143 research outputs found
On Scaling Limits of Power Law Shot-noise Fields
This article studies the scaling limit of a class of shot-noise fields
defined on an independently marked stationary Poisson point process and with a
power law response function. Under appropriate conditions, it is shown that the
shot-noise field can be scaled suitably to have a -stable limit,
intensity of the underlying point process goes to infinity. It is also shown
that the finite dimensional distributions of the limiting random field have
i.i.d. stable random components. We hence propose to call this limte the
- stable white noise field. Analogous results are also obtained for the
extremal shot-noise field which converges to a Fr\'{e}chet white noise field.
Finally, these results are applied to the analysis of wireless networks.Comment: 17 pages, Typos are correcte
Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions
We consider Dirichlet exterior value problems related to a class of non-local
Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein
functions of the Laplacian. We prove elliptic and parabolic
Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain
existence and uniqueness of weak solutions. Next we prove a refined maximum
principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also,
we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier,
valid on compact subsets of the domain, and a full anti-maximum principle by
restricting to fractional Schr\"odinger operators. Furthermore, we show a
maximum principle for narrow domains, and a refined elliptic ABP-type estimate.
Finally, we obtain Liouville-type theorems for harmonic solutions and for a
class of semi-linear equations. Our approach is probabilistic, making use of
the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde
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