A fractional kinetic process describing the intermediate time behaviour of cellular flows

Hairer, Martin; Iyer, Gautam; Koralov, Leonid; Novikov, Alexei; Pajor-Gyulai, Zsolt

doi:10.1214/17-aop1196

Cited by 22 publications

(15 citation statements)

References 46 publications

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“…The recent years have witnessed the ubiquity of such non-Markovian dynamics in relation to the fractional Cauchy problem, see e.g. [27,22,13], and, also due to their central role in diverse physical applications within the field of anomalous diffusion, see e.g. [19], and also for neuronal models for which their long range dependence feature is attractive, see e.g.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

2019

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We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time T of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen timechanged, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the T's within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fréchet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-changed has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to Lévy stable processes is detailed.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

2019

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“…We indicate that the recent years have witnessed the ubiquity of such non-Markovian dynamics in relation to the fractional Cauchy problem, see e.g. [35,25,16], and, also due to their central Thus, the Hahn-Banach theorem yields that we can extend P t as a contraction of L 2 (ν). From now on, when there is no confusion, we denote by P t its extension to L 2 (ν).…”

Section: Introductionmentioning

confidence: 84%

Spectral projections correlation structure for short-to-long range dependent processes

Patie

Srapionyan

2020

Eur. J. Appl. Math

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Let X = (Xt) t≥0 be a stochastic process issued from x ∈ R that admits a marginal stationary measure ν, i.e. νPtf = νf for all t ≥ 0, where Ptf (x) = Ex[f (Xt)]. In this paper we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L 2 (ν)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt) t≥0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in [28], and then time-change by subordinators and their inverses.The authors are grateful to M.Savov for discussion related to long-tailed distributions. 1 role in diverse physical applications within the field of anomalous diffusion, see e.g. [23], as well as for neuronal models for which their long range dependence feature is attractive, see e.g. [22]. We also mention that Leonenko et al. [21] and Mijena and Nane [24] investigate the orthogonal spectral projections correlation structure in the framework of Pearson diffusions, i.e. diffusions with polynomial coefficients. More specifically, in [21], the authors discuss the case when a Pearson diffusion is time-changed by an inverse of an α-stable subordinator, 0 < α < 1. Whereas the authors of [24] consider a Pearson diffusion time-changed by an inverse of a linear combination of independent α-and β-stable subordinators, 0 < α, β < 1. In this work, we start with a general Markov process that admits an invariant measure with its associated semigroup not necessarily being self-adjoint and local, and then we perform a time-change with general subordinators and their inverses.Finally, we emphasize that the notion of long-range dependence, also known as long memory, of stochastic processes has been and it is still a center of great interests in probability theory and its applications in the last decades. We refer for thorough and historical account of this concept to the recent monograph of Samorodnitsky [31]. The definitions of long-range dependence based on the second-order properties of a stationary stochastic process such as asymptotic behavior of covariances, spectral density, and variances of partial sums are among the most developed ...

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“…which is well known in literature and already has a probabilistic interpretation (see Remark 5.4 below for some details). This equation is related with anomalous diffusion (non-Fickian diffusion), see, for example, [21] for a recent application. Let us assume that the process defined in (3.1) is a symmetric CTRW with Mittag-Leffler waiting times and with transition probabilities…”

Section: Convergence To the Variable Order Fractional Diffusionmentioning

confidence: 99%

Semi-Markov Models and Motion in Heterogeneous Media

Ricciuti

Toaldo

2017

J Stat Phys

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Abstract. In this paper we study continuous time random walks (CTRWs) such that the holding time in each state has a distribution depending on the state itself. For such processes, we provide integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel. Particular attention is devoted to the case where the holding times have a power-law decaying density, whose exponent depends on the state itself, which leads to variable order fractional equations. A suitable limit yields a variable order fractional heat equation, which models anomalous diffusions in heterogeneous media.

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A fractional kinetic process describing the intermediate time behaviour of cellular flows

Cited by 22 publications

References 46 publications

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

Spectral projections correlation structure for short-to-long range dependent processes

Semi-Markov Models and Motion in Heterogeneous Media

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