Local fields, Gaussian measures, and Brownian
                    motions

Evans, Steven N.

doi:10.1090/crmp/028/02

Cited by 24 publications

(23 citation statements)

References 58 publications

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“…This formulation shows that the projective Hilbert norm plays the role of the ∞ -norm on ‫ސޔ‬ n−1 . The appearance of ∞ , instead of 2 , agrees with the conventional 'wisdom' that generally in the tropical algebra, 2 is replaced by ∞ [Evans 2001].…”

Section: 2supporting

confidence: 74%

“…Tropicalizations of p-adic Gaussians. Evans [2001] used Kac's Theorem as the definition of Gaussians to extend them to local fields. Local fields are finite algebraic extensions of either the field of p-adic numbers or the field of formal Laurent series with coefficients drawn from the finite field with p elements [Evans 2001].…”

Section: Tropical Analogues Of Gaussiansmentioning

confidence: 99%

“…Evans [2001] used Kac's Theorem as the definition of Gaussians to extend them to local fields. Local fields are finite algebraic extensions of either the field of p-adic numbers or the field of formal Laurent series with coefficients drawn from the finite field with p elements [Evans 2001]. In particular, local fields come with a tropical valuation val, and thus one can define a tropical Gaussian to be the tropicalization of the Gaussian measure on a local field.…”

Section: Tropical Analogues Of Gaussiansmentioning

confidence: 99%

“…They have found diverse applications, from spin glasses, protein dynamics, and genetics, to cryptography and geology; see the recent comprehensive review [Dragovich et al 2017] and references therein. The p-adic Gaussian was originally defined as a step towards building Brownian motions on ‫ޑ‬ p [Evans 2001]. It would be interesting to use tools from tropical algebraic geometry to revisit and expand results involving random p-adic polynomials, such as the expected number of zeroes in a random p-adic polynomial system [Evans 2006], or properties of determinants of matrices with i.i.d p-adic Gaussians [Evans 2002].…”

Section: Proofmentioning

confidence: 99%

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Tropical Gaussians: a brief survey

Tran¹

2020

Alg. Stat.

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Section: 2supporting

confidence: 74%

Section: Tropical Analogues Of Gaussiansmentioning

confidence: 99%

Section: Tropical Analogues Of Gaussiansmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Tropical Gaussians: a brief survey

Tran¹

2020

Alg. Stat.

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“…Diffusion on Cantor sets is not entirely new and has been studied in various contexts mostly as a non-Archimedean field [1], [14], [26]. Del Muto and Figà-Talamanca have generalized this in [13], [18] for locally compact ultrametric spaces, where the group of isometries is transitive and therefore allows to treat the Cantor set as an abelian group.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets

Pearson¹,

Bellissard²

2009

J. Noncommut. Geom.

159

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Abstract. An analogue of the Riemannian Geometry for an ultrametric Cantor set .C; d / is described using the tools of Noncommutative Geometry. Associated with .C; d / is a weighted rooted tree, its Michon tree [29]. This tree allows to define a family of spectral triples .C Lip .C /; H ; D/ using the`2-space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here C Lip .C / denotes the space of Lipschitz continuous functions on .C; d /. The family of spectral triples is indexed by the space of choice functions, which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the family of these spectral triples allows to recover the metric on C . The corresponding -function is shown to have abscissa of convergence, s 0 , equal to the upper box dimension of .C; d /. Taking the residue at this singularity leads to the definition of a canonical probability measure on C , which in certain cases coincides with the Hausdorff measure at dimension s 0 . This measure in turn induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be interpreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace-Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C . This construction is applied to the simplest case, the triadic Cantor set, where: (i) the spectrum and the eigenfunctions of the Laplace-Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimension s 0 , (iii) the corresponding Markov process is shown to have an anomalous diffusion with E.d.X t ; X t Cıt / 2 / ' ıt ln .1=ıt/ as ıt # 0.

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Probability and Real Trees

Evans

2008

Lecture Notes in Mathematics

109

112

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It is intended for PhD students, teachers and researchers who are interested in probability theory, statistics, and in their applications.The duration of each school is 13 days (it was 17 days up to 2005), and up to 70 participants can attend it. The aim is to provide, in three high-level courses, a comprehensive study of some fields in probability theory or Statistics. The lecturers are chosen by an international scientific board. The participants themselves also have the opportunity to give short lectures about their research work.Participants are lodged and work in the same building, a former seminary built in the 18th century in the city of Saint-Flour, at an altitude of 900 m. The pleasant surroundings facilitate scientific discussion and exchange.

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Local fields, Gaussian measures, and Brownian motions

Cited by 24 publications

References 58 publications

Tropical Gaussians: a brief survey

Tropical Gaussians: a brief survey

Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets

Probability and Real Trees

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