Renormalization group and diffusion equation

We study relationship between renormalization group and diffusion equation. We consider the exact renormalization group equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.

A Note on the $$L^p$$ Integrability of a Class of Bochner–Riesz Kernels

For a general compact variety $$\Gamma $$ Γ of arbitrary codimension, one can consider the $$L^p$$ L p mapping properties of the Bochner–Riesz multiplier $$\begin{aligned} m_{\Gamma , \alpha }(\zeta ) \ = \ \mathrm{dist}(\zeta , \Gamma )^{\alpha } \phi (\zeta ) \end{aligned}$$ m Γ , α ( ζ ) = dist ( ζ , Γ ) α ϕ ( ζ ) where $$\alpha > 0$$ α > 0 and $$\phi $$ ϕ is an appropriate smooth cutoff function. Even for the sphere $$\Gamma = {{\mathbb {S}}}^{N-1}$$ Γ = S N - 1 , the exact $$L^p$$ L p boundedness range remains a central open problem in Euclidean harmonic analysis. In this paper we consider the $$L^p$$ L p integrability of the Bochner–Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of $$L^p$$ L p integrability of the kernels differs substantially from the $$L^p$$ L p boundedness range of the corresponding Bochner–Riesz multiplier operator.

Random attractors for 3D Benjamin–Bona–Mahony equations derived by a Laplace-multiplier noise

This paper contributes the dynamics for stochastic Benjamin–Bona–Mahony (BBM) equations on an unbounded 3D-channel with a multiplicative noise. An interesting feature is that the noise has a Laplace-operator multiplier, which seems not to appear in any literature for the study of stochastic PDE. After translating the stochastic BBM equation into a random equation and deducing a random dynamical system, we obtain both existence and semi-continuity of random attractors for this random system in the Sobolev space. The convergence of the system can be verified without the lower bound assumption of the nonlinear derivative. The tail-estimate is achieved by using a square of the usual cutoff function and by a careful analysis of the solution’s biquadrate. A spectrum method is also applied to prove the collective limit-set compactness.

Constructing the Correction

This chapter explains how the correction is constructed, first by considering the transportation of the phase functions. A solution (v, p, R) to the Euler-Reynolds equations is fixed and a correction v₁ = v + V, p₁ = p + P is presented. Here v is an approximation to the “coarse scale velocity” since the solution ultimately achieved by the process will resemble v at a sufficiently coarse scale. The next step is to eliminate the Transport term. A time cutoff function is also introduced, where the time cutoff itself is differentiated in the Transport term. Finally, the chapter describes the High–High Interference term and Beltrami flows, how to construct the corrections Vsubscript I, P₀ in such a way that the Stress term can be reduced to a new stress, and the Stress equation and initial phase directions.

GAUGE AND POINCARÉ INVARIANT REGULARIZATION AND HOPF SYMMETRIES

We consider the regularization of a gauge quantum field theory following a modification of the Pochinski proof based on the introduction of a cutoff function. We work with a Poincaré invariant deformation of the ordinary pointwise product of fields introduced by Ardalan, Arfaei, Ghasemkhani and Sadooghi, and show that it yields, through a limiting procedure of the cutoff functions, to a regularized theory, preserving all symmetries at every stage. The new gauge symmetry yields a new Hopf algebra with deformed costructures, which is inequivalent to the standard one.

NONPERTURBATIVE TACHYON POTENTIAL FROM THE WILSONIAN RENORMALIZATION GROUP

The derivative expansion of the Wilsonian renormalization group generates additional terms in the effective β-functions not present in the perturbative approach. Applied to the nonlinear σ-model, to lowest order the vanishing of the β-function for the tachyon field generates an equation analogous to that found in open string field theory. Although the nonlinear term depends on the cutoff function, this arbitrariness can be removed by a rescaling of the tachyon field.