Author

# Bruno Toaldo

Other affiliations: Catholic University of the Sacred Heart, University of Naples Federico II

Bio: Bruno Toaldo is an academic researcher from Sapienza University of Rome. The author has contributed to research in topic(s): Brownian motion & Fractional calculus. The author has an hindex of 12, co-authored 32 publication(s) receiving 474 citation(s). Previous affiliations of Bruno Toaldo include Catholic University of the Sacred Heart & University of Naples Federico II.

##### Papers

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TL;DR: In this article, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.

Abstract: This paper takes under consideration subordinators and their inverse processes (hitting-times). The governing equations of such processes are presented by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore the concept of time-changed C0-semigroup is discussed in case the time-change is performed by means of the hitting-time of a subordinator. Such time-change gives rise to bounded linear operators governed by integro-differential time-operators. Because these operators are non-local the presence of long-range dependence is investigated.

76 citations

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TL;DR: In this article, the authors construct compositions of vector processes of the form, t > 0,, β ∈ (0, 1),, whose distribution is related to space-time fractional n-dimensional telegraph equations.

Abstract: In this work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes, we present their counterparts as Brownian motion at delayed stable-distributed time.

48 citations

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TL;DR: In this paper, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.

Abstract: In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.

45 citations

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TL;DR: In this paper, a method based on Bernstein functions was proposed to unify three different approaches in the literature, including power law relaxation, semi-Markov process and semi-Maximax relaxation.

Abstract: Exponential relaxation to equilibrium is a typical property of physical systems, but inhomogeneities are known to distort the exponential relaxation curve, leading to a wide variety of relaxation patterns. Power law relaxation is related to fractional derivatives in the time variable. More general relaxation patterns are considered here, and the corresponding semi-Markov processes are studied. Our method, based on Bernstein functions, unifies three different approaches in the literature.

39 citations

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TL;DR: In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.

Abstract: In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.

38 citations

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2,309 citations

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01 Jan 2013

TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

1,794 citations

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TL;DR: In this paper, applied probability and queuing in the field of applied probabilistic analysis is discussed. But the authors focus on the application of queueing in the context of road traffic.

Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

1,038 citations

01 Jan 2016

TL;DR: In this paper, Zuc11b et al. this paper showed that 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15], 1/n [Per17] and 1/m [DFL17] were the most frequent p ≤ p ≥ ∞.

Abstract: (2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

129 citations