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Publication

A Characterisation Of The Continuum Gaussian Free Field In Arbitrary Dimensions

Juhan Aru
2022
Journal Articles

Abstract

We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Official source

Juhan Aru
2022
Journal Articles

Abstract

Official source

Related concepts (16)

Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series.

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces.

Functional (mathematics)

In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space into its field of scalars (that is, they are elements of the dual space ) In functional analysis and related fields, it refers more generally to a mapping from a space into the field of real or complex numbers.

Related publications (24)

A Characterisation Of The Continuum Gaussian Free Field In Arbitrary Dimensions

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A Characterisation Of The Continuum Gaussian Free Field In Arbitrary Dimensions

WILD SOLUTIONS TO SCALAR EULER-LAGRANGE EQUATIONS

Transportation-based functional ANOVA and PCA for covariance operators

Structure and dynamics of liquid water from ab initio simulations: adding Minnesota density functionals to Jacob&apos;s ladder

Structure and dynamics of liquid water from ab initio simulations: adding Minnesota density functionals to Jacob's ladder