Scaling Limits of Solutions of Linear Evolution Equations with Random Initial Conditions

TL;DR

This paper studies the solutions of linear evolution equations with random initial conditions, establishing their existence, uniqueness, and analyzing their behavior under scaling transformations.

Contribution

It introduces a framework for solutions with generalised stationary random initial conditions and explores their scaling limits, extending previous results in the field.

Findings

Existence and uniqueness of solutions for the considered equations.

Characterization of the scaling limits of these solutions.

Extension of the theory to generalised stationary random fields.

Abstract

We consider a linear equation $\partial_t u = \mathcal{L}u$, where $\mathcal{L}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field on $\mathbb{R}^d$. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.