A new global isomorphism theorem is obtained that expresses the local times of transient regular diffusions under P^{x,y}, in terms of related Gaussian processes. This theorem immediately gives an explicit description of the local times of diffusions in terms of 0--th order squared Bessel processes similar to that of N. Eisenbaum and Ray's classical description in terms of certain randomized fourth order squared Bessel processes. The proofs given are very simple. They depend on a new version of Kac's Lemma for h-transformed Markov processes and employ little more than standard linear algebra. The global isomorphism theorem leads to an elementary proof of the Markov property of the local times of diffusions and to other recent results about the local times of general strongly symmetric Markov processes. The new version of Kac's lemma gives simple, short proofs of Dynkin's isomorphism theorem and an unconditioned isomorphism theorem due to Eisenbaum.
Let T(x,r) denote the first hitting time of the disc of radius r centered at x for Brownian motion on the two dimensional torus. We prove that sup_{x} T(x,r)/|log r|^2 --> 2/pi as r --> 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z_n^2 is asymptotic to (2n log n)^2/pi. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied non-rigorously in the physics literature. We also establish a conjecture, due to Kesten and Revesz, that describes the asymptotics for the number of steps needed by simple random walk in Z^2 to cover the disc of radius n.
Let L_n^{X}(x) denote the number of visits to x \in \Z^2 of the simple
planar random walk X, up till step n. Let X' be another simple
planar random walk independent of X.
We show that for any 0
We introduce the concept of capacitary modulus for a set \Lambda\subseteq R^d,
which is a function h that provides simple estimates for the capacity of
\Lambda with respect an arbitrary kernel f, estimates which
depend only on the L^2 inner product
(h, f). We show that for a large class of L\'{e}vy processes, which include the
symmetric stable processes and stable subordinators, a capacitary modulus for the
range of the process is given by it's 1-potential density u^1(x), and
a capacitary modulus for the intersection of the ranges of m independent such
processes is given by the product of their 1-potential densities. The uniformity
of estimates provided by the capacitary modulus allows us to obtain almost-sure
asymptotics for the probability that one such process approaches within \epsilon of the
intersection of m other independent processes, conditional on these latter
processes. Our work generalizes that of Peres, Pemantle and Shapiro on the range of Brownian motion.
We give a simple proof of the necessary and sufficient conditions for the joint continuity of the local times of symmetric stable processes.
We show that the n-th order renormalized self-intersection local time
\gamma_{n}(\mu; t) for the symmetric stable process in R^2, where the n-fold
multiple points are weighted by an arbitrary measure \mu, can be characterized as
the continuous process of zero quadratic variation in the decomposition of a natural
Dirichlet process. This Dirichlet process is the potential of a random measure associated
with \gamma_{n-1}(\mu; t).
Let T(x,r)denote the occupation measure of the disc of radius
r centered at x by planar Brownian
motion run till time 1.
We prove that \sup_{|x| \leq 1}T(x,r)/(r^2|\log r|^2) \to 2 a.s.\
as r \rightarrow 0,
thus solving a problem posed by Perkins and Taylor (1987).
Furthermore, we show that for any a<2,
the Hausdorff dimension of the set of ``perfectly thick points''
x for which \lim_{r \rightarrow 0} T(x,r)/(r^2|\log r|^2)=a, is almost
surely 2-a; this is the correct scaling to obtain a nondegenerate
``multifractal spectrum'' for Brownian occupation measure
in the plane. The proofs rely on a `multiscale refinement'
of the second moment method..
As a consequence of our results on Brownian motion, we prove a conjecture
about simple random walk in Z^2due to Erd\H{o}s and
Taylor (1960):
The number of visits to the most frequently visited lattice site
in the first n steps of the walk, is asymptotic
to (\log n)^2/\pi. We also determine the corresponding
``discrete multifractal spectrum'': For 0
Let \cal{T}(x,r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in R^3.
