Functional measures associated to operators

TL;DR

This paper introduces a measure associated with operators in L^2 spaces, enabling solutions to abstract Cauchy problems and classical PDEs through explicit formulas and perturbative expansions, linking to stochastic integral representations.

Contribution

It develops a novel measure-based framework for solving operator equations and PDEs, including explicit computation methods and extensions to infinite-dimensional paths.

Findings

Operator in L^2 has an associated measure for solution representation.

Explicit formulas for integrals with respect to this measure are derived.

Solutions to diffusion and Fokker-Planck equations can be expressed as stochastic integrals.

Abstract

We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to compute the integral of functions with respect to this measure and develop approximate formulas in terms of a perturbative expansion. We show that this method can be used to represent solutions of classical equations, such as the diffusion and Fokker-Plank equations, as Wiener and Martin-Siggia-Rose-Jansen-de Dominics integrals, and propose an extension to paths in infinite dimensional spaces.