Narrow Results

Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion.

In the present paper, we extend the "torus gauge fixing" approach by Blau and Thompson, which was developed in [10] for the study of Chern–Simons models with base manifolds M of the form M = Σ × S¹, in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M. We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined.

Stimulated by the quantum generalization of the canonical representation theory for Gaussian processes in Ref. 1, we first give the representations (not necessarily canonical) of two stationary Gaussian processes X and Y by means of white noises q_t and p_t with no assumptions on their commutator. We then assume that q_t + ip_t annihilates the vacuum state and prove that the representations are the joint Boson–Fock ones if and only if X and Y have a scalar commutator.

We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

We determine a new explicit representation of strong solutions of Itô-diffusions and elicit its correspondence to the general stochastic transport equation. We apply this formula to deduce an explicit Donsker delta function of a diffusion.

A one-parameter symplectic group {e^tÂ}_t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator U_t implementing such a proper canonical transformation gives a projective unitary representation of {e^tÂ}_t∈ℝ on the Boson–Fock space and that U_t can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent with a real-valued function τ_Â(·) such that .

In this paper, we present a new idea to define the δ-function of an observable within the canonical framework of white noise analysis. We introduce a class of observables called δ-composable observables. We then define the δ-function of such an observable as the generalized operator determined uniquely by the spectral density of the observable. Surprisingly, we find that the δ-function of a δ-composable observable is always positive as a generalized operator. Finally, we show an example of a δ-composable observable and examine properties of its δ-function.

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.¹ Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the -transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.

In this paper, we study the mapping r ↦ δ_r(Q), where Q is an observable satisfying mild conditions and δ_r(Q) is the delta function of Q with r ∈ ℝ. We find that, under certain integrable conditions, the mapping r ↦ δ_r(Q) is just the Radon–Nikodym derivative of Q's spectral measure with respect to Lebesgue measure and, moreover, Q can be represented as an integral of the mapping r ↦ rδ_r(Q) with respect to Lebesgue measure. An example is also given.

A functional integral representation for the weak solution of the Schrödinger equation with polynomially growing potentials is proposed in terms of a white noise functional.

The Radon transform is one of the most useful and applicable tools in functional analysis. First constructed by John Radon in 1917¹¹ it has now been adapted to several settings. One of the principle theorems involving the Radon transform is the Support Theorem. In this paper, we discuss how the Radon transform can be constructed in the white noise setting. We also develop a Support Theorem in this setting.

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space of Potthoff–Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity-dependent potential of the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity-dependent potential gives rise to a generalized Gauss kernel. Besides the propagators, the generating functionals are obtained.

We present the expansion of the multifractional Brownian motion (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover, we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions.

In this paper, we study the self-intersection local times of multifractional Brownian motion (mBm) in higher dimensions in the framework of white noise analysis. We show that when a suitable number of kernel functions of self-intersection local times of mBm are truncated then we obtain a Hida distribution. In addition, we present the expansion of the self-intersection local times in terms of Wick powers of white noises. Moreover, we obtain the convergence of the regularized truncated self-intersection local times in the sense of Hida distributions.

We consider equations describing a barotropic inviscid flow in a channel with topography effects and beta-plane approximation of Coriolis force, in which a large-scale mean flow interacts with smaller scales. Gibbsian measures associated to the first integrals energy and enstrophy are Gaussian measures supported by distributional spaces. We define a suitable weak formulation for barotropic equations, and prove existence of a solution preserving Gibbsian measures, thus providing a rigorous infinite-dimensional framework for the equilibrium statistical mechanics of the model.

We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.

In this paper, we study stochastic currents of Brownian motion $\xi (x)$, $x\in {ℝ}^d$, by using white noise analysis. For $x\in {ℝ}^d\setminus \{0\}$ and for $x=0\in ℝ$ we prove that the stochastic current $\xi (x)$ is a Hida distribution. Moreover for $x=0\in {ℝ}^d$ with $d>1$ we show that the stochastic current is not a Hida distribution.

We consider the (unique) mild solution $u(t,x)$ of a one-dimensional stochastic heat equation on $[0,T]\times ℝ$ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by ${\partial }_{x}u(t,x)$, and its representation as a Feynman–Kac type closed form. The chaos expansion of ${\partial }_{x}u(t,x)$ makes it possible to find its (optimal) Hölder regularity especially in space.

We study an evolution equation associated with the integer power of the Gross Laplacian ${\mathrm{\Delta }}_G^p$ and a potential function V on an infinite-dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [K. J. Hochberg, Ann. Probab.6 (1978) 433.] in the one-dimensional case with $V=0$, as well as by Barhoumi–Kuo–Ouerdiane for the case $p=1$ (See Ref. [A. Barhoumi, H. H. Kuo and H. Ouerdiane, Soochow J. Math.32 (2006) 113.]).