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).

Consider non-linear time-fractional stochastic heat type equations of the following type, 

$$\begin{aligned} \partial ^\beta _tu_t(x)=-\nu (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$

in $(d+1)$ dimensions, where $\nu >0, \beta \in (0,1), \alpha \in (0,2]$. The operator $\partial ^\beta _t$ is the Caputo fractional derivative while $-(-\Delta )^{\alpha /2} $ is the generator of an isotropic stable process and $I^{1-\beta }_t$ is the Riesz fractional integral operator. The forcing noise denoted by $\mathop {F}\limits ^{\cdot }(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $\sigma :\mathbb {R}\rightarrow \mathbb {R}$ is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $\lambda $. In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

/pdf/asymptotic-properties-of-some-space-time-fractional-59bc1jzgf6.pdf

Asymptotic properties of some space-time fractional stochastic equations

Consider non-linear time-fractional stochastic heat type equations of the following type, $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $\sigma$ is assumed to be globally Lipschitz continuous. 
Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $\lambda$. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional derivatives in space, and power law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable L\'evy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index $1<\alpha<2$ to the density of the hitting time of a stable subordinator with index $1/\alpha$, and thereby unify some recent results in the literature. These results also provide a concrete interpretation of Zolotarev duality in terms of the fractional diffusion model.

/pdf/space-time-duality-for-fractional-diffusion-46frvx8mp6.pdf

Space-time duality for fractional diffusion

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by Nigmatullin [25] (Nigmatullin RR (1986) The realization of the generalized transfer in a medium with fractal geometry. Phys Status Solidi B 133:425–430), and Zasalavsky [32] (Zaslavsky G (1994) Fractional kinetic equation for Hamiltonian chaos. Phys D 76:110–122) in ${\mathbb{R}}^{d}$ for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizations on bounded domains D⊂ℝ d obtained in Meerschaert et al.[20, 21](Meerschaert MM, Nane E, Vellaisamy P (2009) Fractional Cauchy problems on bounded domains. Ann Probab 37:979–1007; Meerschaert MM, Nane E, Vellaisamy P (to appear) Distributed-order fractional diffusions on bounded domains. J Math Anal Appl). We also study the solutions of fractional Cauchy problem where the first time derivative is replaced with an infinite sum of fractional derivatives. We point out connections to eigenvalue problems for the fractional time operators considered. The solutions to the eigenvalue problems are expressed by Mittag-Leffler functions and its generalized versions. The stochastic solution of the eigenvalue problems for the fractional derivatives are given by inverse subordinators.

Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results

Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Levy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1 < α < 2 to the density of the hitting time of a stable subordinator with index 1 /α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.

/pdf/space-time-duality-for-fractional-diffusion-34qmnze3uk.pdf

Erkan Nane

Papers

Asymptotic properties of some space-time fractional stochastic equations

Asymptotic properties of some space-time fractional stochastic equations

Space-time duality for fractional diffusion

Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results

Space--time duality for fractional diffusion