Distribution of roots of Mittag-Leffler functions

Popov, A. Yu.; Седлетский, А. М.

doi:10.1007/s10958-013-1255-3

Cited by 63 publications

(25 citation statements)

References 26 publications

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“…It was revisited by Djrbashian [21], and many deep results were derived, especially for the case of α = 2. There are many further refinements [93]; see [83] for an updated account.…”

Section: Mittag-leffler Functionmentioning

confidence: 99%

An inverse problem for a one-dimensional time-fractional diffusion problem

Jin¹,

Rundell²

2012

Inverse Problems

150

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Abstract. Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest.In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems, and a partial list of surprising observations is given below.(a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the preci...

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“…It was revisited by Djrbashian [21], and many deep results were derived, especially for the case of α = 2. There are many further refinements [93]; see [83] for an updated account.…”

Section: Mittag-leffler Functionmentioning

confidence: 99%

An inverse problem for a one-dimensional time-fractional diffusion problem

Jin¹,

Rundell²

2012

Inverse Problems

150

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“…It can be shown that E 1 (z) = e z has no zeros for all non-negative z. Just considering negative z in the case 0 < µ < 1, the proof can be found in Theorem 4.1.1 of Popov and Sedletskii (2013) Corollary 2.5. The discrete compound fractional Poisson process M (t) is DPCP distributed; so too is the fractional Poisson process.…”

Section: Characterizationsmentioning

confidence: 96%

A characterization of signed discrete infinitely divisible distributions

Zhang

Kerns³

2017

Studia Scientiarum Mathematicarum Hungarica

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In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato (2014), Chaumont and Yor (2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter λ in the DPCP class cannot be arbitrarily small.

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“…which has the form n−1 k=0 c k (ς)t n−1−k e tς . Moreover, if ς has large enough modulus, then it is a simple root ( [13], Theorem 2.1.1), giving…”

Section: Distribution Of First Exit Time At Lower Endmentioning

confidence: 99%

“…ς∈Zα,α\U M (0) |Res(H s (z)e tz , ς)| < ∞.By Theorems 2.1.1 and Chapter 6 of[13], M > 0 can be chosen such that all elements in Z α,α \ U M (0) are not real and are simple roots of E α,α . Then for each ς ∈ Z α,α \ U M (0),…”

mentioning

confidence: 99%

Law of two-sided exit by a spectrally positive strictly stable process

Chi

2020

Stochastic Processes and their Applications

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For a spectrally positive strictly stable process with index in (1, 2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper end point, and ii) the joint distribution of the time, the undershoot, and the jump of the process when it makes first exit the other way around. For i), the density of the time of first exit is expressed as an infinite sum of functions, each the product of a polynomial and an exponential function, with all coefficients determined by the roots of a Mittag-Leffler function. For ii), conditional on the undershoot, the time and the jump of first exit are independent, and the marginal conditional densities of the time has similar features as i).the sum runs over the roots and for each root ς, p ς (t) is a polynomial in t whose coefficients are determined by ς and several Mittag-Leffler functions. For all but a finite number of ς, p ς (t) is of order 0. The result provides a connection to some known results on first exit of a standard Brownian motion. It also highlights the importance of gaining more information on the roots of Mittag-Leffler functions [13]. Section 3 considers the joint distribution of the time, the undershoot, and the jump of X when its first exit from [−b, c] occurs at c. It will be shown that conditional on the undershoot, the time and the jump are independent. This allows the joint distribution to be factorized into the marginal p.d.f. of the undershoot, and the marginal conditional p.d.f.'s of the time and the jump, respectively. The expression of the marginal conditional p.d.f. of the time has similar features as the one of first exit at the lower end. This is in contrast to the power series expression of the time of first upward passage of c [1,6,12,16], even though the first passage can be regarded as the limit of first exit from [−b, c] as b → ∞. In both sections, the asymptotics of the time of first exit as t → 0 or ∞ are also considered.The rest of the section fixes notation and collects some background information for later use.Integral transforms. For f ∈ L 1 (R), denote its Laplace transform and Fourier transform, respectively, byThe domain of f is {z ∈ C : e −tz f (t) ∈ L 1 (dt)}. Similarly, for a finite measure µ on R, denote its Laplace transform and Fourier transform, respectively, by µ(z) = e −tz µ(dt) and µ(θ) = µ(−iθ), θ ∈ R.The domain of µ is {z ∈ C : e −tz ∈ L 1 (dt)}. Let z 0 ∈ C. Denote U r (z 0 ) = {z ∈ C : |z − z 0 | < r}. If function g is analytic in U r (z 0 ) \ {z 0 } for some r > 0 and has z 0 as a pole, possibly removable, then the residual of g at z 0 is Res(g(z), z 0 ) = c 1 =

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Distribution of roots of Mittag-Leffler functions

Cited by 63 publications

References 26 publications

An inverse problem for a one-dimensional time-fractional diffusion problem

An inverse problem for a one-dimensional time-fractional diffusion problem

A characterization of signed discrete infinitely divisible distributions

Law of two-sided exit by a spectrally positive strictly stable process

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