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

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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Methods Of Modern Mathematical Physics

Stochastic differential equations and diffusion processes

Introduction to lie algebras and representation theory

Formulae for the Derivatives of Heat Semigroups

We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.

Formulae for the derivatives of heat semigroups

Construction of connections.- The infinitesimal generators and associated operators.- Decomposition of noise and filtering.- Application: Analysis on spaces of paths.- Stability of stochastic dynamical systems.- Appendices.

/pdf/on-the-geometry-of-diffusion-operators-and-stochastic-flows-145jy7xhks.pdf

On the Geometry of Diffusion Operators and Stochastic Flows

For a wide class of local martingales (M

 t
 ) there is a default function, which is not identically zero only when (M

 t
 ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup
 s≤t
 
M

 s
 in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times.

/pdf/the-importance-of-strictly-local-martingales-applications-to-21qqeg9r6s.pdf

The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes

We investigate the effective behaviour of a small transversal perturbation of order to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions Hi, i = 1, ..., n. An averaging principle is shown to hold and the action component of the solution converges, as → 0, to the solution of a deterministic system of differential equations when the time is rescaled at 1/. An estimate for the rate of the convergence is given. In the case when the perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at 1/2 converges to that of a limiting stochastic differentiable equation.

/pdf/an-averaging-principle-for-a-completely-integrable-53vq6p1npt.pdf

Xue-Mei Li

Papers

Formulae for the Derivatives of Heat Semigroups

Formulae for the derivatives of heat semigroups

On the Geometry of Diffusion Operators and Stochastic Flows

The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes

An averaging principle for a completely integrable stochastic Hamiltonian system