Strictly two-dimensional self-avoiding walks: Thermodynamic properties revisited

The density crossover scaling of various thermodynamic properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse-grained bead-spring model. In the semidilute regime we confirm over an order of magnitude of the monomer density ?? the expected power law scaling for the interaction energy between different chains e _int ?? ?? ^21/8, the total pressure P ?? ?? ³ and the dimensionless compressibility gT = lim _q??0 S(q) ?? 1/?? ². Various elastic contributions associated to the affine and non-affine response to an infinitesimal strain are analyzed as functions of density and sampling time. We show how the size ??(??) of the semidilute blob may be determined experimentally from the total monomer structure factor S(q) characterizing the compressibility of the solution at a given wave vector q . We comment briefly on finite persistence length effects.      

On the occurrence of polygon-shaped patterns in vibrated cylindrical granular beds

We report experimental observations of polygon-shaped patterns formed in a vertically vibrated bed of circular cross-section. A phase map is determined, showing that the polygon pattern is established for ?? = A(2??f)²/g ? 10 . The sensitivity of the polygon structure to bed parameters was tested by studying beds of different particle sizes and fill levels. It was hypothesized that the polygon pattern observed in cylindrical beds is the corresponding pattern to the formation of arches in square-shaped beds. The close relationship between these two patterns was demonstrated by two observations: i) the radii of the arches of a corresponding square bed and the inner radius of the cylindrical bed were found to be very similar and ii) the boundary lengths of the two patterns were in good agreement.      

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation ? u(x, t)/? t = κ Δ u(x, t) + ξ(x, t)u(x, t), x ∈ ? ^d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u ₀(x), x ∈ ? ^d , is typically taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d κ, split into two at rate ξ ∨ 0, and die at rate (?ξ) ∨ 0. In earlier work we looked at the Lyapunov exponents

$$ \lambda _{p}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t} \log \mathbb {E} ([u(0,t)]^{p})^{1/p}, \quad p \in \mathbb{N} , \qquad \lambda _{0}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t}\log u(0,t). $$

For the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ. In the present paper we investigate what happens when κΔ is replaced by Δ^??, where ?? = {??(x, y) : x, y ∈ ? _d , x ～ y} is a collection of random conductances between neighbouring sites replacing the constant conductances κ in the homogeneous model. We show that the associated annealed Lyapunov exponents λ _p (??), p ∈ ?, are given by the formula

$$ \lambda _{p}(\mathcal{K} ) = \text{sup} \{\lambda _{p}(\kappa ) : \, \kappa \in \text{Supp} (\mathcal{K} )\}, $$

where, for a fixed realisation of ??, Supp(??) is the set of values taken by the ??-field. We also show that for the associated quenched Lyapunov exponent λ ₀(??) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(??) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ, and for all ?? satisfying a certain clustering property, namely, there are arbitrarily large balls where ?? is almost constant and close to any value in Supp(??). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of ?? where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of ?? where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

     

Phenomenological implications of light stop and higgsinos

We examine the phenomenological implications of light $\tilde t_R $ and higgsinos in the Minimal Supersymetric Standard Model, assuming tan² ??<m _t m _b and heavy $\tilde t_L $ and gauginos. In this simplified setting, we study the contributions to ??m _{B d},?? _K,BR(b??s??),R _b???(Z??bb) /??(Z??hadrons),BR(t??bW), and their interplay.     

Parity-violating asymmetry in the reactions 6Li(n,\alpha) 3H and 10B(n,\alpha)^{7}Li* \rightarrow 7Li + \gamma

We present two experiments measuring the parity-violating asymmetry (P-odd) of tritons emission in the reaction ⁶Li(n,??)³H and ??-quanta emission in the reaction ¹⁰B(n,??)⁷Li* ?? ⁷Li + ?? with polarized cold neutrons, aiming at estimation of the weak meson-nucleon coupling constants. These experiments started in the PNPI, Gatchina and continued in the ILL, Grenoble. We describe an integral method to measure P-odd asymmetry and the two experiments. The results of our measurements are: $\alpha{}_{P-odd}^{6Li}=(-8.8\pm2.1)\cdot10^{-8}$ and $\alpha{}_{P-odd}^{10B}=(0.8\pm3.9)\cdot10^{-8}$ .     

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant ??, which is related to energy density of cosmological vacuum by $ \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda $ . In the Veneziano ghost theory, the vacuum energy density ?? _?? is expressed by absolute value of the product of quark vacuum condensate and quark current mass: $ \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q < 0|:\bar qq:|0 > | $ . We calculate the quark local vacuum condensates ??0|: $ \bar q $ q: |0?? by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator S _f (p) with an effective gluon propagator G _???? ^ab (q). The quark current mass m _q is predicted by use of chiral perturbation theory. Our theoretical result of ??, with the resulting ??0|: 471-4 q: |0?? = ?(235 MeV)³ and light quark current mass m _q ? 3.29?C6.15 MeV, is in a good agreement with the observable of the ?? ?? 10^?52 m^?2 used widely in a great amount of literatures.     

