The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

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Multidimensional Stochastic Processes as Rough Paths

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

Persistence and First-Passage Properties in Non-equilibrium Systems

Lectures on stochastic differential equations and Malliavin calculus

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

An introduction to numerical methods for stochastic differential equations

We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].

Asymptotic expansion of stochastic flows

Nous nous interessons aux approximations numeriques des solutions fortes d'une equation differentielle stochastique (EDS), utilisant un pas de temps fixe, et les increments de la trajectoire Brownienne. Nous utilisons l'approche developpee par Ben Arous, Castell, et Hu, qui permet d'approcher en temps petit la solution d'une EDS, par la solution d'une equation differentielle ordinaire (EDO) inhomogene en temps. Nous obtenons ainsi des EDO's, qui lorsque le pas de temps diminue, fournissent une suite d'approximations de la solution de l'EDS asymptotiquement efficace, au sens de Clark et Newton. Nous distinguons d'une part le cas d'une EDS conduite par un Brownien de dimension 1, ou satisfaisant la condition de commutativite; d'autre part le cas d'une EDS conduite par un Brownien multi-dimensionnel, et ne satisfaisant pas la condition de commutativite. Lorsque les EDO's obtenues sont resolues numeriquement de facon suffisamment precise, la propriete d'efficacite asymptotique est preservee. Les methodes exposees dans cet article sont des methodes alternatives et facilement generalisables d'approximation des solutions fortes d'une EDS.

The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations

An efficient approximation method for stochastic differential equations by means of the exponential Lie series

Let {Sk, k ≥ 0} be a symmetric random walk on \({\mathbb Z}^d\), and \(\{\eta(x), x\in {\mathbb Z^d\}}\) an independent random field of centered i.i.d. random variables with tail decay \(P(\eta(x)> t)\approx\exp(-t^{\alpha})\). We consider a random walk in random scenery, that is \(X_n=\eta(S_0)+\dots+\eta(S_n)\). We present asymptotics for the probability, over both randomness, that {Xn > nβ} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process \(\sum_x l_n^2(x)\), where ln(x) is the number of visits of site x up to time n.

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

We prove large deviations principles in large time, for the Brownian occupation time in random scenery \({{\frac{{1}}{{t}} \int_0^t \xi(B_s) \, ds}}\) The random field is constant on the elements of a partition of ℝd into unit cubes These random constants, say \({{{{\left\lbrace{{ \xi(j), j \in \mathbb{{Z}}^d}}\right\rbrace}} }}\) consist of iid bounded variables, independent of the Brownian motion {Bs,s≥0} This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery We prove large deviations principles in ``quenched'' and ``annealed'' settings

Fabienne Castell

Papers

Asymptotic expansion of stochastic flows

The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations

An efficient approximation method for stochastic differential equations by means of the exponential Lie series

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

Large deviations for Brownian motion in a random scenery