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

Convergence of Probability Measures

Fractional Differential Equations

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

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Finite Time Blowup in L2 Sense of Solutions to SPDEs with Bernstein Functions of the Laplacian

We will look at reaction–diffusion type equations of the following type, $$\begin{aligned} \partial ^\beta _tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }]. \end{aligned}$$
 We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $$0<\eta \leqslant \eta _c$$
 , there is no global solution other than the trivial one while for $$\eta >\eta _c$$
 , non-trivial global solutions do exist. The critical parameter $$\eta _c$$
 is shown to be $$\frac{1}{\eta ^*}$$
 where $$\begin{aligned} \eta ^*:=\sup _{a>0}\left\{ \sup _{t\in (0,\,\infty ),x\in \mathbb {R}^d}t^a\int _{\mathbb {R}^d}G(t,\,x-y)V_0(y)\,\mathrm{d}y<\infty \right\} \end{aligned}$$
 and $$G(t,\,x)$$
 is the heat kernel of the corresponding unforced operator. $$V_0$$
 is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.

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Critical parameters for reaction–diffusion equations involving space–time fractional derivatives

Correlation structure of time-changed Pearson diffusions

We will look at reaction-diffusion type equations of the following type, $$\partial^\beta_tV(t,x)=-(-\Delta)^{\alpha/2} V(t,x)+I^{1-\beta}_t[V(t,x)^{1+\eta}]$$ We first study the equation on the whole space by making sense of it via an integral equation Roughly speaking, we will show that when $0 \eta_c$, non-trivial global solutions do exist We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution

Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. 
We study both the initial value and the final value problem. 
Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. 
For $t=0$, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Holder type.

Erkan Nane

Papers

Finite Time Blowup in L2 Sense of Solutions to SPDEs with Bernstein Functions of the Laplacian

Critical parameters for reaction–diffusion equations involving space–time fractional derivatives

Correlation structure of time-changed Pearson diffusions

Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders