Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

Hochberg, Kenneth J.; Orsingher, Enzo

doi:10.1007/bf02214661

Cited by 44 publications

(46 citation statements)

References 7 publications

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“…In Sec. V B we shall consider telegraph processes [85,86,98,105,106]. A remarkable duality shows that fractional and iterated Brownian motions can be identified.…”

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

confidence: 99%

“…These equations, anyway, come from composite processes just as their hyperbolic counterparts. In particular, they describe diffusion associated with the original formulation of the IBM [87,106] (but not with IBM as later formulated and presented above) or, in the case of odd-order kinetic operators ∇ 2n+1 x , with the composition of Brownian motion with stable processes [106].…”

Section: F Other Higher-order Diffusion Equationsmentioning

confidence: 99%

“…The return probability is defined as the spatial average of P , i.e., 8 Parabolic equations of the type (87) were studied in Refs. [86,[104][105][106][110][111][112] and generalized to higher-order operators ∇ n x and several other forms. the trace of the heat kernel per unit volume,…”

Section: G Spectral and Walk Dimensionsmentioning

confidence: 99%

“…(87). Other combinations of telegraph and Brownian processes are possible, leading to different quartic diffusion equations [105,106].…”

Section: B Fractional Telegraph Processmentioning

confidence: 99%

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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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“…In Sec. V B we shall consider telegraph processes [85,86,98,105,106]. A remarkable duality shows that fractional and iterated Brownian motions can be identified.…”

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

confidence: 99%

Section: F Other Higher-order Diffusion Equationsmentioning

confidence: 99%

Section: G Spectral and Walk Dimensionsmentioning

confidence: 99%

“…(87). Other combinations of telegraph and Brownian processes are possible, leading to different quartic diffusion equations [105,106].…”

Section: B Fractional Telegraph Processmentioning

confidence: 99%

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Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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“…This idea can, of course, be extended to more complicated, multi-dimensional diffusions and recent work (see e.g. Hochberg and Orsingher [9]) connecting densities of certain stochastic processes with higher order linear parabolic equations suggests that similar applications of our results with higher order parabolic equations may be in the offing. Still, our personal motivation for our results was applications in stochastic partial differential equations and the stochastic averaging of parabolic equations with random coefficients for which we refer the reader to Dawson and Kouritzin [5].…”

Section: Nmentioning

confidence: 78%

Averaging for Fundamental Solutions of Parabolic Equations

Kouritzin

1997

Journal of Differential Equations

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Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order, =-dependent parabolic partial differential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of aǹ`a veraged'' parabolic equation, we bring forth a novel approach to comparing x-derivatives of(as = Ä 0) under general regularity conditions and our basic hypothesis thatfor each x, t (i.e., pointwise). The flexibility afforded by studing fundamental vis-aÁ -vis specific solutions of these equations not only permits =-dependent Cauchy data and provides a unified method of treating all x-derivatives of u = up to order 2p&1 but also proves an invaluable tool when considering related problems of stochastic averaging. Our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial differential equations. However, it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modified set of conditions must be used.

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Complex-Valued Time Series Models and Their Relations to Directional Statistics

Shiohama

2023

Research Papers in Statistical Inference for Time Series and Related Models

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Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

Cited by 44 publications

References 7 publications

Diffusion in multiscale spacetimes

Diffusion in multiscale spacetimes

Averaging for Fundamental Solutions of Parabolic Equations

Complex-Valued Time Series Models and Their Relations to Directional Statistics

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