Enzo Orsingher

Bio: Enzo Orsingher is an academic researcher from Sapienza University of Rome. The author has contributed to research in topics: Brownian motion & Fractional calculus. The author has an hindex of 30, co-authored 189 publications receiving 3251 citations. Previous affiliations of Enzo Orsingher include University of Salerno.

State-dependent Fractional Point Processes

Abstract: The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{ u_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0< u_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{ u_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $ u_k$ differs from that constructed from the fractional state equations (in the case $ u_k = u$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.

Random motions at finite velocity in a non-Euclidean space

Abstract: In this paper telegraph processes on geodesic lines of the Poincare half-space and Poincare disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincare half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

Space–Time Fractional Equations and the Related Stable Processes at Random Time

Abstract: In this paper, we consider the general space–time fractional equation of the form $$\sum _{j=1}^m \lambda _j \frac{\partial ^{ u _j}}{\partial t^{ u _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)$$ , for $$ u _j \in \left( 0,1 \right] $$ and $$\beta \in \left( 0,1 \right] $$ with initial condition $$w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)$$ . We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{ u _1, \ldots , u _m} (t) \right) $$ , $$t>0$$ , where $$\varvec{S}_n^{2\beta }$$ is an isotropic stable process independent from $$\mathcal {L}^{ u _1, \ldots , u _m}(t)$$ , which is the inverse of $$\mathcal {H}^{ u _1, \ldots , u _m} (t) = \sum _{j=1}^m \lambda _j^{1/ u _j} H^{ u _j} (t)$$ , $$t>0$$ , with $$H^{ u _j}(t)$$ independent, positively skewed stable random variables of order $$ u _j$$ . The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{ u _1, \ldots , u _m} (t) \right) $$ , $$t>0$$ , supplies a probabilistic representation for the solutions of the fractional equations above and coincides for $$\beta = 1$$ with the n-dimensional Brownian motion at the random time $$\mathcal {L}^{ u _1, \ldots , u _m} (t)$$ , $$t>0$$ . The iterated process $$\mathfrak {L}^{ u _1, \ldots , u _m}_r (t)$$ , $$t>0$$ , inverse to $$\mathfrak {H}^{ u _1, \ldots , u _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/ u _j} \, _1H^{ u _j} \left( \, _2H^{ u _j} \left( \, _3H^{ u _j} \left( \ldots \, _{r}H^{ u _j} (t) \ldots \right) \right) \right) $$ , $$t>0$$ , permits us to construct the process $$\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{ u _1, \ldots , u _m}_r (t) \right) $$ , $$t>0$$ , the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For $$r \rightarrow \infty $$ and $$\beta = 1$$ , we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation $$\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)$$ . Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.

Time-inhomogeneous jump processes and variable order operators

Abstract: In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Levy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernstein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.

Convergence of Probability Measures

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

Table Of Integrals Series And Products

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Lévy processes and infinitely divisible distributions

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.