LisMath Seminar 

21/04/2022, 12:00 — 12:30 — Online
Pedro Cardoso, Instituto Superior Técnico, Universidade de Lisboa

Hydrodynamic limit of symmetric exclusion processes with long jumps

We discuss the hydrodynamic behavior of the long jumps symmetric exclusion process with a slow barrier. When jumps occur between a negative site and a non-negative site, the rates are slowed down by a factor of $\alpha n^{-\beta}$, where $\alpha >0$ and $\beta \geq 0$. The jump rates are given by a symmetric transition probability $p(\cdot)$. In [4], we study the case where $p(\cdot)$ has finite variance and obtain the heat equation with various boundary conditions, analogously to [6]. On other hand, in [5] we study the case where $p(\cdot)$ has infinite variance and obtain diverse partial differential equations given in terms of the fractional Laplacian and the regional fractional Laplacian, with different boundary conditions, similarly to [1] and [2]. Finally, we present how we can derive a fractional porous media equation from the symmetric exclusion process, as it is described in [3].

References:

[1] Bernardin C., Cardoso P., Gonçalves P., Scotta S., “Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion,” arXiv preprint arXiv:2007.01621, (2021).

[2] Bernardin C., Gonçalves P., Jiménez-Oviedo B., “A microscopic model for a one parameter class of fractional Laplacians with Dirichlet boundary conditions,” Archive for Rational Mechanics and Analysis, 239, No. 1, 1–48 (2021).

[3] Cardoso P., de Paula R., Gonçalves P., “Derivation of the fractional porous media equation from a microscopic dynamics,” In preparation, (2022).

[4] Cardoso P., Gonçalves P., Jiménez-Oviedo B., “Hydrodynamic be- havior of long-range symmetric exclusion with a slow barrier: diffusive regime,” arXiv preprint arXiv:2111.02868, (2021).

[5] Cardoso P., Gonçalves P., Jiménez-Oviedo B., “Hydrodynamic be- havior of long-range symmetric exclusion with a slow barrier: superdif- fusive regime,” arXiv preprint arXiv:2201.10540, (2022).

[6] Franco T., Gonçalves P., Neumann A., “Phase transition of a heat equation with Robin’s boundary conditions and exclusion process,” Transactions of the American Mathematical Society, 367, No. 9, 6131– 6158 (2015).