Inelastic Character of Solitons of Slowly Varying gKdV Equations

In this paper we study soliton-like solutions of the variable coefficients, the subcritical gKdV equation $$u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0,\quad {\rm in} \quad \mathbb{R}_t\times\mathbb{R}_x, \quad m=2,3\,\, { \rm and }\,\, 4,$$ with ${\lambda\geq 0, a(\cdot ) \in (1,2)}$ a strictly increasing, positive and asymptotically flat potential, and ${\varepsilon}$ small enough. In previous works (Mu?oz in Anal PDE 4:573?C638, 2011; On the soliton dynamics under slowly varying medium for generalized KdV equations: refraction vs. reflection, SIAM J. Math. Anal. 44(1):1?C60, 2012) the existence of a pure, global in time, soliton u(t) of the above equation was proved, satisfying $$\lim_{t\to -\infty}\|u(t) - Q_1(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0,\quad 0\leq \lambda<1,$$ provided ${\varepsilon}$ is small enough. Here R(t, x) := Q _c (x ? (c ? ??)t) is the soliton of R _t +? (R _xx ??? R + R ^m ) _x =?0. In addition, there exists ${\tilde \lambda \in (0,1)}$ such that, for all 0?<??? <?1 with ${\lambda\neq \tilde \lambda}$ , the solution u(t) satisfies $$\sup_{t\gg \frac{1}{\varepsilon}}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}\lesssim \varepsilon^{1/2}.$$ Here ${{\rho'(t) \sim (c_\infty(\lambda) -\lambda)}}$ , with ${{\kappa(\lambda)=2^{-1/(m-1)}}}$ and ${{c_\infty(\lambda)>\lambda}}$ in the case ${0<\lambda<\tilde\lambda}$ (refraction), and ${\kappa(\lambda) =1}$ and c _??(??)?<??? in the case ${\tilde \lambda<\lambda<1}$ (reflection). In this paper we improve our preceding results by proving that the soliton is far from being pure as t ?? +???. Indeed, we give a lower bound on the defect induced by the potential a(·), for all ${{0<\lambda<1, \lambda\neq \tilde \lambda}}$ . More precisely, one has $$\liminf_{t\to +\infty}\| u(t) - \kappa_m(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}>rsim \varepsilon^{1 +\delta},$$ for any ${{\delta>0}}$ fixed. This bound clarifies the existence of a dispersive tail and the difference with the standard solitons of the constant coefficients, gKdV equation.     

Aging effect in CaLaBa{Cu1???xFex}3O7???�� with 0 �� x �� 0.07 studied by M?ssbauer spectroscopy

In this work, we study the long-term aging effect caused by Fe atoms in the superconductor CaLaBa{Cu_1???xFe_x}₃O_7????? with 0 ?? x ?? 0.07. XRD confirms that this system has a YBCO-like structure. The critical temperature (T_c) is strongly affected by aging and depends on the amount of Fe in the structure. Room temperature Mössbauer spectroscopy reveals the presence of the typical species A, B?CB ??, C and new species E ?? and F. Interestingly; A, which corresponds to the Fe^3?+? atom located in the Cu(1) of the chains with spin S _z = 3/2, shows a drastic reduction which means migration to the species B, B ?? and C. Species B and B ?? correspond to the Fe^3?+? in the Cu(2) site forming planar quasi-octahedral and planar square pyramidal, while the C specie is a square pyramidal with O(5) respectively (spin S_z = 3/2 in all these cases). Aging causes loss of superconductivity in the samples with 5 and 7% of iron content.     

Supremum of the Airy2 Process Minus a Parabola on a Half Line

Let $\mathcal {A}_{2}(t)$ be the Airy₂ process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as the one-point marginal of the Airy_2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution F _GUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution F _GOE(4^1/3 x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay $e^{-\frac{4}{3} x^{3/2} }$ .     

Neutron-rich hypernuclei in the chiral soliton model

The binding energies of neutron-rich strangeness S = ?1 hypernuclei are estimated in the chiral soliton approach using the bound state rigid oscillator version of the SU(3) quantization model. Additional binding of strange hypernuclei in comparison with nonstrange neutron-rich nuclei takes place at not large values of atomic (baryon) numbers, A = B ?? ??10. This effect becomes stronger with increasing isospin of nuclides, and for the ??nuclear variant?? of the model with rescaled Skyrme constant e. Binding energies of _?? ⁸ He and recently discovered _?? ⁶ H satisfactorily agree with data. Hypernuclei _?? ⁷ H, _?? ⁹ He are predicted to be bound stronger in comparison with their nonstrange analogues ⁷H, ⁹He; hypernuclei _?? ¹⁰ Li, _?? ¹¹ LI, _?? ¹² Be, _?? ¹³ Be, etc. are bound stronger in the nuclear variant of the model.     

