Brownian subordinators and fractional Cauchy problems

Baeumer, Boris; Meerschaert, Mark M.; Nane, Erkan

doi:10.1090/s0002-9947-09-04678-9

Cited by 93 publications

(109 citation statements)

References 62 publications

(95 reference statements)

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“…In this case, it is an important observation [45,46,[50][51][52][53] that the probability density (31) can also be obtained as a solution of an ordinary partial differential equation including higher-order spatial derivatives, once suitable source terms are included. In this section we generalize these ideas in order to obtain positive-definite solutions for the UV limit of HL gravity in three and four spacetime dimensions.…”

Section: Diffusion On Anisotropic Spacetimesmentioning

confidence: 99%

Probing the quantum nature of spacetime by diffusion

2013

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Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semidefinite probability densities, applicable to asymptotically safe gravity, Hořava-Lifshitz gravity and loop quantum gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.

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Section: Diffusion On Anisotropic Spacetimesmentioning

confidence: 99%

Probing the quantum nature of spacetime by diffusion

2013

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“…An example of particular importance for us is iterated Brownian motion (IBM) [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. To illustrate iterated Brownian motion, we set α = 1 and consider the ordinary D-dimensional higher-order operator ∇ n , Eq.…”

Section: E Iterated Brownian Motion (β = 1 γ = 2 S = 0)mentioning

confidence: 99%

“…(70) is a solution of both (71) and (84). In general, there exists a triple connection between fractional diffusion equations with fractional time σ , higher-order diffusion equations with integer time, and iterated processes [95,97,98,100,102,104].…”

Section: E Iterated Brownian Motion (β = 1 γ = 2 S = 0)mentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.

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“…A successful one is that based on evolution equations of fractional order and the name fractional diffusion follows. This approach has been maintained by analytical methods, e.g., [6,7,37], by random walks, e.g., [15,20,21,22], by the continuous time random walk (CTRW) through a well-scaled passage to the diffusion limit, e.g., [16,17,18,23], by the parametric subordination, e.g., [19], and by the inverse stable subordination, e.g., [1,42].…”

Section: Introductionmentioning

confidence: 99%

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Pagnini

2013

fcaa

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The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported.MSC 2010 : Primary 33E20; Secondary 26A33, 33E30, 44A35, 60G18, 60G22

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Brownian subordinators and fractional Cauchy problems

Cited by 93 publications

References 62 publications

Probing the quantum nature of spacetime by diffusion

Probing the quantum nature of spacetime by diffusion

Diffusion in multiscale spacetimes

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

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