Generators of Brownian motions on abstract Wiener spaces

Kei Harada

Displaying similar documents to “Generators of Brownian motions on abstract Wiener spaces”

The number of absorbed individuals in branching brownian motion with a barrier

Pascal Maillard (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$ . At the point $x g t; 0$ , we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_{0}$ , such that this process becomes extinct almost surely if and only if $c \geq c_{0}$ . In this case, if $Z_{x}$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P (Z_{x} = n)$ as $n$ goes to infinity....

Remarks on q-CCR relations for |q| > 1

Marek Bożejko (2007)

Banach Center Publications

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In this paper we give a construction of operators satisfying q-CCR relations for q > 1: $A (f) A * (g) - A * (g) A (f) = q^{N} ⟨ f, g ⟩ I$ and also q-CAR relations for q < -1: $B (f) B * (g) + B * (g) B (f) = {| q |}^{N} ⟨ f, g ⟩ I$ , where N is the number operator on a suitable Fock space $ℱ_{q} ()$ acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

Distributions of truncations of the heat kernel on the complex projective space

Nizar Demni (2014)

Annales mathématiques Blaise Pascal

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Let ${(U_{t})}_{t \geq 0}$ be a Brownian motion valued in the complex projective space $ℂ P^{N - 1}$ . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $| U_{t}^{1} |^{2}$ and of $(| U_{t}^{1} |^{2}, | U_{t}^{2} |^{2})$ , and express them through Jacobi polynomials in the simplices of $ℝ$ and $ℝ^{2}$ respectively. More generally, the distribution of $(| U_{t}^{1} |^{2}, \dots, | U_{t}^{k} |^{2}), 2 \leq k \leq N - 1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰 (N - k + 1)$ yet computations become tedious. We also revisit the approach initiated in [] and...

Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case

Krzysztof Dębicki, Grzegorz Sikora (2011)

Applicationes Mathematicae

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Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $ℙ (s u p_{t \in [0, T]} (\sum_{i = 1}^{n} λ_{i} B_{H_{i}} (t) - c t) > u)$ , where $B_{H_{i}} (t) : t \geq 0$ , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_{i} \in (0, 1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $m i n_{i = 1, . . ., n} H_{i}$ .

Three examples of brownian flows on $ℝ$

Yves Le Jan, Olivier Raimond (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $d X_{t} = 1_{{X_{t} g t; 0}} W^{+} (d t) + 1_{{X_{t} l t; 0}} d W^{-} (d t),$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by $ϕ^{\pm}$ . The flow $ϕ^{\pm}$ is a Wiener solution of the SDE. Moreover, $K^{+} = 𝖤 [δ_{ϕ^{\pm}} | W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $K_{s, t}^{+} f (x) = f (x) + \int_{s}^{t} K_{s, u} (1_{ℝ}^{+} f^{'}) (x) W^{+} (d u) + \frac{1}{2} \int_{s}^{t} K_{s, u} f ` ` (x) d u$ for $s l t; t$ , $x \in ℝ$ and $f$ a twice continuously differentiable function. A third flow $ϕ^{+}$ can be constructed out of the $n$ -point motions of $K^{+}$ . This flow is coalescing and its $n$ -point motion...

Characterizations of processes with stationary and independent increments under $G$ -expectation

Yongsheng Song (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Our purpose is to investigate properties for processes with stationary and independent increments under $G$ -expectation. As applications, we prove the martingale characterization of $G$ -Brownian motion and present a pathwise decomposition theorem for generalized $G$ -Brownian motion.

Perturbing the hexagonal circle packing: a percolation perspective

Itai Benjamini, Alexandre Stauffer (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the hexagonal circle packing with radius $1 / 2$ and perturb it by letting the circles move as independent Brownian motions for time $t$ . It is shown that, for large enough $t$ , if $𝛱_{t}$ is the point process given by the center of the circles at time $t$ , then, as $t \to \infty$ , the critical radius for circles centered at $𝛱_{t}$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly...

The Dyson Brownian Minor Process

Mark Adler, Eric Nordenstam, Pierre Van Moerbeke (2014)

Annales de l’institut Fourier

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Consider an $n \times n$ Hermitean matrix valued stochastic process ${H_{t}}_{t \geq 0}$ where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k \times k$ minors in the upper left corner...

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

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Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for $M = S^{m}$ , the m-dimensional sphere. Let $Y (B); B \in ℬ (S^{m})$ be the Gaussian random measure on $S^{m}$ , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on $S^{m}$ , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for $B_{i}$ , i = 1,2,..., $B_{i} \cap B_{j} = \emptyset$ ,...

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik (2003)

Colloquium Mathematicae

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It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Gap universality of generalized Wigner and $β$ -ensembles

László Erdős, Horng-Tzer Yau (2015)

Journal of the European Mathematical Society

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We consider generalized Wigner ensembles and general $β$ -ensembles with analytic potentials for any $β \geq 1$ . The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $β$ -ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal....

Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup

R. Macías, C. Segovia, J. L. Torrea (2006)

Studia Mathematica

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We obtain weighted $L^{p}$ boundedness, with weights of the type $y^{δ}$ , δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${ℒ_{k}^{α}}_{k}$ , when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong...