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

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A Microscopic Model for a One Parameter Class of Fractional Laplacians with Dirichlet Boundary Conditions C. Bernardin, P. Gonçalves & B. Jiménez-Oviedo Communicated by A. Garroni

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 ∈ R. Our results are valid for θ ≤ 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 → 0 is the one that we should obtain when taking small values of θ > 0.

1. Introduction Normal (diffusive) transport phenomena are described by standard random walk models. Anomalous transport, in particular transport phenomena giving rise to superdiffusion, are nowadays encapsulated in the Lévy flights or Lévy walks framework [7,8] and appear in physics, finance and biology. The term “Lévy flight” was coined by Mandelbrot and is nothing but a random walk in which the steplengths have a probability distribution that is heavy tailed. A (one-dimensional) Lévy walker moves with a constant velocity v for a heavy-tailed random time τ on a distance x = vτ in either direction with equal probability and then chooses a new direction and moves again. One then easily shows that for Lévy flights or Lévy walks, the space-time scaling limit P(x, t) of the probability distribution of

C. Bernardin et al.

the particle position x(t) is solution of the fractional diffusion equation ∂t P = −c(−)γ /2 P,

(1.1)

where c is a constant and γ ∈ (1, 2). In physics, the description of anomalous transport phenomena by Lévy walks instead of Lévy flights is sometimes preferred despite the two models having the same scaling limit form provided by (1.1) because the first ones have a finite speed of propagation (see [7] for more details). While Lévy walks and Lévy flights are today well known and popular models to describe superdiffusion in infinite systems in various application fields, there have recently been several physical studies pointing out that it would be desirable to have a better understanding of Lévy walks in bounded domains. For bounded domains, boundary conditions and exchange with reservoirs or environment have to be taken into account. A particular interest for this problem is related to the description of anomalous diffusion of energy in low-dimensional lattices [9,19] in contact with reservoirs [10,11,18,20]. It is for

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