# mSQG equations in distributional spaces and point vortex approximation

## TL;DR

This paper proves the existence of distributional solutions for the modified Surface Quasi-Geostrophic (mSQG) equations for almost all initial conditions, using a point vortex approximation approach based on Gaussian measures and the Central Limit Theorem.

## Contribution

It introduces a novel method of constructing solutions to mSQG equations via a limit of rescaled random point vortices, connecting stochastic and PDE analysis.

## Key findings

- Existence of distributional solutions for mSQG for μ-almost every initial condition.
- Construction of stationary solutions with white noise marginals.
- Validation of point vortex approximation through probabilistic methods.

## Abstract

Existence of distributional solutions of a modified Surface Quasi-Geostrophic equation (mSQG) is proven for $\mu$-almost every initial condition, where $\mu$ is a suitable Gaussian measure. The result is the by-product of existence of a stationary solution with white noise marginal. This solution is constructed as a limit of random point vortices, uniformly distributed and rescaled according to the Central Limit Theorem.