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).

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

We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.

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On the infinite divisibility of distributions of some inverse subordinators

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type,

 $$ \partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)] $$
 in (d + 1) dimensions, where ν > 0,β ∈ (0, 1), α ∈ (0, 2]. The operator $\partial ^{\beta }_{t}$
 is the Caputo fractional derivative while − (−Δ)α/2 is the generator of an isotropic α-stable Levy process and I1−β is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel {\cdot }{F}(t,x)$
 is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on b, σ and the initial condition. Our results complement those of P. Chow in (Commun. Stoch. Anal. 3(2):211–222, 2009), Chow (J. Differential Equations 250(5):2567–2580, 2011), and Foondun et al. in (2016), Foondun and Parshad (Proc. Amer. Math. Soc. 143(9):4085–4094, 2015) among others.

Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Consider the following class of conformable time-fractional stochastic equation $$T_{\alpha,t}^a u(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0 0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $\lambda$. We also show that when the non-linear term $\sigma$ grows faster than linear, the energy of the solution blows-up at finite time for all $\alpha\in (0,1)$.

Asymptotic behavior of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

We consider the following fractional stochastic partial differential equation on a bounded, open subset B of R d for d ≥ 1
 ∂ t u t ( x ) = L u t ( x ) + ξ σ ( u t ( x ) ) F ˙ ( t , x ) , where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F ˙ ( t , x ) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α-stable Levy process in R d . We study the behaviour of the solution with respect to the parameter ξ.We show that under zero exterior boundary conditions, in the long run, the pth-moment of the solution grows exponentially fast for large values of ξ. However when ξ is very small we observe eventually an exponential decay of the pth-moment of this same solution.

/pdf/some-properties-of-non-linear-fractional-stochastic-heat-1m77zoqao4.pdf

Erkan Nane

Papers

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

On the infinite divisibility of distributions of some inverse subordinators

Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Asymptotic behavior of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

Some properties of non-linear fractional stochastic heat equations on bounded domains