# Mosco convergence of nonlocal to local quadratic forms

## TL;DR

This paper proves that nonlocal quadratic forms related to jump processes converge to local gradient forms in bounded domains, using Mosco convergence, and establishes density of smooth functions in nonlocal spaces.

## Contribution

It demonstrates Mosco convergence of nonlocal to local quadratic forms and unifies bounded and unbounded operators within this framework.

## Key findings

- Mosco convergence established for nonlocal to local forms
- Density of smooth functions in nonlocal spaces proven
- Framework applies to both bounded and unbounded operators

## Abstract

We study sequences of nonlocal quadratic forms and function spaces that are related to Markov jump processes in bounded domains with a Lipschitz boundary. Our aim is to show the convergence of these forms to local quadratic forms of gradient type. Under suitable conditions we establish the convergence in the sense of Mosco. Our framework allows bounded and unbounded nonlocal operators to be studied at the same time. Moreover, we prove that smooth functions with compact support are dense in the nonlocal function spaces under consideration.