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This paper derives the weak-sense Gaussian solution to a family of\ud fractional-in-time and multifractional-in-space stochastic partial differential\ud equations, driven by fractional-integrated-in-time spatiotemporal white\ud noise. Some fundamental results on the theory of pseudodifferential operators\ud of variable order, and on the Mittag-Leffler function are applied to\ud obtain the temporal, spatial and spatiotemporal H¨older continuity, in the\ud mean-square sense, of the derived solu...
We introduce a non-homogeneous fractional Poisson process by replacing the time variable in the fractional Poisson process of renewal type with an appropriate function of time. We characterize the resulting process by deriving its non-local governing equation. We further compute the first and second moments of the process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from th...
We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We construct the multifractal, such as log-gamma, log-tempered stable, or log-normal tempered stable scenarios.
This paper derives the stochastic solution of a Cauchy problem\ud for the distribution of a fractional diffusion process. The governing equation\ud involves the Bessel-Riesz derivative (in space) to model heavy tails of the\ud distribution, and the Caputo-Djrbashian derivative (in time) to depicts the\ud memory of the diffusion process. The solution is obtained as Brownian motion\ud with time change in terms of the Bessel-Riesz subordinator on the inverse\ud stable subordinator. This stochast...
Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic proper...
Principal support vector machine was proposed recently by Li,\ud Artemiou and Li (2011) to combine L$1$ support vector machine and\ud sufficient dimension reduction. We introduce the principal L$q$ support vector\ud machine as a unified framework for linear and nonlinear sufficient\ud dimension reduction. By noticing that the solution of L$1$ support\ud vector machine may not be unique, we set $q>1$ to ensure the\ud uniqueness of the solution. The asymptotic distribution of the\ud proposed es...
Continuous time random walks have random waiting times between particle jumps. We define\ud the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions\ud (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme\ud model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but\ud diffusion processes with non-independent increments. The waiting times are selected from the do...
A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a non-linear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Lo\'eve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-It\^o stochastic integra...
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