Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Influence of left ventricular relaxation on the pressure half time of aortic regurgitation

BACKGROUND The severity of aortic regurgitation can be estimated using pressure half time (PHT) of the aortic regurgitation flow velocity, but the correlation between regurgitant fraction and PHT is weak. AIM To test the hypothesis that the association between PHT and regurgitant fraction is substantially influenced by left ventricular relaxation. METHODS In 63 patients with aortic regurgitation, subdivided into a group without (n = 22) and a group with (n = 41) left ventricular hypertrophy, regurgitant fraction was calculated using the difference between right and left ventricular cardiac outputs. Left ventricular relaxation was assessed using the early to late diastolic Doppler tissue velocity ratio of the mitral annulus (E/ADTI), the E/A ratio of mitral inflow (E/AM), and the E deceleration time (E-DT). Left ventricular hypertrophy was assessed using the M mode derived left ventricular mass index. RESULTS The overall correlation between regurgitant fraction and PHT was weak (r = 0.36, p < 0.005). In patients without left ventricular hypertrophy, there was a significant correlation between regurgitant fraction and PHT (r = 0.62, p < 0.005), but not in patients with left ventricular hypertrophy. In patients with a left ventricular relaxation abnormality (defined as E/ADTI< 1, E/AM< age corrected lower limit, E-DT >/= 220 ms), no associations between regurgitant fraction and PHT were found, whereas in patients without left ventricular relaxation abnormalities, the regurgitant fraction to PHT relations were significant (normal E/AM: r = 0.57, p = 0.02; E-DT< 220 ms: r = 0.50, p < 0.001; E/ADTI < 1: r = 0.57, p = 0.02). CONCLUSIONS Only normal left ventricular relaxation allows a significant decay of PHT with increasing aortic regurgitation severity. In abnormal relaxation, which is usually present in left ventricular hypertrophy, wide variation in prolonged backward left ventricular filling may cause dissociation between the regurgitant fraction and PHT. Thus the PHT method should only be used in the absence of left ventricular relaxation abnormalities.

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary

Time-Delayed Theory of the Neolithic Transition in Europe

The classical wave-of-advance model of the neolithic transition (i.e., the shift from hunter-gatherer to agricultural economies) is based on Fisher's reaction-diffusion equation. Here we present an extension of Einstein's approach to Fickian diffusion, incorporating reaction terms. On this basis we show that second-order terms in the reaction-diffusion equation, which have been neglected up to now, are not in fact negligible but can lead to important corrections. The resulting time-delayed model agrees quite well with observations

Retention half times in the skeleton of plutonium and 90Sr from above-ground nuclear tests: a retrospective study of the Swiss population.

Plutonium and (90)Sr are considered to be among the most radiotoxic nuclides produced by the nuclear fission process. In spite of numerous studies on mammals and humans there is still no general agreement on the retention half time of both radionuclides in the skeleton in the general population. Here we determined plutonium and (90)Sr in human vertebrae in individuals deceased between 1960 and 2004 in Switzerland. Plutonium was measured by sensitive SF-ICP-MS techniques and (90)Sr by radiometric methods. We compared our results to the ones obtained for other environmental compartments to reveal the retention half time of NBT fallout (239)Pu and (90)Sr in trabecular bones of the Swiss population. Results show that plutonium has a retention half time of 40+/-14 years. In contrast (90)Sr has a shorter retention half time of 13.5+/-1.0 years. Moreover (90)Sr retention half time in vertebrae is shown to be linked to the retention half time in food and other environmental compartments. These findings demonstrate that the renewal of the vertebrae through calcium homeostatic control is faster for (90)Sr excretion than for plutonium excretion. The precise determination of the retention half time of plutonium in the skeleton will improve the biokinetic model of plutonium metabolism in humans.

Retention half times in the skeleton of plutonium and Sr-90 from above-ground nuclear tests: A retrospective study of the Swiss population

Plutonium and Sr-90 are considered to be among the most radiotoxic nuclides produced by the nuclear fission process. In spite of numerous studies on mammals and humans there is still no general agreement on the retention half time of both radionuclides in the skeleton in the general population. Here we determined plutonium and Sr-90 in human vertebrae in individuals deceased between 1960 and 2004 in Switzerland. Plutonium was measured by sensitive SF-ICP-MS techniques and Sr-90 by radiometric methods. We compared our results to the ones obtained for other environmental compartments to reveal the retention half time of NBT fallout Pu-239 and Sr-90 in trabecular bones of the Swiss population. Results show that plutonium has a retention half time of 40 +/- 14 years. In contrast Sr-90 has a shorter retention half time of 13.5 +/- 1.0 years. Moreover Sr-90 retention half time in vertebrae is shown to be linked to the retention half time in food and other environmental compartments. These findings demonstrate that the renewal of the vertebrae through calcium homeostatic control is faster for Sr-90 excretion than for plutonium excretion. The precise determination of the retention half time of plutonium in the skeleton will improve the biokinetic model of plutonium metabolism in humans. (C) 2010 Elsevier Ltd. All rights reserved.