Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

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

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Stochastic equations in infinite dimensions

DISCRETE-PARAMETER RANDOM FIELDS * Discrete Parameter Martingales * Two Applications in Analysis * Random Walks * Multiparameter Walks * Gaussian Random Walks * Limit Theorems CONTINUOUS-PARAMETER RANDOM FIELDS * Continuous Parameter Martingales * Constructing Markov Processes * Generation of Markov Processes * Probabilistic Potential Theory * Multiparameter Markov Processes * Brownian Sheet and Potential Theory APPENDICES * Kolmogorov's Consistency Theorem * Laplace Transforms * Hausdorff Dimensions and Measures * Energy and Capacity

Multiparameter Processes: An Introduction to Random Fields

A Primer on Stochastic Partial Differential Equations- The Stochastic Wave Equation- Application of Malliavin Calculus to Stochastic Partial Differential Equations- Some Tools and Results for Parabolic Stochastic Partial Differential Equations- Sample Path Properties of Anisotropic Gaussian Random Fields

A Minicourse on Stochastic Partial Differential Equations

The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance. The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a ``random noise,'' also known as a ``generalized random field.'' At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe. The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals a la Norbert Wiener, an infinite-dimensional Ito-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts. There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.

Analysis Of Stochastic Partial Differential Equations

We consider nonlinear parabolic 	SPDEs of the form 	$\partial_t u={\cal L} u + \sigma(u)\dot w$, where 	$\dot w$ denotes space-time white noise, 	$\sigma:R\to R$ is [globally] Lipschitz continuous, 	and $\cal L$ is the $L^2$-generator of a L'evy process. 	We present precise criteria for existence 	as well as uniqueness of solutions. 	More significantly, we prove that these solutions grow 	in time with at most a precise exponential rate.	 	We establish also that when $\sigma$ is globally Lipschitz 	and asymptotically sublinear, the solution 	to the nonlinear heat equation is ``weakly intermittent,'' 	provided that the symmetrization 	of $\cal L$ is recurrent and the initial data is sufficiently 	large. 	Among other things, 	our results lead to general 	formulas for the upper second-moment 	Liapounov exponent of the parabolic 	Anderson model for $\cal L$ in dimension $(1+1)$. When 	${\cal L}=\kappa\partial_{xx}$ for $\kappa>0$, these formulas 	agree with the earlier results of statistical physics 	(Kardar (1987), Krug and Spohn (1991), Lieb and Liniger (1963)), and also probability theory (Bertini and Cancrini (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is 	when $u_0=\delta_0$ or $u_0\equiv 1$; in those cases 	the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).

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Intermittence and nonlinear parabolic stochastic partial differential equations

We consider a system of d coupled non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution {u(t , x)}t∈R+,x∈[0 ,1], in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem. AMS 2000 subject classifications: Primary: 60H15, 60J45; Secondary: 60G60.

Davar Khoshnevisan

Papers

Multiparameter Processes: An Introduction to Random Fields

A Minicourse on Stochastic Partial Differential Equations

Analysis Of Stochastic Partial Differential Equations

Intermittence and nonlinear parabolic stochastic partial differential equations

Hitting probabilities for systems of non-linear stochastic heat equations with additive noise