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The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

Francesco Mainardi, Antonio Mura, Gianni Pagnini
2010 International Journal of Differential Equations  
Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order.  ...  stochastic processes that we generally refer to as time-fractional diffusion processes.  ...  We note that, for q ≥ 4 , Eq. (3.12) is akin to the hyper-Airy differential equation of order q−1 , see e.g. [3] .  ... 
doi:10.1155/2010/104505 fatcat:cip36mkudfdrlgro6czoid3534

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation [article]

Promit Ghosal, Guilherme L. F. Silva
2022 arXiv   pre-print
In turn, such universal limits are described by the integro-differential Painlev\'e II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the  ...  KPZ equation at any finite time.  ...  A Model Riemann-Hilbert Problem In this section we discuss a model Riemann-Hilbert Problem that will be used in the construction of a local parametrix in the asymptotic analysis for the orthogonal polynomials  ... 
arXiv:2201.12941v1 fatcat:mcqthczxlrd5thhzrnsl3deusi

Some historical aspects of error calculus by Dirichlet forms [chapter]

Nicolas Bouleau
2014 Festschrift Masatoshi Fukushima  
These two characteristics imply a peculiar differential calculus for the propagation of errors through models.  ...  Some works of Laplace are related to this approach and also the paper of Cauchy [8]; c) The calculus of finite probabilistic errors where the errors are represented by random variables, which has been  ...  measures, diffusions defined by stochastic differential equations).  ... 
doi:10.1142/9789814596534_0005 fatcat:bxccky6itfa4fazqtucruuplce

Some Historical Aspects of Error Calculus by Dirichlet Forms [article]

Nicolas Bouleau
2014 arXiv   pre-print
that make the use of Dirichlet forms more relevant and efficient.  ...  We discuss the main stages of development of the error calculation since the beginning of XIX-th century by insisting on what prefigures the use of Dirichlet forms and emphasizing the mathematical properties  ...  measures, diffusions defined by stochastic differential equations).  ... 
arXiv:1401.2644v1 fatcat:bg4l5msbmna5llkltmosicmw5y

Models of anomalous diffusion: the subdiffusive case

A. Piryatinska, A.I. Saichev, W.A. Woyczynski
2005 Physica A: Statistical Mechanics and its Applications  
The model describes asymptotic behavior of a jump (anomalous random walk) process with random jump sizes and random interjump time intervals with infinite means (and variances) which do not satisfy the  ...  In the case when these intervals have a fractional exponential p.d.f., the fractional Komogorov-Feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions  ...  Zaslavsky for arising his interest in the intriguing world of fractional diffusion.  ... 
doi:10.1016/j.physa.2004.11.003 fatcat:l4ksp27v4fdl5ntbtmepth2g4y

Lecture notes : Spectral properties of non-self-adjoint operators

Johannes Sjöstrand
2009 Journées Équations aux dérivées partielles  
set of spectral instability, by the following simplified version of a theorem of Roch and Silberman: The result becomes more sublte if we use the more traditional definition with a non-strict inequality  ...  The first part gives some old and recent results on non-self-adjoint differential operators.  ...  This allows us in principle to consider more general random perturbations and will be used in Section 9.  ... 
doi:10.5802/jedp.54 fatcat:wd6nje5u5bgark4fljpce7hvtm

Fourier's Method of Linear Programming and Its Dual

H. P. Williams
1986 The American mathematical monthly  
97I30 Sequences and series 97I40 Differential calculus 97I50 Integral calculus 97I60 Functions of several variables 97I70 Functional equations 97I80 Complex analysis 97I99 None of the above  ...  differential equations of infinite order 35R60 Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35R70 Partial differential equations with multivalued  ...  relativity 83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems) 83C10 Equations of motion 83C15 Exact solutions 83C20 Classes of solutions; algebraically special solutions  ... 
doi:10.2307/2322281 fatcat:yvhgyh2epbcwdoqdhuaopkcrue

The M-Wright function in time-fractional diffusion processes: a tutorial survey [article]

Francesco Mainardi, Antonio Mura, Gianni Pagnini
2010 arXiv   pre-print
Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order.  ...  stochastic processes that we generally refer to as time-fractional diffusion processes.  ...  Gorenflo and the anonymous referees for useful comments. This analysis can be compared to that described with a different language in papers by Meerschaert et al. [51, 52] .  ... 
arXiv:1004.2950v1 fatcat:i4lblxt6e5djzhvs5ovfefyc5m

Reviews and Descriptions of Tables and Books

1967 Mathematics of Computation  
These data were computed by the author on a desk calculator "many years ago" and were subsequently misplaced.  ...  16 [A].-Rudolph Ondrejka, The First 100 Exact Double Factorials, ms. of 12 handwritten sheets (undated) deposited in the UMT file.  ...  A. Tumarkin, Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equation t(py')' + (q + er) y = /, translated by D. E.  ... 
doi:10.1090/s0025-5718-67-99891-2 fatcat:l7yy2lo24jevbnkguypncloi7u

Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions [article]

Tristan Gautié
2021 arXiv   pre-print
We present a combined analysis of these systems, employing tools of random matrix theory and stochastic calculus as well as tools of quantum mechanics, in order to solve some original problems.  ...  Lastly, the fourth chapter, centred on the particular case of bridge processes, allows for a joint treatment of scalar and matrix models; therein, we develop a generalization of the Ferrari-Spohn problem  ...  an ordinary differential equation.  ... 
arXiv:2111.05737v1 fatcat:k7wukqyslrghjmpqsnu3sxch6a

Reviews and Descriptions of Tables and Books

1958 Mathematics of Computation  
There is a selection of starter, predictor, and corrector formulae for use in the numerical integration of differential equations (5 pages). F. Differential equations.  ...  , U2(s, a) of a generalized Airy equation U"(s) 4-s"U(s) = 0, in which, if complications are to be avoided, a + 2 (supposed positive) must not be the reciprocal of an integer.  ... 
doi:10.1090/s0025-5718-58-99281-0 fatcat:zozjkhxa3zavlfkedcvuqrak7y

Averaging along irregular curves and regularisation of ODEs [article]

R. Catellier, M. Gubinelli
2016 arXiv   pre-print
We consider the ordinary differential equation (ODE) dx_t =b(t,x_t ) dt+ dw_t where w is a continuous driving function and b is a time-dependent vector field which possibly is only a distribution in the  ...  In the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H ∈ (0,1), we prove that almost surely the ODE admits a solution for all b in the Besov-HÃlder  ...  Allowing random b could open the way to the study of a general class of stochastic transport equations where the drift itself depends on the solution.  ... 
arXiv:1205.1735v4 fatcat:eiw6gslo6nbzhmw672qjpsf6te

A growth model in a random environment

Janko Gravner, Craig A. Tracy, Harold Widom
2002 The Annals of Probability  
We assume that p x are chosen independently at random with a common distribution F and that the initial state is such that the origin is far above the other sites.  ...  This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory  ...  Throughout, we denote by · integration with respect to dF and by p a generic random variable with distribution F .  ... 
doi:10.1214/aop/1029867130 fatcat:a7ylswri6bglfcydfj23rm56h4

A growth model in a random environment [article]

Janko Gravner, Craig A. Tracy, Harold Widom
2001 arXiv   pre-print
We assume that p_x are chosen independently at random with a common distribution F, and that the initial state is such that the origin is far above the other sites.  ...  This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory  ...  This means that the maximum is achieved at a convex ψ.  ... 
arXiv:math/0011150v2 fatcat:7afx6m66s5ex7lzafscgkv254a

Algorithmic Foundations for the Diffraction Limit [article]

Sitan Chen, Ankur Moitra
2020 arXiv   pre-print
For more than a century and a half it has been widely-believed (but was never rigorously shown) that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system  ...  In particular we show that there is a phase transition where the sample complexity goes from polynomial to exponential.  ...  Finally by querying ρ at carefully chosen points along a random direction v we can use the matrix pencil method [Moi15] to set up a generalized eigenvalue problem to recover the centers of the Airy disks  ... 
arXiv:2004.07659v2 fatcat:5f6ojb4ghjdxzlxs2kwmsdvyv4
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