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Annular substructures in protoplanetary discs, ubiquitous in sub-mm observations, can be caused by gravitational coupling between a disc and its embedded planets. Planetary density waves inject angular momentum into the disc leading to gap opening only after travelling some distance and steepening into shocks (in the absence of linear damping); no angular momentum is deposited in the planetary co-orbital region, where the wave has not shocked yet. Despite that, simulations show mass evacuation from the co-orbital region even in inviscid discs, leading to smooth double-trough gap profiles. Here, we consider the early time-dependent stages of planetary gap opening in inviscid discs. We find that an often-overlooked contribution to the angular momentum balance caused by the time-variability of the specific angular momentum of the disc fluid (caused, in turn, by the time-variability of the radial pressure support) plays a key role in gap opening. Focusing on the regime of shallow gaps with depths of |$\lesssim 20~{{\ \rm per\ cent}}$|⁠ , we demonstrate analytically that early gap opening is a self-similar process, with the amplitude of the planet-driven perturbation growing linearly in time and the radial gap profile that can be computed semi-analytically. We show that mass indeed gets evacuated from the co-orbital region even in inviscid discs. This evolution pattern holds even in viscous discs over a limited period of time. These results are found to be in excellent agreement with 2D numerical simulations. Our simple gap evolution solutions can be used in studies of dust dynamics near planets and for interpreting protoplanetary disc observations. [ABSTRACT FROM AUTHOR]

While the statistical inference of first-order autoregressive processes driven by independent and identically distributed noises has a long history, the statistical analysis for first-order autoregressive processes driven by dependent noises is more recent. This paper considers the problem of estimating all the unknown parameters in first-order autoregressive processes driven by the fractional Gaussian noise, which is a self-similar stochastic process used to model persistent or anti-persistent dependency structures in observed time series. The estimation procedure is built upon the marriage of the log-periodogram regression and the method of moments. The usual asymptotic properties, including consistency and asymptotic normality, are established under some mild conditions. Monte Carlo simulations are performed to demonstrate the feasibility and effectiveness of the proposed method. Finally, the estimation methods are applied to model some realized volatility time series, where we find that the realized volatility is rough. Moreover, the proposed model is compared with some alternative models, including the first-order autoregressive process driven by the standard Gaussian noise, the heterogeneous autoregression model with lag structure (1,5,21) (HAR(1,5,21)), the autoregressive fractionally integrated moving average model (ARFIMA(1,d,0)) and the scaled fractional Brownian motion, in forecasting the logarithmic realized volatility. [ABSTRACT FROM AUTHOR]

We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model. These systems are characterized by a parameter that controls the strength of the interaction, k > 0. Despite being a one-dimensional system, this system is interesting due to the properties that arise from the long-range interaction between particles. It is a well-known fact that when k is small enough, particle collisions occur almost surely, while when k is large, collisions never occur. However, aside from this fact, there was no characterization of the collision times until now. In this paper, we derive the fractal (Hausdorff) dimension of the set of collision times by generalizing techniques introduced by L. Liu and Y. Xiao [Probab. Math. Stat. 18(2), 369–383 (1998)] to study the return times to the origin of self-similar Markov processes. In our case, we consider the return times to configurations where at least one collision occurs, which is a condition that defines unbounded sets as opposed to a single point, namely, the origin. We find that the fractal dimension characterizes the collision behavior of these systems and establishes a clear delimitation between the colliding and non-colliding regions in a way similar to that of a thermodynamic function. [ABSTRACT FROM AUTHOR]

The self-healing capacities of a GCL with natural bentonite (NB) as core (NB-GCL) and a GCL with a polymerized bentonite (PB) as core (PB-GCL) were investigated under simulated field conditions, i.e. geomembrane-GCL-a clayey subsoil layer composite liner system with a damage hole on the geomembrane and the GCL. The clayey subsoils tested had initial water contents of 22.2% and 26.8%. The liquids used were deionized (DI) water and 0.3 M NaCl solution. The test results indicate that for both the PB-GCL and NB-GCL only hydrated on the subsoils for 1–3 months, a damage hole of 15 mm in diameter was almost not self-healed. For cases of applying a constant liquid head on the top of the geomembrane of 100 mm, the sizes of damage holes were self-healed in term of diameter is about half of the values reported in the literature for tests with plenty liquid supply to the damaged GCLs. Further, for the conditions tested and for the cases self-healed, DI water leaked into the subsoil was less than about 60 g, and 0.3 M NaCl solution leaked was less than 150 g. • Self-healing capacity of GCLs was newly studied under a simulated field condition. • GCLs hydrated on clayey subsoils had very limited self-healing capacity. • Self-heling capacity of GCLs with limited liquid supply was clarified. • Leakage through a hole on GCL before it was self-healed was novelly investigated. [ABSTRACT FROM AUTHOR]

On post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$ , are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate. [ABSTRACT FROM AUTHOR]

We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers. [ABSTRACT FROM AUTHOR]

In this work, we theoretically and numerically investigate Rayleigh–Taylor dynamics with constant acceleration. On the side of theory, we employ the group theory approach to directly link the governing equations to the momentum model, and to precisely derive the buoyancy and drag parameters for the bubble and spike in the linear, nonlinear, and mixing regimes. On the side of simulations, we analyze numerical data on Rayleigh–Taylor mixing by applying independent self-similar processes associated with the growth of the bubble amplitude and with the bubble merger. Based on the obtained results, we reveal the constituents governing Rayleigh–Taylor dynamics in the linear, nonlinear, and mixing regimes. We outline the implications of our considerations for experiments in plasmas, including inertial confinement fusion. [ABSTRACT FROM AUTHOR]

Except in the trivial case of spatially uniform flow, the advection-diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection-diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue-eigenfunction pairs on a multiscale parameter q = 2ik Pe, where k is the non-dimensional wavenumber of the tracer in the streamwise direction, and Pe is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions. [ABSTRACT FROM AUTHOR]

We study the large deviation behaviors of a stochastic fluid queue with an input being a generalized Riemann–Liouville (R–L) fractional Brownian motion (FBM), referred to as GFBM. The GFBM is a continuous mean-zero Gaussian process with non-stationary increments, extending the standard FBM with stationary increments. We first derive the large deviation principle for the GFBM by using the weak convergence approach. We then obtain the large deviation principle for the stochastic fluid queue with the GFBM as the input process as well as for an associated running maximum process. Finally, we study the long-time behavior of these two processes; in particular, we show that a steady-state distribution exists and derives the exact tail asymptotics using the aforementioned large deviation principle together with some maximal inequality and modulus of continuity estimates for the GFBM. [ABSTRACT FROM AUTHOR]

Context. The development of information and communication technologies has led to an increase in the volume of information sent over the network. Media service platforms play an important role in the creation and processing of bitrate in the information network. Therefore, there is a need to develop a methodology for predicting bitrate in various media service platforms by creating an effective algorithm that minimizes the forecast error. Objective. The aim of the work is to synthesize in analytical form the state transition matrix of the Kalman filter for nonstationary self-similar processes when predicting the bitrate in telecommunication networks. Method. A methodology has been developed for predicting teletraffic in media service platforms, based on a modification of the Kalman filter for non-Gaussian processes. This methodology uses an original procedure for calculating statistics, which makes it possible to reduce the filtering and forecast error that arises due to the uncertainty of the analytical model of the process under study. The methodology does not require knowledge of the analytical model of the process, as well as strict restrictions on its stochastic characteristics. Results. A methodology for estimating and forecasting bitrate in telecommunication systems is proposed. This methodology was used to study teletraffic processes in the media service platforms Google Meet, Zoom, Microsoft Teams. The passage of real bitrate through the specified media service platforms was studied. A comparison of real teletraffic with predicted teletraffic was carried out. The influence of the order of the state transition matrix of the Kalman filter on the error of estimation and prediction has been studied. It has been established that even a low (second) order of the state transition matrix allows one to obtain satisfactory forecast results. It is shown that the use of the proposed methodology makes it possible to predict traffic with a relative error of the order of 3-4%. Conclusions. An original algorithm for assessing and forecasting the characteristics of media traffic has been developed. Recommendations for improving the technology for building media service platforms are formulated. It is shown that the bitrates generated by various media service platforms, in the case of applying the proposed estimation and forecasting methodology, are invariant with respect to the type of stochastic processes being processed [ABSTRACT FROM AUTHOR]

Earthquake dynamic rupture requires a nucleation process to provide sufficient energy to overcome the fracture energy. Large earthquakes may occur via a cascading rupture process, which includes many triggering processes that cascade from small to large sections of the fault system. During such a process, the nucleation of a large section of the fault plane may occur dynamically via the propagating rupture from a small section of the fault plane. A quasi-static view of seismic nucleation has been widely discussed in earthquake seismology; however, the dynamic nucleation process remains poorly known. Here, we investigate one aspect of the dynamic nucleation process by focusing on the rupture propagation velocity during the nucleation process. We simplify this process as self-similar crack propagation at a constant rupture velocity in a finite nucleation zone within a target region that possesses a uniform fracture energy. We numerically solve this elastodynamic problem in two dimensions for both the anti-plane and in-plane cases using the Boundary Integral Equation method. As the rupture velocity increases, the critical ratio of the fracture energy step to continue the rupture increases and the critical size of the dynamic nucleation zone decreases. The rapid increase in the ratio of the fracture energy step toward infinity could explain why earthquakes never propagate at slow rupture velocities. However, the effect on the size of the nucleation zone is rather limited, with the size of the dynamic nucleation zone decreasing to ~ 70% of the static nucleation zone size. However, such a small difference would result in a significant overall difference if such a dynamic nucleation process repeatedly occurred in the cascading rupture process of a large earthquake, which would be a difficult situation for earthquake early warning. [ABSTRACT FROM AUTHOR]

This paper presents and explores a diffusion model that generalizes Brownian motion (BM). On the one hand, as BM: the model's mean square displacement grows linearly in time, and the model is Gaussian and selfsimilar (with Hurst exponent 1 2 ). On the other hand, in sharp contrast to BM: the model is not Markov, its increments are not stationary, and its non-overlapping increments are not independent. Moreover, the model exhibits a host of statistical properties that are dramatically different than those of BM: aging and anti-aging, positive and negative momenta, correlated velocities, persistence and anti-persistence, aging Wiener–Khinchin spectra, and more. Conventionally, researchers resort to anomalous-diffusion models—e.g. fractional BM and scaled BM (both with Hurst exponents different than 1 2 )—to attain such properties. This model establishes that such properties are attainable well within the realm of diffusion. As it is seemingly Brownian yet highly non-Brownian, the model is termed Weird BM. [ABSTRACT FROM AUTHOR]

The retarded solutions to the Maxwell-like equations of Weyl gravitoelectromagnetism (GEM) are obtained in a similar way to those of electromagnetism. These solutions keep an analogous structure as their electromagnetic counterparts and they become the Jefimenko-type equations of GEM. [ABSTRACT FROM AUTHOR]

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r>0 for an arbitrary compactly supported Borel probability measure \nu is given by its Rényi dimension at the point q_{r} where the L^{q}-spectrum of \nu and the line through the origin with slope r intersect. In particular, this proves the continuity of r\mapsto \overline {D}_{r}(\text {\nu)} as conjectured by Lindsay [ Quantization dimension for probability distributions , ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L^{q}-spectrum restricted to \left (0,1\right). We give sufficient conditions in terms of the L^{q}-spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a \mathcal {C}^{1}-self-conformal iterated function system on \mathbb {R}^{d} without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved. [ABSTRACT FROM AUTHOR]

We consider the generalized Langevin equation driven by a fractional non-Gaussian noise, the so-called Hermite process. We prove the existence and the uniqueness of the solution for this equation as well as various properties. An useful link between the solution to the Langevin equation with Hermite noise and a class of self-similar stochastic processes with stationary increments is given in our work. Based on this relationship, we deduce the regularity of the sample paths of the solution and the behavior of its p -variation. Some applications concerning the statistical estimation of the noise intensity parameter or the Euler method for the generalized Langevin equation are also discussed. [ABSTRACT FROM AUTHOR]

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity. [ABSTRACT FROM AUTHOR]

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised k th order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised k th order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory. [ABSTRACT FROM AUTHOR]

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval (0 , 1). We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

Self-similar constructions of transformations preserving a sigma-finite measure are considered and their properties and the spectra of the induced Gaussian and Poisson dynamical systems are studied. The orthogonal operator corresponding to such a transformation has the property that some power of this operator is a nontrivial direct sum of operators isomorphic to the original one. The following results are obtained. For any subset of the set of positive integers, in the class of Poisson suspensions, sets of spectral multiplicities of the form are realized. A Gaussian flow is presented such that the set of spectral multiplicities of the automorphisms is if and if . A Gaussian flow such that the automorphisms have distinct spectral types for but all automorphisms , , are pairwise isomorphic is constructed. [ABSTRACT FROM AUTHOR]

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly α -stable process at its first exit time from (0 , ∞). We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes. [ABSTRACT FROM AUTHOR]

The Friction Stir Welding (FSW) process depends entirely upon mechanical contact between the tool and the workpiece. As a result of this, all process phenomena and process outcomes such as weld geometry and mechanical properties are governed by FSW's frictional system. The following work characterizes this system with a focus on process initialization, heat input and material flow. For this purpose, an experimental program for the isolated investigation of the frictional system was carried out. Short-term effects such as contact initiation, run-in behavior and frictional transitions are considered as well as the influences of process parameters and geometry. The system and its behavior are analyzed quantitatively and qualitatively by experiments altering the normal pressure, relative velocity, and tool geometry. The experiments demonstrate a self-similar behavior of the process, including an important wear transition which initiates the material flow, and a subsequent equilibrium of forces, heat balance, and temperatures. The interaction between the tool and the welded material is described, as is the link between the frictional interface and material flow initialization. Based on these findings, recommendations are provided for process optimization and tool design. [ABSTRACT FROM AUTHOR]

Two former presidents of the ISST provide illuminating discussions of Fraser's work and the new directions in which time studies has evolved (appropriate in light of Fraser's evolutionary theory of time). I first met Julius Fraser, along with his wife Jane, in late summer 1995. Many ISST members no doubt have in their possession a typed postcard (part of the collection that Julius amassed during his travels) in which greetings and well wishes were interspersed with directives regarding ISST tasks. [Extracted from the article]

High-pressure and temperature deformation experiments interfaced with acoustic emission (AE) monitoring have been conducted to study transformational faulting in Mn 2 GeO 4 olivine, which transforms to the β phase, isostructural to wadsleyite. Metastable Mn 2 GeO 4 olivine exhibits a marked embrittlement behavior at temperatures between 800 and 1100 K, emitting numerous AEs. At each temperature, brittle deformation is characterized by a two-stage process: (1) a "preparation" stage with numerous diffusedly located low-magnitude AEs and large b values (>2), and (2) a failure stage where larger-magnitude AEs form a planar distribution with b values about 1. Microstructure analysis reveals extensive kink band development in olivine grains in the recovered samples. Kink band boundaries (KBBs), with a typical thickness of ∼100 nm, are filled with a nanometric β-Mn 2 GeO 4 "gouge". A dense array of secondary shear localizations is often present within the kink bands, suggesting significant shear deformation therein. The combined observations suggest that faulting in metastable Mn 2 GeO 4 olivine is a self-similar process, from grain-scale to the sample-scale. Both observed embrittlement behavior and the microstructure of metastable Mn 2 GeO 4 olivine are essentially identical to those in Mg 2 GeO 4 olivine we have reported previously, indicating that the physical mechanism of faulting in metastable olivine is insensitive to the specific crystallographic structure of the high-pressure phase. The low b values (about 1) observed in the faulting process in our experiments are similar to those of deep focus earthquakes in cold subduction zones. Our observed mechanism explains deep focus seismicity in cold metastable mantle wedges, provided that the self-similarity assumption holds to geological scales. • Mn 2 GeO 4 olivine rock faults when transforming to wadsleyite structure under pressure. • Transformational faulting, detected by AEs, is associated with kink band formation. • Faulting is preceded by a preparation stage with high b values and low magnitude AEs. • Main faults are characterized by b ≈ 1, with AEs distributed in a planar fashion. • Fault development is through a self-similar process. [ABSTRACT FROM AUTHOR]

Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order d, which extends FBM, is a paradigmatic example of a stochastic process belonging to the dth Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when d≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d\ge 3$$\end{document}. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent of those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order d. He solved it in this same paper in the particular case d=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d=2$$\end{document} in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case d≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d\ge 3$$\end{document}. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations. [ABSTRACT FROM AUTHOR]

• Reveal the self-similarity (Mach-number-invariance) of the normalized time-averaged surface heat flux in an obliquely impinging jet in a range of the nozzle exit Mach numbers from 0.1 to 1.0. • Conduct temperature sensitive paint (TSP) measurements in an obliquely impinging jet in a range of the Mach numbers from 0.1 to 1.0. • Elucidate the origin of this self-similarity based on a semi-empirical model proposed for the maximum surface temperature difference used for normalization,. • Exhibit self-similarity is also found in the normalized root-mean-square (RMS) fields of the heat flux fluctuation and the normalized skin friction fields. This study reveals self-similarity (Mach-number-invariance) of the normalized time-averaged surface heat flux fields measured using temperature sensitive paint (TSP) in an obliquely impinging jet in a range of the nozzle exit Mach numbers from 0.1 to 1.0. Further, this self-similarity is also found in the normalized root-mean-square (RMS) fields of the heat flux fluctuation and the normalized skin friction fields extracted from the surface temperature and heat flux fields. To elucidate the origin of this self-similarity, a semi-empirical model is proposed for the maximum surface temperature difference used for normalization, indicating that the Mach number effect on the impinging jet heat transfer is mainly represented by the recovery temperature at the impingement point. [ABSTRACT FROM AUTHOR]

Subordination principle, Randomly scaled Gaussian processes, 35R11, 60G22, 60J25, 60J25, Time-fractional evolution equations, Time-changed Markov processes, Hille-Phillips functional calculus, Feynman-Kac formulae, Generalized grey Brownian motion, 35C15, 47D06, 47D08, 47A60 Keywords: Time-fractional evolution equations; Subordination principle; Feynman-Kac formulae; Randomly scaled Gaussian processes; Generalized grey Brownian motion; Time-changed Markov processes; Hille-Phillips functional calculus; 35R11; 35C15; 47D06; 47D08; 47A60; 60G22; 60J25 EN Time-fractional evolution equations Subordination principle Feynman-Kac formulae Randomly scaled Gaussian processes Generalized grey Brownian motion Time-changed Markov processes Hille-Phillips functional calculus 35R11 35C15 47D06 47D08 47A60 60G22 60J25 1818 1836 19 10/11/22 20221001 NES 221001 Introduction ged fractional Ornstein-Uhlenbeck processFract. [Extracted from the article]

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all turbulent flows. Finer observations however prove that this picture is too simplistic, owing to intermittency of energy dissipation and high velocity derivatives. In this review, we discuss how such intermittency can be explained and fitted into a new picture of turbulence. We first discuss how the concept of multi-fractality (invented by Parisi and Frisch in 1982) enables to generalize the concept of self-similarity in a non-homogeneous environment and recover a universality in turbulence. We further review the local extension of this theory, and show how it enables to probe the most irregular locations of the velocity field, in the sense foreseen by Lars Onsager in 1949. Finally, we discuss how the multi-fractal theory connects to possible singularities, in the real or in the complex plane, as first investigated by Frisch and Morf in 1981. [ABSTRACT FROM AUTHOR]

We present a numerical approach for analysing simultaneous amplification and compression of a chirped hyperbolic secant soliton using a tapered Yb-doped photonic crystal fiber (PCF) based self-similar compressor. Theoretical studies reveal the conservation of input soliton during the amplification process due to self-similar property, which in turn helps for high quality compression of amplified soliton. In order to satisfy the self-similar condition, we note that the Yb-doped fiber should exhibit a decreasing group velocity dispersion (GVD). Such a constraint is met using a tapered PCF in our work, which is modeled using a fully vectorial effective index method (FVEIM). Influence of spatial and spectral dependent gain, saturation energy, amplified spontaneous emission (ASE), dispersion and nonlinearity on pulse propagation are analyzed using the combined solution of dynamic rate equations (DRE) and the generalized nonlinear Schrödinger equation (GNLSE). Numerical outcomes upon solving the aforementioned equations show that a 1 ps pulse is compressed down to a pulsewidth of 55.4 fs with a peak power of 1.03 kW in a total fiber length of $L=2.5$ m and taper ratio of 2.34. Overall quality of the compression scheme is evidenced by a relatively high compression factor (Fc) = 18.12, and low percentage of pedestal energy ($PE$) = 10.95 % [ABSTRACT FROM AUTHOR]

We characterize the cumulative distribution functions and copulas of two-dimensional self-similar Ito processes, with randomly correlated Wiener margins, as solutions of certain elliptic partial differential equations.

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically. [ABSTRACT FROM AUTHOR]

We study how memory impacts passages at the origin for a so-called elephant random walk in the diffusive regime. We observe that the number of zeros always grows asymptotically like the square root of the time, despite the fact that, depending on the memory parameter, first return times to 0 may have a finite expectation or a fat tail with exponent less than 1/2. We resolve this apparent paradox by recasting the questions in the framework of scaling limits for Markov chains and self-similar Markov processes. [ABSTRACT FROM AUTHOR]

In contrast to their seemingly simple and shared structure of independence and stationarity, Lévy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari, we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a Lévy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal–Getoor index. [ABSTRACT FROM AUTHOR]

This paper is devoted to the characterization of spectrum candidates with a new tree structure to be the spectra of a spectral self-similar measure μ N , D generated by the finite integer digit set D and the compression ratio N − 1 . The tree structure is introduced with the language of symbolic space and widens the field of spectrum candidates. The spectrum candidate considered by Łaba and Wang is a set with a special tree structure. After showing a new criterion for the spectrum candidate with a tree structure to be a spectrum of μ N , D , three sufficient and necessary conditions for the spectrum candidate with a tree structure to be a spectrum of μ N , D were obtained. This result extends the conclusion of Łaba and Wang. As an application, an example of spectrum candidate Λ (N , B) with the tree structure associated with a self-similar measure is given. By our results, we obtain that Λ (N , B) is a spectrum of the self-similar measure. However, neither the method of Łaba and Wang nor that of Strichartz is applicable to the set Λ (N , B) . [ABSTRACT FROM AUTHOR]

This article proposes a simple approach to estimate quantiles of daily financial returns directly from high-frequency data. We denote the resulting estimator as realized quantile (RQ) and use it to forecast tail risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES). The RQ estimator is built on the assumption that financial logarithm prices are subordinated self-similar processes in intrinsic time. The intrinsic time dimension stochastically transforms the clock time in order to capture the real "heartbeat" of financial markets in accordance with their trading activity and/or riskiness. The self-similarity assumption allows to compute daily quantiles by simply scaling up their intraday counterparts, while the subordination technique can easily accommodate numerous empirical features of financial returns, such as volatility persistence and fat-tailedness. Our method, which is built on a flexible assumption, is simple to implement and exploits the rich information content of high-frequency data from another time perspective than the classical clock time. In a comprehensive empirical exercise, we show that our forecasts of VaR and ES are more accurate than the ones from a large set of up-to-date comparative models, for both, stocks and foreign exchange rates. [ABSTRACT FROM AUTHOR]

We investigate the controllability analysis of nonlinear fractional order neutral-type stochastic integro-differential system with nonGaussian process. We stress out the stochastic term of our system driven by the uncomplicated non-Gaussian Hermite process known as the Rosenblatt process, which is named after by Murray Rosenblatt who first devised this introduced concept. This process is self-similar with consistent accretion and beside emerged as restriction in the non-central limit theorem, and it exists in the second wiener chaos. The necessary and sufficient conditions for the controllability are verified by employing fixed point techniques. At end, we present illustrative examples to clarify the abstract results. [ABSTRACT FROM AUTHOR]

In chemical or physical reaction dynamics, it is essential to distinguish precisely between reactants and products for all times. This task is especially demanding in time-dependent or driven systems because therein the dividing surface (DS) between these states often exhibits a nontrivial time-dependence. The so-called transition state (TS) trajectory has been seen to define a DS which is free of recrossings in a large number of one-dimensional reactions across time-dependent barriers and thus, allows one to determine exact reaction rates. A fundamental challenge to applying this method is the construction of the TS trajectory itself. The minimization of Lagrangian descriptors (LDs) provides a general and powerful scheme to obtain that trajectory even when perturbation theory fails. Both approaches encounter possible breakdowns when the overall potential is bounded, admitting the possibility of returns to the barrier long after the trajectories have reached the product or reactant wells. Such global dynamics cannot be captured by perturbation theory. Meanwhile, in the LD-DS approach, it leads to the emergence of additional local minima which make it difficult to extract the optimal branch associated with the desired TS trajectory. In this work, we illustrate this behavior for a time-dependent doublewell potential revealing a self-similar structure of the LD, and we demonstrate how the reflections and side-minima can be addressed by an appropriate modification of the LD associated with the direct rate across the barrier. [ABSTRACT FROM AUTHOR]

In this paper, we perform the joint study of scale-invariant operators and self-similar processes of complex order. More precisely, we introduce general families of scale-invariant complex-order fractional-derivation and integration operators by constructing them in the Fourier domain. We analyze these operators in detail, with special emphasis on the decay properties of their output. We further use them to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst exponents. These random processes are expressed via their characteristic functionals over the Schwartz space of functions. They are therefore defined as generalized random processes in the sense of Gel'fand. Beside their self-similarity and stationarity, we study the Sobolev regularity of the proposed random processes. Our work illustrates the strong connection between scale-invariant operators and self-similar processes, with the construction of adequate complex-order scale-invariant integration operators being preparatory to the construction of the random processes. [ABSTRACT FROM AUTHOR]

We study certain renormalised first passage bridges of self-similar processes, generalising the "Brownian co-ascent process" discussed by Panzo (Sém. Prob. L, 2019) and introduced by Rosenbaum and Yor (Sém. Prob. XLVI, 2014). We provide a characterisation of co-ascent processes via Palm measures, namely that the co-ascent of a self-similar process is the process under the Palm distribution associated with its record measure. We use this representation to derive a distributional identity for α -stable Lévy-subordinators with α ∈ (0 , 1). [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane H , obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in (Bertoin et al. Probab Theory Relat Field 172:663–724, 2018). [ABSTRACT FROM AUTHOR]

We consider undirected graphs that grow through the successive combination of component sub-graphs. For any well-behaved functions defined for such graphs, taking values in a Banach space, we show that there must exist a scaling law applicable when successive copies of the same component graph are combined. Crucially, we extend the approach introduced in previous work to the successive combination of component random sub-graphs. We illustrate this by generalizing the preferential attachment operation for the combination of stochastic block models. We discuss a further wide range of random graph combination operators to which this theory now applies, indicating the ubiquity of growth scaling laws (and asymptotic decay scaling laws) within applications, where the modules are quite distinct, yet may be considered as instances drawn from the same random graph. This is a type of statistically self-similar growth process, as opposed to a deterministic growth process incorporating exact copies of the same motif, and it represents a natural, partially random, growth processes for graphs observed in the analysis of social and technology contexts. [ABSTRACT FROM AUTHOR]

The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the F d H 3  continuum using local fractional differential operators is obtained. The mapping of a bending problem for a self-similar beam into the corresponding problem for a fractal continuum is defined. Displacements, rotations, bending moments and shear forces as functions of fractal parameters of the beam are estimated, allowing the mechanical response for self-similar beams to be established. An example of the structural behavior of a cantilever beam with a load at the free end is considered to study the influence of fractality on the mechanical properties of beams. [ABSTRACT FROM AUTHOR]

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The hierarchical organization and emergence of scaling laws in complex systems—geophysical, biological, technological, and socioeconomic—have been the topic of extensive research at the turn of the twentieth century. Although significant progress has been achieved, the mathematical origin of and relation among scaling laws for different system attributes remain unsettled. Paradigmatic examples are the Gutenberg–Richter law of seismology and Horton's laws of geomorphology. We review the results that clarify the appearance, parameterization, and implications of scaling laws in hierarchical systems conceptualized by tree graphs. A recently formulated theory of random self-similar trees yields a suite of results on scaling laws for branch attributes, tree fractal dimension, power-law distributions of link attributes, and power-law relations between distinct attributes. Given the relevance of power laws to extreme events and hazards, our review informs related theoretical and modeling efforts and provides a framework for unified analysis in hierarchical complex systems. [ABSTRACT FROM AUTHOR]

We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is introduced, based on which we obtain approximately asymptotically consistent algorithms for clustering locally asymptotically self-similar stochastic processes. In the simulation study, clustering data sampled from fractional and multifractional Brownian motions with distinct Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets' equity indexes returns and sovereign CDS spreads provides a successful real world application. Implementations in MATLAB of the proposed algorithms and the simulation study are publicly shared in GitHub. [ABSTRACT FROM AUTHOR]

A non-trivial finitely generated pro-p group G is said to be strongly hereditarily selfsimilar of index p if every non-trivial finitely generated closed subgroup of G admits a faithful selfsimilar action on a p-ary tree. We classify the solvable torsion-free p-adic analytic pro-p groups of dimension less than p that are strongly hereditarily self-similar of index p. Moreover, we show that a solvable torsion-free p-adic analytic pro-p group of dimension less than p is strongly hereditarily self-similar of index p if and only if it is isomorphic to the maximal pro-p Galois group of some field that contains a primitive pth root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free p-adic analytic pro-p groups that admit a faithful self-similar action on a p-ary tree, completing the classification of the 3-dimensional torsion-free p-adic analytic pro-p groups that admit such actions. [ABSTRACT FROM AUTHOR]