Verallgemeinerungen eines Satzes von DOBRUSCHIN. I

Abstract

SCHIN (vgl. [ 2 ] , [I21 [I 51) mit wachsendem TL schwachgegen eine Mischung P stationarer PoIssoNscher Verteilungsgesetze P, , wenn L stationar sowie Iin Fall G = Rs strebt 11 1) In [I31 untersucht Sztlsz einen wesentlich allgemeineren Zufallsmechanisnius. Seine Resultate beriihren sich jedoch kaum mit den hier abgeleiteten Ergebnissen.

Spectral analysis of non-local Schrödinger operators

“…where the supremum is taken over k ∈ R d . Since a(0) = 1 (see (6)), we have that sup Re a(z) > 1/2 when |τ | is small enough. Then (56) is valid in all the cases.…”

“…There is a duality between these two terms. If H is rewritten in the form H = (L + 1) + (v(x) − 1), then after the Fourier transform, the first term becames an operator of multiplication by function a(k) decaying at infinity (see (6)) and the second term becomes a convolution operator. If v is smooth, then the spectrum of the second term is given by Lemma 2.1, and it coincides with the segment [−1, max v(x) − 1].…”

“…Function a(y) is the density of transition from x to x + y at the moment of the jump. Operators H of the form (1) appear in many applications, such as models of population dynamics that include the KPP type processes (where the offspring start at the location of the parent particle [9], [4], [5], [6]) and contact processes (where the locations of the new particles are randomly distributed around the location of the parent [10], [12]). Another area of application is the theory of phase transitions for homopolymers, [2], [3].…”

“…Relations (6) imply that the closure of the range of the function χ( a(k) − 1) is [−a, 0], where χ ≤ a ≤ 2χ. Remark.…”

An infinite particle system with additive interactions

“…is transient then there is a non-trivial stationary distribution which is translation invariant. Using results of Debes, Kerstan, Liemant and Matthes (1970) we can then identify all the translationinvariant stationary distributions with finite intensity, and give conditions on the initial distribution A which guarantee that starting from A TJt converges in distribution as t~oo. The condition which is sufficient for convergence is satisfied by all translation-invariant point processes with finite intensity.…”