Fractional diffusion-type equations with exponential and logarithmic differential operators

We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density of a stable process (see Mainardi et al. (2001)): the first equation considered here is obtained by adding an exponential differential (or shift) operator expressed in terms of the Riesz–Feller derivative. We prove that this produces a random component in the time-argument of the corresponding stable process, which is represented by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional diffusion equation, a logarithmic differential operator involving the Riesz-derivative, we obtain, as a solution, the transition semigroup of a stable process subordinated by an independent gamma subordinator with drift. Finally, we show that an extension of the space-fractional diffusion equation, containing both the fractional shift operator and the Feller integral, is satisfied by the transition density of the process obtained by time-changing the stable process with an independent linear birth process with drift.     

Exact Small Ball Constants for Some Gaussian Processes under the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Norm

We find some logarithmic and exact small deviation asymptotics for the L ²-norms of certain Gaussian processes closely connected with a Wiener process. In particular, processes obtained by centering and integrating Brownian motion and Brownian bridge are examined. Bibliography: 28 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 5–21.     

Gaussian Limiting Behavior of the Rescaled Solution to the Linear Korteweg–de Vries Equation with Random Initial Conditions

We analyze the asymptotic behavior of the rescaled solution to the linear Korteweg–de Vries equation when the initial conditions are supposed to be random and weakly dependent. By means of the method of moments we prove the Gaussianity of the limiting process and we present its correlation function. The same technique is applied to the analysis of another third-order heat-type equation.     

Fractional Gamma and Gamma-Subordinated Processes

We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0. In the case ν > 1, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index 1/ν. For ν less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., [42 Van Noortwijk, J.M., 2009. A survey of the application of Gamma processes in maintenance. Reliability Engineering System Safety 94: 2–21.[Crossref], [Web of Science ®] , [Google Scholar]] and the references therein).

As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications.     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

On the Maximum of Some Conditional and Integrated Gaussian Fields and their Statistical Applications

We evaluate upper bounds for the maximal distributions of some Gaussian random fields, which arise in the study of the asymptotic behavior of various two-dimensional empirical processes, with random index. Some of them are generalizations of well-known conditional Brownian fields, while the others are obtained by their integration. We present also some possible statistical applications of our results.     

On the maximum of the generalized Brownian bridge

We present some extensions of the distributions of the maximum of the Brownian bridge in [0,t] when the conditioning event is placed at a future timeu>t or at an intermediate timeu<t. The standard distributions of Brownian motion and Brownian bridge are obtained as limiting cases. These results permit us to derive also the distribution of the first-passage time of the Brownian bridge. Similar generalizations are carried out for the Brownian bridge with drift μ; in this case, it is shown that the maximal distribution is independent of μ (whenu≥t). Finally, the case of the two-sided maximal distribution of Brownian motion in [0,t], conditioned onB(u)=η (for bothu>t andu<t), is considered. Dip. di Statistica, Probabilità e Stat. Applicate, Università di Roma “La Sapienza,” Piazzale Aldo Moros, 00185 Roma, Italy. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 200–213, April–June, 1999.     

On the Solutions of Linear Odd-Order Heat-Type Equations with Random Initial Conditions

In this paper odd-order heat-type equations with different random initial conditions are examined. In particular, we give rigorous conditions for the existence of the solutions in the case where the initial condition is represented by a strictly ϕ –subGaussian harmonizable process η = η (x). Also the case where η is represented by a stochastic integral with respect to a process with independent increment is studied. Partially supported by the NATO grant PST.CLG.980408.     

Space-Fractional Versions of the Negative Binomial and Polya-Type Processes

In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.