175 research outputs found
The dimension of the range of a transient random walk
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of
the range of an arbitrary transient walk in Z^d. This endeavor solves a problem
of Barlow and Taylor (1991).Comment: 37 pages, 5 figure
Strong invariance and noise-comparison principles for some parabolic stochastic PDEs
We consider a system of interacting diffusions on the integer lattice.
By letting the mesh size go to zero and by using a suitable scaling, we
show that the system converges (in a strong sense) to a solution of the
stochastic heat equation on the real line. As a consequence, we obtain
comparison inequalities for product moments of the stochastic heat equa-
tion with different nonlinearities
Dynamical percolation on general trees
H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph . When is a tree they derived a necessary and
sufficient condition for percolation to exist at some time . In the case
that is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time in a given target set . The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time , in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field
A macroscopic multifractal analysis of parabolic stochastic PDEs
It is generally argued that the solution to a stochastic PDE with
multiplicative noise---such as , where denotes
space-time white noise---routinely produces exceptionally-large peaks that are
"macroscopically multifractal." See, for example, Gibbon and Doering (2005),
Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved
that the spatial peaks of the solution to the mentioned stochastic PDE indeed
form a random multifractal in the macroscopic sense of Barlow and Taylor (1989;
1992). The main result of the present paper is a proof of a rigorous
formulation of the assertion that the spatio-temporal peaks of the solution
form infinitely-many different multifractals on infinitely-many different
scales, which we sometimes refer to as "stretch factors." A simpler, though
still complex, such structure is shown to also exist for the
constant-coefficient version of the said stochastic PDE.Comment: 41 page
Fractal-dimensional properties of subordinators
This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely limδ→0U(δ)N(t,δ)=t , where N(t,δ) is the minimal number of boxes of size at most δ needed to cover a subordinator’s range up to time t, and U(δ) is the subordinator’s renewal function. Our main result is a central limit theorem (CLT) for N(t,δ) , complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N(t,δ) , but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator’s jumps of size greater than δ . This new process can be manipulated with remarkable ease in comparison with N(t,δ) , and allows better understanding of the box-counting dimension of a subordinator’s range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process
Patterns in random walks and Brownian motion
We ask if it is possible to find some particular continuous paths of unit
length in linear Brownian motion. Beginning with a discrete version of the
problem, we derive the asymptotics of the expected waiting time for several
interesting patterns. These suggest corresponding results on the
existence/non-existence of continuous paths embedded in Brownian motion. With
further effort we are able to prove some of these existence and non-existence
results by various stochastic analysis arguments. A list of open problems is
presented.Comment: 31 pages, 4 figures. This paper is published at
http://link.springer.com/chapter/10.1007/978-3-319-18585-9_
Breadth first search coding of multitype forests with application to Lamperti representation
We obtain a bijection between some set of multidimensional sequences and this
of -type plane forests which is based on the breadth first search algorithm.
This coding sequence is related to the sequence of population sizes indexed by
the generations, through a Lamperti type transformation. The same
transformation in then obtained in continuous time for multitype branching
processes with discrete values. We show that any such process can be obtained
from a dimensional compound Poisson process time changed by some integral
functional. Our proof bears on the discretisation of branching forests with
edge lengths
Composition of processes and related partial differential equations
In this paper different types of compositions involving independent
fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial
differential equations governing the distributions of
I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and
J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods
and compared with those existing in the literature and with those related to
B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0
is examined in detail and its moments are calculated. Furthermore for
J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following
factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0.
A series of compositions involving Cauchy processes and fractional Brownian
motions are also studied and the corresponding non-homogeneous wave equations
are derived.Comment: 32 page
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