Research

In this thesis, we investigate the long-range symmetric exclusion process in one dimension with heavy-tailed jumps, dependent on a parameter alpha which determines the heaviness of the tail. We show that for alpha between 0 and 2, the stationary density fluctuations converge to a non-local Ornstein-Uhlenbeck process governed by the Regional Fractional Laplacian, in particular showing the existence and uniqueness of the process. Similar methods can be used to show the well-posedness for singular SPDE given by the stochastic Buger's equation (with Regional Fractional Laplacian). This is based on recent work by Cardoso and Gonçalves, as well as Gräfner, Perkowski, Popat.

Furthermore, as alpha goes to 2, we show that the nonlocal Ornstein-Uhlenbeck process converges to the classical Ornstein-Uhlenbeck process for the Laplacian with Dirichlet boundary conditions.

We investigate the convergence of nonlocal quadratic forms and associated function spaces linked to Markov jump processes in bounded domains. As part of the thesis, we demonstrate their convergence to local gradient-type quadratic forms (such as the Dirichlet energy) in the sense of Mosco, a generalisation of the famous concept of Gamma-convergence. This thesis is based on a paper by Gounoue, Kassmann and Voigt