Generalized Second Order Differential Operators

Löbus, Jörg-Uwe

doi:10.1002/mana.19911520119

Cited by 19 publications

(14 citation statements)

References 14 publications

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“…As discussed in [29], [30], the operator −D m D x with Dirichlet b.c. The number b is called derivative number and is denoted F − (0) (see Section 4 for further details).…”

Section: Model and Resultsmentioning

confidence: 99%

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Faggionato

2012

Electron. J. Probab.

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We consider a sequence X-(n), n >= 1, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X((n)) with Dirichlet conditions outside (0, n) to the analogous problem for a suitable generalized second order differential operator - D-mn D-x, with Dirichlet conditions outside a given interval. If the measures dm(n) weakly converge to some measure dm(infinity), we prove a limit theorem for the eigenvalues and eigenfunctions of - D-mn D-x to the corresponding spectral quantities of - D-m infinity D-x. As second result, we prove the Dirichlet-Neumann bracketing for the operators - D-m D-x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self-similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics

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“…As discussed in [29], [30], the operator −D m D x with Dirichlet b.c. The number b is called derivative number and is denoted F − (0) (see Section 4 for further details).…”

Section: Model and Resultsmentioning

confidence: 99%

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Faggionato

2012

Electron. J. Probab.

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“…Note that if W i (x i ) = x i , we obtain the equations in (1). This notion of generalized derivative has been studied by several authors in the literature, see for instance, [1,4,[6][7][8]. We also call attention to [1] since it provides a detailed study of such notion.…”

Section: Introductionmentioning

confidence: 99%

W-Sobolev spaces

Simas

Valentim

2011

Journal of Mathematical Analysis and Applications

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Fix strictly increasing right continuous functions with left limits and periodic increments,. Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W -generalized elliptic equations, and uniqueness results for W -generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.

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“…The process ξ that we study is a quasidiffusion, so results on quasidiffusions apply in this context -but it is a distinguished quasidiffusion among the many possible quasidiffusions taking values in T. Quasidiffusions can exhibit behavior considerably different from that of ξ -for instance, Feller and McKean [22] described a quasidiffusion that has all of R as its state space, but spends all its time in Q . These processes (with killing and appropriate boundary conditions) were shown by Löbus [54], extending work by Feller [15,18,20] to be the only Markov processes taking values in R whose generators are in some sense local, and satisfy a certain maximum principle. Various authors [47,48,51] have given beautiful spectral representations of quasidiffusions using Kreȋn's theory of strings [13,19,41].…”

Section: Introductionmentioning

confidence: 82%

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Bhamidi¹,

Evans²,

Peled³

et al. 2008

Institute of Mathematical Statistics Collections

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Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be "continuous" in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator f → 1 2 f ′′ . We then consider the special case where the state space is the self-similar set {±q k : k ∈ Z} ∪ {0} for some q > 1. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's "lost" notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, q-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local timeboth again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve q-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

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Generalized Second Order Differential Operators

Cited by 19 publications

References 14 publications

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

W-Sobolev spaces

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

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