New approach to analyzing and evaluating cross sections for partial photoneutron reactions

The presence of substantial systematic discrepancies between the results of different experiments devoted to determining cross sections for partial photoneutron reactions??first of all, (??, n), (??, 2n), and (??, 3n) reactions??is a strong motivation for studying the reliability and authenticity of these data and for developing methods for taking into account and removing the discrepancies in question. In order to solve the first problem, we introduce objective absolute criteria involving transitional photoneutron-multiplicity functions F ₁, F ₂, F ₃, ??; by definition, their values cannot exceed 1.0, 0.5, 0.33, ??, respectively. With the aim of solving the second problem, we propose a new experimental-theoretical approach. In this approach, reaction cross sections are evaluated by simultaneously employing experimental data on the cross section for the total photoneutron yield, ?? ^expt(??, xn) = ?? ^expt(??, n) + 2?? ^expt(??, 2n) + 3?? ^expt(??, 3n) + ??, which are free from drawbacks plaguing experimental methods for sorting neutrons in multiplicity, and the results obtained by calculating the functions F _theor ¹ , F _theor ² , F _theor ³ , ?? on the basis of the modern model of photonuclear reactions. The reliability and authenticity of data on the cross sections for (??, n), (??, 2n), and (??, 3n) partial reactions???? ^eval(??, in) = F _i ^theor ?? ^expt(??, xn)??were evaluated for the ⁹⁰Zr, ¹¹⁵In, ^{112,114,116,117,118,119,120,122,124}Sn, ¹⁵⁹Tb, and ¹⁹⁷Au nuclei.     

Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise

Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model??the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points $\mathcal{N}_{M}(x)$ above a level $\frac{x}{2}V_{m}$ , with V _m =2lnM standing for the leading-order typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of $\mathcal{N}_{M}(x)$ for 0<x<2. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level x, results in an important difference between the mean and the typical values of $\mathcal{N}_{M}(x)$ . This can be further used to determine the typical threshold x _m of extreme values in the pattern which turns out to be given by $x_{m}^{(\mathit{typ})}=2-c\ln\ln M /\ln M $ with $c=\frac{3}{2}$ . Such observation provides a rather compelling explanation of the mechanism behind universality of c. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum p _max of intensity is to be given by $-\ln p_{\mathit{max}}=\alpha_{-}\ln M +\frac{3}{2f'(\alpha_{-})}\ln\ln M+O(1)$ , where f(??) is the corresponding singularity spectrum positive in the interval ????(?? _?,?? ₊) and vanishing at ??=?? _?>0. For the 1/f noise case we further study asymptotic values of the prefactors in scaling laws for the moments of the counting function. Our numerics shows however that one needs prohibitively large sample sizes to reach such asymptotics even with a moderate precision. This motivates us to derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.     

Smectic order parameters via liquid crystal NMR spectroscopy: Application to a partial bilayer smectic A phase

Solute molecules were dissolved in the liquid crystal 4-cyano-4??-n-octyloxybiphenyl (8OCB), known to form a partial bilayer smectic-A phase. Through measurement of solutes?? and solvent??s orientational order parameters via nuclear magnetic resonance spectroscopy, and their analysis via a statistical thermodynamic density functional theory, values of the solvent??s positional order parameters and solutes?? positional-orientational distribution functions were obtained. Near to the transition to the nematic phase, the main positional order parameter of the smectic liquid crystal turned out to be comprised in the interval 0.4?C0.6, though the quality of the fittings assuming the phase as nematic all across the temperature range investigated was only slightly worse. This may be ascribed to the looseness of the partial bilayer smectic structure. Solutes were found to preferentially lie in those regions where liquid crystal molecule terminal chains are located.      

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Contribution of vector resonances to the \bar{B}_{d}^{0}\to\bar {K}^{*0}\mu^{+}\mu^{-} decay

The fully differential angular distribution for the rare flavor-changing neutral current decay $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) \mu^{+}\mu^{-} $ is studied. The emphasis is placed on accurate treatment of the contribution from the processes $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) V $ with intermediate vector resonances V=??(770),??(782),?(1020),J/??,??(2S),?? decaying into the ?? ⁺ ?? ^? pair. The dilepton invariant-mass dependence of the branching ratio, longitudinal polarization fraction f _L of the $\bar{K}^{*0}$ meson, and forward?Cbackward asymmetry A _FB is calculated and compared with data from Belle, CDF and LHCb. It is shown that inclusion of the resonance contribution may considerably modify the branching ratio, calculated in the SM without resonances, even in the invariant-mass region far from the so-called charmonia cuts applied in the experimental analyses. This conclusion crucially depends on values of the unknown phases of the B ⁰??K ^?0 J/?? and B ⁰??K ^?0 ??(2S) decay amplitudes with zero helicity.     

