Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLevy mixing related to distributed order calculus, subordinators and slow diffusions
107
0
0.0
(
0
)
Authors
Bruno Toaldo
Ask ChatGPT about the research
No Arabic abstract
The study of distributed order calculus usually concerns about fractional derivatives of the form $int_0^1 partial^alpha u , m(dalpha)$ for some measure $m$, eventually a probability measure. In this paper an approach based on Levy mixing is proposed. Non-decreasing Levy processes associated to Levy triplets of the form $l a(y), b(y), u(ds, y) r$ are considered and the parameter $y$ is randomized by means of a probability measure. The related subordinators are studied from different point of views. Some distributional properties are obtained and the interplay with inverse local times of Markov processes is explored. Distributed order integro-differential operators are introduced and adopted in order to write explicitly the governing equations of such processes. An application to slow diffusions is discussed.
rate research
Read More
We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$.$ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).
Consider a spectrally positive Stable($1+alpha$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning sizes varying during the lifetime. As for Crump-Mode-Jagers processes (with characteristics), we consider for each level the collection of individuals alive. We arrange their sizes at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable($1+alpha$) process, this yields new theorems of Ray-Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson--Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
For a difference approximations of multidimensional diffusion, the truncated local limit theorem is proved. Under very mild conditions on the distribution of the difference terms, this theorem provides that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess a densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy a uniform diffusion-type estimates. The proof is based on the new version of the Malliavin calculus for the product of finite family of measures, that may contain non-trivial singular components. An applications for uniform estimates for mixing and convergence rates for difference approximations to SDEs and for convergence of difference approximations for local times of multidimensional diffusions are given.
In this paper, we construct a Malliavin derivative for functionals of square-integrable Levy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Levy process.
Among Markovian processes, the hallmark of Levy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Levy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
