35 research outputs found
Self-intersection local time of order k for Gaussian processes in S′(Rd)
AbstractWe study existence and continuity of self-intersection local time of any order of Gaussian S′(Rd) processes. In particular, we give results for Wiener and Ornstein–Uhlenbeck processes. We study processes associated with several classes of covariances, which arise in examples mainly as fluctuation limits of particle systems
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
A long range dependence stable process and an infinite variance branching system
We prove a functional limit theorem for the rescaled occupation time
fluctuations of a -branching particle system [particles
moving in according to a symmetric -stable L\'{e}vy
process, branching law in the domain of attraction of a -stable law,
, uniform Poisson initial state] in the case of intermediate
dimensions, . The limit is a process of
the form , where is a constant, is the Lebesgue
measure on , and is a -stable
process which has long range dependence. For , there are two long
range dependence regimes, one for , which coincides with
the case of finite variance branching , and another one for
, where the long range dependence depends on the value
of . The long range dependence is characterized by a dependence exponent
which describes the asymptotic behavior of the codifference of
increments of on intervals far apart, and which is for the
first case (and for ) and for the
second one. The convergence proofs use techniques of -valued processes.Comment: Published at http://dx.doi.org/10.1214/009117906000000737 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Number variance for hierarchical random walks and related fluctuations
We study an infinite system of independent symmetric random walks on a
hierarchical group, in particular, the c-random walks. Such walks are used,
e.g., in population genetics. The number variance problem consists in
investigating if the variance of the number of "particles" N_n(L) lying in the
ball of radius L at a given time n remains bounded, or even better, converges
to a finite limit, as . We give a necessary and sufficient
condition and discuss its relationship to transience/recurrence property of the
walk. Next we consider normalized fluctuations of N_n(L) around the mean as
and L is increased in an appropriate way. We prove convergence of
finite dimensional distributions to a Gaussian process whose properties are
discussed. As the c-random walks mimic symmetric stable processes on R, we
compare our results to those obtained by Hambly and Jones (2007,2009), where
the number variance problem for an infinite system of symmetric stable
processes on R was studied. Since the hierarchical group is an ultrametric
space, corresponding results for symmetric stable processes and hierarchical
random walks may be analogous or quite different, as has been observed in other
contexts. An example of a difference in the present context is that for the
stable processes a fluctuation limit process is a centered Gaussian process
which is not Markovian and has long range dependent stationary increments, but
the counterpart for hierarchical random walks is Markovian, and in a special
case it has independent increments