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).

This paper develops solutions of fractional Fokker-Planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non-Gaussian L\'evy processes, with space-time-dependent drift, diffusion and jump coefficients, thus significantly extends Magdziarz and Zorawik's result in "M. Magdziarz and T. Zorawik, Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, L\'evy noise and space-time-dependent coefficients. Proc. Amer. Math. Soc., Accepted (2015)." Fractional Fokker-Planck equation describing subdiffusion is solved by our result in full generality from perspective of stochastic representation.

Stochastic Solution of Fractional Fokker-Planck Equations with Space-Time-Dependent Coefficients

This paper studies stabilities of stochastic differential equation (SDE) driven by time-changed Levy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for solution of time-changed SDE to be stable in different senses will be established. Connection between stability of solution to time-changed SDE and that to corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Levy noise, where the time-change processes are inverse of general Levy subordinators. These results are important improvements of the results in "Q. Wu, Stability of stochastic differential equation with respect to time-changed Brownian motion, 2016.".

Stability of stochastic differential equation driven by time-changed Lévy noise

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$ 
in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and $d 0$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, Ecoled'ete de Probabilites de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.

Space-time fractional stochastic partial differential equations with L\'evy Noise.

We introduce a class of stochastic processes based on symmetric $\alpha$-stable processes. 
These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $\alpha$-stable process. We call them $\alpha$-time processes. They generalize Brownian time processes studied in \cite{allouba1, allouba2, allouba3}, and they introduce new interesting examples. We establish the connection of 
$\alpha-$time processes to some higher order PDE's for $\alpha$ rational. We also study the exit problem for $\alpha$-time processes as they exit regular domains and connect them to elliptic PDE's. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

Higher order PDE's and iterated Processes

We study for the first time the Cauchy problem for semilinear fractional elliptic equation. This paper is concerned with the Gaussian white noise model for the initial Cauchy data. We establish the ill-posedness of the problem. Then, under some assumption on the exact solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Some convergence rates between the exact solution and the regularized solution is established in $L^2$ and $H^q$ norms.

Erkan Nane

Papers

Stochastic Solution of Fractional Fokker-Planck Equations with Space-Time-Dependent Coefficients

Stability of stochastic differential equation driven by time-changed Lévy noise

Space-time fractional stochastic partial differential equations with L\'evy Noise.

Higher order PDE's and iterated Processes

Approximation of mild solutions of a semilinear fractional elliptic equation with random noise