# Bessel SPDEs with general Dirichlet boundary conditions

## TL;DR

This paper extends the theory of Bessel stochastic partial differential equations (SPDEs) to include general Dirichlet boundary conditions by generalizing integration by parts formulae and constructing associated dynamics.

## Contribution

It generalizes integration by parts for Bessel bridges and processes with arbitrary boundary and initial conditions, enabling the formulation of Bessel SPDEs with general Dirichlet boundary conditions.

## Key findings

- Extended integration by parts formulae for Bessel bridges and processes.
- Formulated Bessel SPDEs with arbitrary boundary conditions.
- Constructed weak gradient dynamics for 2-dimensional Bessel bridges.

## Abstract

We generalise the integration by parts formulae obtained in arXiv:1811.00518v5 [math.PR] to Bessel bridges on $[0,1]$ with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of arXiv:1811.00518v5 [math.PR] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension $2$, by providing a weak construction of the gradient dynamics corresponding to a $2$-dimensional Bessel bridge.