A Microscopic Model for a One Parameter Class of Fractional Laplacians with Dirichlet Boundary Conditions

Abstract

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $\alpha$ at the left of the system and $\beta$ at the right of the system. The strength of the reservoirs is ruled by $\kappa N^{-\theta }>0$. Here N is the size of the system, $\kappa >0$ and $\theta \in {{\mathbb {R}}}$. Our results are valid for $\theta \le 0$. For $\theta =0$, we obtain a collection of fractional reaction–diffusion equations indexed by the parameter $\kappa$ and with Dirichlet boundary conditions. Their solutions also depend on $\kappa$. For $\theta <0$, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $\theta > 0$ is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $\theta =0$ when we send the parameter $\kappa$ to zero. Indeed, we conjecture that the limiting profile when $\kappa \rightarrow 0$ is the one that we should obtain when taking small values of $\theta >0$.

Appendix A: Computations Involving the Generator

Lemma A.1

For any $x \ne y \in \Lambda _N$, we have

• (i) $L_N^0 (\eta _x \eta _y) = \eta _x L_N^0 \eta _y + \eta _y L_N^0 \eta _x - p(y-x) (\eta _y -\eta _x)^2,$

• (ii) $L_N^r (\eta _x \eta _y) = \eta _x L_N^r \eta _y + \eta _y L_N^r \eta _x,$

• (iii) $L_N^\ell (\eta _x \eta _y) = \eta _x L_N^\ell \eta _y + \eta _y L_N^\ell \eta _x.$

Proof

For (i) we have, by definition of $L_N^0$, that

$$\begin{aligned} \begin{aligned} L_N^0 (\eta _x \eta _y) =&\, \frac{1}{2}\sum _{{\bar{x}},{\bar{y}}\in \Lambda _{N}}p({\bar{y}}-{\bar{x}})\left[ (\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{x}(\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{y}-\eta _{x}\eta _{y}\right] \\ =&\,\frac{1}{2}\sum _{{\bar{x}},{\bar{y}}\in \Lambda _{N}}p({\bar{y}}-{\bar{x}})\left[ ((\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{x}\eta _{y}-\eta _{x}\eta _{y})+((\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{y}\eta _{x}-\eta _{x}\eta _{y})\right. \\&\left. +\,(\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{x}(\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{y}-(\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{x}\eta _{y}-(\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{y}\eta _{x}+\eta _{x}\eta _{y}\right] \\ =&\,\eta _x L_N^0 \eta _y \\&+\, \eta _y L_N^0 \eta _x + \frac{1}{2}\sum _{{\bar{x}},{\bar{y}}\in \Lambda _{N}}p({\bar{y}}-{\bar{x}})\left[ (\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{x}-\eta _{x}\right] \left[ (\sigma ^{{\bar{x}},{\bar{y}}}\eta )_{y}-\eta _{y}\right] \\ =&\, \eta _x L_N^0 \eta _y + \eta _y L_N^0 \eta _x - p(y-x) (\eta _y -\eta _x)^2. \end{aligned} \end{aligned}$$

In order to prove (ii), note that $\left[ (\sigma ^{\bar{x}}\eta )_{x}-\eta _{x}\right] \left[ (\sigma ^{\bar{x}}\eta )_{y}-\eta _{y}\right]$ is equal to zero, for all ${\bar{x}} \in {{\mathbb {Z}}}$. Thus, by definition of $L_N^r$, we have that

$$\begin{aligned} \begin{aligned} L_N^r (\eta _x \eta _y)&=\sum _{{\bar{x}}\in \Lambda _{N},{\bar{y}}\ge N}p({\bar{y}}-{\bar{x}})\left[ \eta _{{\bar{x}}}(1-\beta )+(1-\eta _{{\bar{x}}})\beta \right] \left[ (\sigma ^{{\bar{x}}}\eta )_{x}(\sigma ^{{\bar{x}}}\eta )_{y}-\eta _{x}\eta _{y}\right] \\&= \eta _x L_N^r \eta _y + \eta _y L_N^r \eta _x \\&\quad +\,\sum _{{\bar{x}}\in \Lambda _{N},{\bar{y}}\ge N}p({\bar{y}}-{\bar{x}})\\&\qquad \left[ \eta _{{\bar{x}}}(1-\beta )+(1-\eta _{{\bar{x}}})\beta \right] \left[ (\sigma ^{{\bar{x}}}\eta )_{x}-\eta _{x}\right] \left[ (\sigma ^{{\bar{x}}}\eta )_{y}-\eta _{y}\right] \\&=\eta _x L_N^r \eta _y + \eta _y L_N^r \eta _x. \end{aligned} \end{aligned}$$

The proof of the third expression is analogous. $\quad \square$