[Paper Review] A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions

Cédric Bernardin, Patrícia Gonçalves+1 more|arXiv (Cornell University)|Mar 2, 2018

Stochastic processes and statistical mechanics8 references3 citations

TL;DR

This paper proposes a microscopic stochastic model for a one-parameter family of fractional Laplacians with Dirichlet boundary conditions, derived from a symmetric exclusion process with long-range jumps and reservoirs. It establishes a hydrodynamic limit yielding a fractional reaction-diffusion equation with a singular reaction term that enforces boundary densities α and β, leading to a new class of regional fractional Laplacians indexed by κ.

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 $α$ at the left of the system and $β$ at the right of the system. The strength of the reservoirs is ruled by $κ$N --$θ$ > 0. Here N is the size of the system, $κ$ > 0 and $θ$ $\in$. Our results are valid for $θ$ $\le$ 0. For $θ$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $κ$ and with Dirichlet boundary conditions. Their solutions also depend on $κ$. For $θ$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $θ$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $θ$ = 0 when we send the parameter $κ$ to zero. Indeed, we conjecture that the limiting profile when $κ$ $ ightarrow$ 0 is the one that we should obtain when taking small values of $θ$ > 0.

Motivation & Objective

• To develop a microscopic stochastic model that generates fractional Laplacians with Dirichlet boundary conditions in bounded domains.
• To analyze the hydrodynamic limit of a symmetric exclusion process with long-range jumps and reservoirs in contact with particle reservoirs at densities α and β.
• To characterize the resulting macroscopic PDE, which depends on the reservoir coupling strength parameter κ and exhibits a singular reaction term enforcing boundary conditions.
• To clarify the physical interpretation of non-local operators like the restricted and regional fractional Laplacians by linking them to particle systems with long jumps and reservoirs.
• To investigate the limiting behavior of the solution as the coupling parameter κ → 0, conjecturing its relation to small positive θ values in the open θ > 0 case.

Proposed method

• Model the system as a symmetric exclusion process on a one-dimensional lattice of size N with transition rates p(z) ∼ |z|^{-(1+γ)} for 1 < γ < 2, representing long jumps.
• Introduce reservoirs at the left and right boundaries with densities α and β, coupled via a rate κN^{-θ} with θ ≤ 0.
• Prove the hydrodynamic limit for θ ≤ 0, showing convergence of the particle density to a weak solution of a fractional PDE with a singular reaction term.
• Define the operator L_κ = L - κV₁, where V₁(u) = c_γ γ^{-1}(u^{-γ} + (1−u)^{-γ}), which generates the limiting fractional Laplacian with Dirichlet conditions.
• Use the Lax-Milgram theorem to establish existence and uniqueness of the weak solution to the limiting PDE by proving coercivity and continuity of the bilinear form a^κ(F,G) = ⟨F,G⟩_{γ/2} + κ⟨F,G⟩_{V₁}.
• Analyze the limit as κ → 0, conjecturing that it corresponds to the small-θ > 0 regime, which remains open.

Experimental results

Research questions

• RQ1 How does the hydrodynamic limit of a symmetric exclusion process with long-range jumps and reservoirs depend on the coupling strength parameter κ and the exponent θ in the reservoir rate κN^{-θ}?
• RQ2 What is the macroscopic PDE that describes the density profile in the hydrodynamic limit for θ ≤ 0, and how does it incorporate Dirichlet boundary conditions?
• RQ3 Can the singular reaction term in the limiting PDE be interpreted as a boundary condition rather than a reaction, and what does this imply for the underlying non-local operator?
• RQ4 What is the relationship between the κ-dependent regional fractional Laplacian L_κ and known fractional Laplacian variants such as the restricted and spectral fractional Laplacians?
• RQ5 What is the limiting behavior of the solution as κ → 0, and how does it relate to the open case θ > 0?

Key findings

• For θ = 0, the hydrodynamic limit yields a fractional reaction-diffusion equation with a singular reaction term that enforces Dirichlet boundary conditions at α and β, and the solution depends explicitly on the parameter κ.
• The limiting PDE is governed by a new class of regional fractional Laplacians L_κ = L - κV₁ on [0,1], where V₁(u) = c_γ γ^{-1}(u^{-γ} + (1−u)^{-γ}), and L is the generator of the symmetric stable process.
• For κ = 1, the operator L_κ recovers the restricted fractional Laplacian; in the limit κ → 0, it converges to the regional fractional Laplacian.
• The weak solution of the limiting PDE exists and is unique, established via the Lax-Milgram theorem applied to the coercive and continuous bilinear form a^κ(F,G) = ⟨F,G⟩_{γ/2} + κ⟨F,G⟩_{V₁}.
• The solution converges to the regional fractional Laplacian limit as κ → 0, suggesting that this limit captures the behavior expected for small positive θ in the open case θ > 0.
• The singular reaction term V₁ is interpreted as a boundary condition mechanism, providing a physical interpretation for non-local operators in bounded domains.