Narrow Results

In this paper, we reduce the problem of quantization of the Yang–Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections on ${ℝ}^3$. We suggest a formally self-adjoint expression for the quantized Yang–Mills Hamiltonian as an operator on the corresponding Lebesgue $L^2$-space. In the case when the Yang–Mills field is associated to the abelian group $U(1)$, we define the probability measure which depends on two real parameters $m>0$ and $c\ne 0$. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is $\{0\}\cup [\frac{1}{2}m,\infty )$, i.e. it has a gap.

We construct measures on the space , n≤4, of Bruhat–Schwartz distributions over the field of p-adic numbers, corresponding to finite volume polynomial interactions in a p-adic analog of the Euclidean quantum field theory. In contrast to earlier results in this direction, our choice of the free measure is the Gaussian measure corresponding to an elliptic pseudo-differential operator over . Analogs of the Euclidean P(φ)-theories with free and half-Dirichlet boundary conditions are considered.

We prove that the optimal transportation mapping that takes a Gaussian measure γ on an infinite dimensional space to an equivalent probability measure g·γ satisfies the Monge–Ampère equation provided that log g∈L¹(γ) and g log g∈L¹(γ).

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.

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian⁷ and Poissonian¹² cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup ${\mathrm{\text{e}}}^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers $L^N$ of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the L_q space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the L_q space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the L_q space for 1 ≤ q < ∞. Also, in the L_q space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

A method named the Gaussian equivalent representation and developed to calculate path integrals over a Gaussian measure, is presented. As an example partition functions for simple liquids and proton plasma are calculated by this method. Free energy and a pair correlation function in the lowest and next approximations are obtained.