Publication

Singularity of solutions to singular SPDEs

2024

Abstract

Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the d-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the Φ43-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Φ4-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.

Official source

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2024

Abstract

Official source

Related concepts (20)

Marginal distribution

In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained.

Joint probability distribution

Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).

Singularity theory

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string.