1) is contained in Theorem 6.8, but the exposition below is independent of this proof, see however Remark 7.7 at the end of this chapter.
René Schilling

Chapter 8. Two Special Lévy Processes

We will now study the structure of the paths of a Lévy process. We begin with two extreme cases: Lévy processes which only grow by jumps of size 1 and Lévy processes with continuous paths.
René Schilling

Chapter 9. Random Measures

We continue our investigations of the paths of càdlàg Lévy processes. Independently of Chapters 5 and 6 we will show in Theorem 9.12 that the processes constructed in Theorem 7.5 are indeed all Lévy processes; this gives also a new proof of the Lévy–Khintchine formula, cf.
René Schilling

Chapter 10. A Digression: Stochastic Integrals

In this chapter we explain how one can integrate with respect to (a certain class of) random measures. Our approach is based on the notion of random orthogonal measures and it will include the classical Itô integral with respect to squareintegrable martingales.
René Schilling

Chapter 11. From Lévy to Feller Processes

We have seen in Lemma 4.8 that the semigroup \(P_{t}f(x)\;:=\;\mathbb{E}^{x}f(X_{t})=\mathbb{E}f(X_{t}+x)\) of a Lévy process \((X_{t}){_t\geqslant0}\) is a Feller semigroup.
René Schilling

Chapter 12. Symbols and Semimartingales

So far, we have been treating the symbol \(q(x,\xi)\) of (the generator of) a Feller process X as an analytic object. On the other hand, Theorem 11.13 indicates, that there should be some probabilistic consequences. In this chapter we want to follow this lead, show a probabilistic method to calculate the symbol and link it to the semimartingale characteristics of a Feller process.
René Schilling

Chapter 13. Dénouement

It is well known that the characteristic exponent \(\psi(\xi)\) of a Lévy process \(L\;=\;(L_t)_{t\geqslant0}\) can be used to describe many probabilistic properties of the process.
René Schilling

Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations


Chapter 14. White Noise

We begin our description of noisy equations with an analysis of the underlying noise. Throughout these notes, we use only space-time white noise. There is a very nice noise theory that is “white in time and colored in space”, as very well described in the book of Sanz-Solé [51] on the Malliavin calculus.We will not cover such noises here, however. The material of this chapter borrows heavily from the paper [34] with Kunwoo Kim.
Davar Khoshnevisan

Chapter 15. Lévy Processes

Before we continue our discussion of stochastic partial differential equations, we pause to recall a few facts from the theory of Lévy processes on LCA groups.
Davar Khoshnevisan

Chapter 16. SPDEs

The “SPDEs” in the title is shorthand for stochastic partial differential equations or more appropriately still, stochastic partial integro-differential equations. This is the main topic of these lecture notes.
Davar Khoshnevisan

Chapter 17. An Invariance Principle for Parabolic SPDEs

Throughout this chapter we suppose that J1, J2, \(\ldots\) are independent, identically distributed random variables, with values in \(\mathbb{Z}\), and assume that there exist constants \(\kappa, \alpha\;>0\) such that the characteristic function \(\phi\) of the Ji’s satisfies
$$\phi(z) := Ee^{izJ_{1}}=\;1-\kappa |Z|^{\alpha}\;+\;o(|z|^{\alpha}) \;\mathrm{as}\;z\rightarrow0$$
Davar Khoshnevisan

Chapter 18. Comparison Theorems

In this chapter we outline very briefly some of the main applications of the invariance principle from the preceding chapter. More specifically, we plan to prove that for a large family of SPDEs: (1) the solution is always positive as long as the initial value is positive; and (2) one can sometimes compute “moment functionals” of solutions to various SPDEs to one another. The material of this chapter is borrowed from Joseph–Khoshnevisan–Mueller [30].
Davar Khoshnevisan

Chapter 19. A Dash of Color

A great portion of the literature on SPDEs is concerned, in one form or another, with stochastic partial differential equations that are driven by noises showing some form of correlations. As of this time, there seems to be no unified theory of nonlinear SPDEs driven by correlated (henceforth, “colored”) noise, except in certain special cases; see, for example, Carmona–Molchanov [5] and Hu–Lu– Nualart [24] and their combined bibliography. By contrast, there is a very general approach to linear SPDEs, thanks to a rich theory of Gaussian processes. I will conclude these notes by describing, very briefly, the theory of linear SPDEs that are driven by quite general Gaussian noises.
Davar Khoshnevisan


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