Aspects of Brownian Motion

Mansuy, Roger; Yor, Marc

doi:10.1007/978-3-540-49966-4

Cited by 94 publications

(94 citation statements)

References 56 publications

(77 reference statements)

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“…Proof. We first note the following identity since its right-hand side also equals to that of (2.4) with σ replaced by √ −1 (see, for example, [14]):…”

Section: Reduction To Ikeda-taniguchi's Constructionmentioning

confidence: 99%

Tau functions of KP solitons realized in Wiener space

Aihara¹,

Akahori²,

Fujii³

et al. 2013

Bulletin of the London Mathematical Society

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“…Proof. We first note the following identity since its right-hand side also equals to that of (2.4) with σ replaced by √ −1 (see, for example, [14]):…”

Section: Reduction To Ikeda-taniguchi's Constructionmentioning

confidence: 99%

Tau functions of KP solitons realized in Wiener space

Aihara¹,

Akahori²,

Fujii³

et al. 2013

Bulletin of the London Mathematical Society

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“…The property (20) uses (19) and the orthogonal basis expansion of the function 1 {g} : h → 1 {g=h} .…”

Section: Reminders On Dirichlet Characters the Content Of This Sectimentioning

confidence: 99%

Random Characters under the $L$-Measure, I: Dirichlet Characters

Barhoumi-Andréani

2017

International Mathematics Research Notices

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Abstract. We define the L-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group.We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalised characters converge in law to Gaussians after rescaling (Kerov Central Limit Theorem The goal of this article is to study the properties of characters of G q when q → +∞ and when the characters are selected at random according to the L-measure that we introduce in definition 1.1.This article focuses on the case of Dirichlet characters modulo q that are multiplicative group morphisms χ : G q → C × extended to Z/qZ by setting χ(n) = 0 if n ∈ (Z/qZ) \ (Z/qZ) × and finally extended by periodicity to Z by setting χ(n) := χ(n mod q) (see e.g. [15, § 5]). These maps were used by Dirichlet to prove his celebrated theorem on the infinitude of primes in arithmetic progressions.Dirichlet characters have the following properties :(1) χ is periodic modulo q : χ(n + q) = χ(n) for all m, n ∈ N, (2) χ is completely multiplicative :χ(n) = 0 if and only if gcd(n, q) = 1. Note that these properties imply that χ(1) = 1, as χ(1) = χ(1 × 1) = χ(1) 2 and χ(1) = 0 since gcd(1, q) = 1.We define G q to be the set of Dirichlet characters modulo q. Apart from the value 0, these characters take values in the ϕ(q)-roots of unity e 2iπk/ϕ(q) , k 0 , where ϕ is Euler's totient function. There are exactly ϕ(q) such characters.The L-function attached to a Dirichlet character χ is the following function defined for all s ∈ {Re > 1} by (see e.g. [28])We consider these functions as linear forms in the character χ, hence the choice of notation compared with the usual one that writes L(s, χ).This measure can be written in the following way using the infamous Euler formula (12) :where P is the set of prime numbers. We can thus interpret (3) in the setting of statistical mechanics as a system of particles on a circle (whose positions are given by the angles of the character evaluated in prime numbers) submitted to a particular logarithmic confinement potential at inverse temperature 2. RANDOM DIRICHLET CHARACTERS 3Let χ ≡ χ (q) denote the canonical evaluation on G q in the Dynkin formalism, namelyWe will be interested in the behaviour of the random variables χ k for a fixed integer k. The main theorem of this paper states Theorem 1.2 (Convergence in law of the evaluations). Let k ∈ 2, q be a fixed integer and s ∈ (1, +∞). Then, under P s,q , the following convergence in law is satisfiedwhere C is a standard Cauchy-distributed random variable of densityThis theorem is proven in section 3.1. An immediate striking comparison with the winding number of the complex Brownian motion can be made...

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“…On the other hand, classical Brownian motion remains a topic of intense study for classical probabilists as it continues to yield deep and fascinating secrets (see e.g. [24] for an account of recent progress.) Is quantum probability missing an opportunity here?…”

Section: Quantum Brownian Motionmentioning

confidence: 99%