# A natural extension of Markov processes and applications to singular SDEs

## TL;DR

This paper introduces a novel method for extending Markov processes to include singular points as polar sets, enabling the construction of solutions to SDEs and SPDEs from all initial conditions, even in infinite-dimensional spaces.

## Contribution

It presents a new extension technique for Markov processes that improves upon trivial extensions, allowing solutions to stochastic differential equations from all starting points.

## Key findings

- Extension method makes added points polar, avoiding process trapping.
- Enables construction of solutions for SDEs with singular coefficients.
- Applicable to Markov processes from Dirichlet forms and martingale problems.

## Abstract

We develop a general method for extending Markov processes to a larger state space such that the added points form a polar set. The so obtained extension is an improvement on the standard trivial extension in which case the process is made stuck in the added points, and it renders a new technique of constructing extended solutions to S(P)DEs from all starting points, in such a way that they are solutions at least after any strictly positive time. Concretely, we adopt this strategy to study SDEs with singular coefficients on an infinite dimensional state space (e.g. SPDEs of evolutionary type), for which one often encounters the situation where not every point in the space is allowed as an initial condition. The same can happen when constructing solutions of martingale problems or Markov processes from (generalized) Dirichlet forms, to which our new technique also applies.