The asymptotic distribution of the eigenvalues for a class of Markov operators

“…The eigenfunctions ϕ n are continuous and bounded on D. In addition, λ 1 is simple and the corresponding eigenfunction ϕ 1 , often called the ground state eigenfunction, is strictly positive on D. For more general properties of the semigroups {T D t } t≥0 , see [24], [12], [16]. It is well known (see [4], [16], [17], [27]) that if D is a bounded connected Lipschitz domain and α = 2, or that if D is a bounded connected domain for 0 < α < 2, then {T D t } t≥0 is intrinsically ultracontractive.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

Bañuelos

Kulczycki

2006

Communications in Partial Differential Equations

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Let D ⊂ R 2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b > 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λ n } ∞ n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap:where C is an absolute constant. The estimate is obtained by proving new weighted Poincaré inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem established in [5].

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“…For all fractional orders β, the singular values decay exponentially, but the decay rate increases dramatically with the increase of the fractional order β and the terminal time T . Hence, anomalous superdiffusion does not change the exponentially ill-posed nature of the backward problem, but numerically it does enable recovering more Fourier modes of the initial data v. Last, we note that for other choices of the fractional derivative, e.g., the Riemann-Liouville fractional derivative and the fractional Laplacian [6,55], the magnitude of eigenvalues of the operator also tends to infinity, and the growth rate increases with the fractional order β. Therefore, the preceding observations on the space fractional backward problem are expected to be valid for these choices as well.…”

Section: 2mentioning

confidence: 86%

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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The asymptotic distribution of the eigenvalues for a class of Markov operators

Cited by 69 publications

References 4 publications

Trace estimates for stable processes

Trace estimates for stable processes

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

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

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