Z c (4025) as the hadronic molecule with hidden charm

We have studied the loosely bound $D^{*}\bar{D}^{*}$ system. Our results indicate that the recently observed charged charmonium-like structure Z _c (4025) can be an ideal $D^{*}\bar{D}^{*}$ molecular state. We have also investigated its pionic, dipionic, and radiative decays. We stress that both the scalar isovector molecular partner Z _c0 and three isoscalar partners ${\tilde{Z}}_{c0,c1,c2}$ should also exist if Z _c (4025) is a $D^{*}\bar{D}^{*}$ molecular state in the framework of the one-pion-exchange model. Z _c0 can be searched for in the channel e ⁺ e ^?→Y→Z _c0(4025)(ππ)_P-wave where Y can be Y(4260) or any other excited 1^?? charmonium or charmonium-like states such as Y(4360), Y(4660), etc. The isoscalar $D^{*}\bar{D}^{*}$ molecular states ${\tilde{Z}}_{c0,c2}$ with 0⁺(0⁺⁺) and 0⁺(2⁺⁺) can be searched for in the three pion decay channel $e^{+}e^{-}\to Y \to {\tilde{Z}}_{c0,c2} (3\pi)^{I=0}_{\text{P-wave}}$ . The isoscalar molecular state ${\tilde{Z}}_{c1}$ with 0^?(1^+?) can be searched for in the channel ${\tilde{Z}}_{c1}\eta$ . Experimental discovery of these partner states will firmly establish the molecular picture.     

On the Standard Model prediction for \mathcal{B}(B_{s,d} \to \mu^{+} \mu^{-})

The decay B _s →μ ⁺ μ ^? is one of the milestones of the flavor program at the LHC. We reappraise its Standard Model prediction. First, by analyzing the theoretical rate in the light of its main parametric dependence, we highlight the importance of a complete evaluation of higher-order electroweak corrections, at present known only in the large-m _t limit, and leaving sizable dependence on the definition of electroweak parameters. Using insights from a complete calculation of such corrections for $K\to\pi\nu\bar{\nu}We present O(?? _s ) results on the decays of polarized W ^± and Z bosons into massive quark pairs. The NLO QCD corrections to the polarized decay functions are given up to the second order in the quark mass expansion. We find a surprisingly strong dependence of the NLO polarized decay functions on finite quark mass effects even at the relatively large mass scale of the W ^± and Z bosons. As a main application we consider the decay t??b+W ⁺ involving the helicity fractions ?? _mm of the W ⁺ boson followed by the polarized decay $W^{+}(\uparrow)\to q_{1}\bar{q}_{2}$ for which we determine the O(?? _s ) polar angle decay distribution. We also discuss NLO polarization effects in the production/decay process $e^{+}e^{-}\to Z(\uparrow)\to q\bar{q}$ .     

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation ? _t ?? $\mathcal{D}$ Δ?=λS ²?, where S(x, t) is a Gaussian stochastic field with covariance C(x?x′, t, t′), and x∈ $\mathbb{R}$ ^d . It is shown that the coupling λ _cN (t) at which the N-th moment <? ^N (x, t)> diverges at time t, is always less or equal for $\mathcal{D}$ >0 than for $\mathcal{D}$ =0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫ ^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$ =0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$ →i $\mathcal{D}$ , the case of interest for backscattering instabilities in laser-plasma interaction.     

Physics beyond the Standard Model through b → s μ ?+? μ ? transition

A comprehensive study of the impact of new-physics operators with different Lorentz structures on decays involving the b ?? s ?? ^?+? ?? ^? transition is performed. The effects of new vector?Caxial vector (VA), scalar?Cpseudoscalar (SP) and tensor (T) interactions on the differential branching ratios, forward?Cbackward asymmetries (A _FB??s), and direct CP asymmetries of ${\bar B}_{\rm s}^0 \to \mu^+ \mu^-$ , ${\bar B}_{\rm d}^0 \to$ $ X_{\rm s} \mu^+ \mu^-$ , ${\bar B}_{\rm s}^0 \to \mu^+ \mu^- \gamma$ , ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ , and ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ are examined. In ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ , we also explore the longitudinal polarization fraction f _L and the angular asymmetries $A_{\rm T}^{(2)}$ and A _LT, the direct CP asymmetries in them, as well as the triple-product CP asymmetries $A_{\rm T}^{\rm (im)}$ and $A^{\rm (im)}_{\rm LT}$ . While the new VA operators can significantly enhance most of the observables beyond the Standard Model predictions, the SP and T operators can do this only for A _FB in ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ .     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.