# The Donsker delta function and local time for McKean-Vlasov processes and applications

## TL;DR

This paper develops a stochastic differential equation framework for the Donsker delta measure and local time of McKean-Vlasov processes, providing explicit formulas for specific cases and linking the delta function to local time.

## Contribution

It introduces a novel SDE for the Donsker delta measure of McKean-Vlasov processes and derives explicit formulas for their delta functions and local times.

## Key findings

- Derived SDE for the Donsker delta measure of McKean-Vlasov processes
- Established the relationship between the delta function and local time
- Provided explicit formulas for certain McKean-Vlasov processes

## Abstract

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean-Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon-Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean-Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.