Bruno Toaldo

Bio: Bruno Toaldo 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 12, co-authored 32 publications receiving 474 citations. Previous affiliations of Bruno Toaldo include Catholic University of the Sacred Heart & University of Naples Federico II.

Space-time fractional equations and the related stable processes at random time

Abstract: In this paper we consider the general fractional equation \sum_{j=1}^m \lambda_j \frac{\partial^{ u_j}}{\partial t^{ u_j}} w(x_1,..., x_n ; t) = -c^2 (-\Delta)^\beta w(x_1,..., x_n ; t), for u_j \in (0,1], \beta \in (0,1] with initial condition w(x_1,..., x_n ; 0)= \prod_{j=1}^n \delta (x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \bm{S}_n^{2\beta} \mathcal{L} c^2 {L}^{ u_1,..., u_m} (t) \r, t>0, where \bm{S}_n^{2\beta} is an isotropic stable process independent from {L}^{ u_1,..., u_m}(t) which is the inverse of {H}^{ u_1,..., 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 r.v.'s 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 \bm{S}_n^{2\beta} (c^2 {L}^{ u_1,..., u_m} (t)), 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 time {L}^{ u_1,..., u_m} (t), t>0. The iterated process {L}^{ u_1,..., u_m}_r (t), t>0, inverse to {H}^{ u_1,..., u_m}_r (t) =\sum_{j=1}^m \lambda_j^{1/ u_j} _1H^{ u_j} (_{2}H^{ u_j} (_3H^{ u_j} (... _{r}H^{ u_j} (t)...))), t>0, permits us to construct the process \bm{S}_n^{2\beta} (c^2 {L}^{ u_1,..., u_m}_r (t)), t>0, the distribution of which solves a space-fractional generalized telegraph equation. For r \to \infty and \beta = 1 we obtain a distribution which represents the n-dimensional generalisation of the Gauss-Laplace law and solves the equation \sum_{j=1}^m \lambda_j w(x_1,..., x_n) = c^2 \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2} w(x_1,..., x_n).

Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation

Abstract: In this article integro-differential Volterra equations whose convolution kernel depends on the vector variable are considered and a connection of these equations with a class of semi-Markov processes is established. The variable order $\alpha(x)$-fractional diffusion equation is a particular case of our analysis and it turns out that it is associated with a suitable (non-independent) time-change of the Brownian motion. The resulting process is semi-Markovian and its paths have intervals of constancy, as it happens for the delayed Brownian motion, suitable to model trapping effects induced by the medium. However in our scenario the interval of constancy may be position dependent and this means traps of space-varying depth as it happens in a disordered medium. The strength of the trapping is investigated by means of the asymptotic behaviour of the process: it is proved that, under some technical assumptions on $\alpha(x)$, traps make the process non-diffusive in the sense that it spends a negligible amount of time out of a neighborhood of the region $\text{argmin}(\alpha(x))$ to which it converges in probability under some more restrictive hypotheses on $\alpha(x)$.

Population models at stochastic times

Abstract: In this article, we consider time-changed models of population evolution $\mathcal{X}^f(t)=\mathcal{X}(H^f(t))$, where $\mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $\mathcal{X}$ is a pure birth process, we study the form of the distribution, the intertimes between successive jumps and the condition of explosion (also in the case of killed subordinators). We also investigate the case where $\mathcal{X}$ represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size $n_0$. Finally, the subordinated linear birth-death process is considered. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

Population models at stochastic times

Abstract: In this paper we consider time-changed models of population evolution X f ( t ) = X ( H f ( t )), where X is a counting process and H f is a subordinator with Laplace exponent f . In the case where X is a pure birth process, we study the form of the distribution, the intertimes between successive jumps, and the condition of explosion (also in the case of killed subordinators). We also investigate the case where X represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n 0 . Finally, the subordinated linear birth–death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

On semi-Markov processes and their Kolmogorov's integro-differential equations

Abstract: Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward equations for a large class of homogeneous semi-Markov processes, having the form of a Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.

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.

Applied Probability and Queues

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

Cerebrovascular Consequences of Obstructive Sleep Apnea

Abstract: Obstructive sleep apnea (OSA) is defined by interrupted breathing during sleep due to airway obstruction with an ongoing respiratory effort. In adults, OSA most commonly is caused by decreased muscle tone (required for patency) in the soft tissues of the upper airway.[1][1] Symptoms of OSA include