Symmetric Stable Levy Process Research Articles

Statistical inference on the drift parameter in symmetric stable Lévy process with a deterministic drift

ABSTRACTIn statistical inference on the drift parameter in the symmetric stable Levy process with the parameter of stability with a deterministic drift , there are some simple options how to do it. We may, for example, base this inference on the properties of the stable distribution. Although this method is very simple, it turns out that it is more appropriate to use inverse methods. For the hypotheses testing about the drift parameter , it is more proper to standardize the observed process and to use inverse methods based on the first exit time of the observed process of a prespecified interval until some given time. These procedures are illustrated, and their times of decision are compared with the direct approach. Other generalizations are possible when the random part is a stochastic integral of a known, deterministic function with respect to the symmetric stable Levy process or a symmetric stochastic integral of a random (but observable) process with respect to the symmetric stable Levy process.

Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.

Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction

We consider a random field ϕ: {1, ... , N} → ℝ with Laplacian interaction of the form Σ i V(Δϕ i ), where Δ is the discrete Laplacian and the potential V(·) is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward e ≥ 0 each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime (e e c ), where 0 < e c < oo. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance N 3/2 , while in the localized regime the distance is just O((log N) 2 ). A subtle scenario shows up in the critical regime (e = e c ), where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5. Our approach is based on Markov renewal theory.

Harnack?s Inequality for Stable L�vy Processes

We prove Harnack’s inequality for harmonic functions of a symmetric stable Levy process on Rd without the assumption that the density function of its Levy measure is locally bounded from below.

Probabilistic Proof of Boundary Harnack Principle for α-Harmonic Functions

The boundary Harnack principle for fractional Laplacian is now known to hold in Lipschitz domains [5]. It states that if two nonnegative functions, harmonic with respect to a symmetric stable Levy process vanish continuously outside a Lipschitz domain, near a part of its boundary, then the ratio of the functions is bounded inside the domain, near this part of the boundary. We give a probabilistic proof of the assertion using elementary properties of the stable process.