We prove that
\sup_{|x| \leq 1}\cal{T}(x,r)/(r^2|\log r|) \rightarrow 16/\pi^2 a.s.
as r\rightarrow 0,
thus solving a problem posed by Taylor in 1974. Furthermore, for any a \in (0,16/\pi^2), the Hausdorff dimension of the set of ``thick points'' x for which \limsup_{r \rightarrow 0} \cal{T}(x,r)/(r^2|\log r|)=a, is almost
surely 2-a\pi^2/8; this is the correct scaling to obtain a nondegenerate ``multifractal spectrum'' for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension d>3.
These results are related to the LIL of Ciesielski and Taylor (1962) for the Brownian occupation measure of small balls, in the same
way that Levy's uniform modulus of continuity, and the formula of Orey and Taylor (1974) for the dimension of ``fast points'', are related to the usual LIL.
We also show that the \liminf scaling of \cal{T}(x,r) is quite different: we exhibit non-random c_1,c_2>0, such that
c_1 < \sup_x \liminf_{r \to 0} \cal{T}(x,r)/r^2 < c_2 \; \, a.s.
In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of `limsup type'.
Sufficient conditions are obtained for the continuity of
renormalized self-intersection
local times for the multiple intersections of a large class of strongly
symmetric L\'{e}vy processes
in R^m, m=1,2. In R^2 these
include Brownian motion and stable processes of index greater than 3/2, as well
as many processes in their domains of attraction. In R^1 these include stable
processes of index 3/4<\beta\le 1 and many processes in their domains of
attraction.
Let
(\Omega,\FF(t),X(t), P^{x}) be one of these radially symmetric L\'evy processes
with 1-potential density u^1(x,y). Let G^{2n}_F denote the class of positive
finite measures \mu on R^m for which
\[
\int \int (u^1(x,y))^{2n}\,d\mu(x)\,d\mu(y)<\infty. \] For \mu\in G^{2n}_F, let
\bea &&\alpha_{n,\epsilon}(\mu,\lambda)
\stackrel{def}{=}\nn
\int \int_{\{0\leq t_1\leq \cdots \leq t_n\leq \lambda\}}\\&&\qquad
f_{\epsilon}(X(t_1)-x)\prod_{j=2}^n f_{\epsilon}(X(t_j)- X(t_{j-1}))\,dt_1\cdots\,dt_n
\,d\mu(x)\nn
\eea where f_{\epsilon} is an approximate delta-function at zero and \lambda is an
random exponential time, with mean one, independent of X (with probability
measure P_\lambda). The renormalized self-intersection local time of X with respect
to the measure \mu is defined as
\[
\gamma_{n}(\mu)=\lim_{\epsilon\rightarrow
0}\,\sum_{k=0}^{n-1}(-1)^{k}\(\stackrel{n-1}{k}\)(u^1_{\epsilon}(0))^{k}
\alpha_{n-k,\epsilon}(\mu,\la)
\] where, u^1(x)\st u^1(x+z,z), for all z\in R^m, and
u^1_{\epsilon}(x)\st\int f_{\epsilon}(x-y)u^1(y)\,dy. Conditons are obtained under which
this limit exists in L^2(\Omega\times R^+,P^y_\la) for all y\in R^m, where
P^y_\la\st P^y\times P_\lambda.
Let \{\mu_x,x\in R^m\} denote the set of translates of the measure \mu. The
main result in this paper is a sufficient condition for the continuity of
\{\gamma_{n}(\mu_x),\,x\in R^m\}, namely that this process is continuous P^y_\lambda
almost surely for all y\in R^m, if the corresponding 2n-th Wick power chaos
process, \{:G^{2n}\mu_x:,\,x\in R^m\} is continuous almost surely. This chaos
process is obtained in the following way. A Gaussian process G_{x,\delta} is defined
which has covariance u^1_\delta(x,y), where \lim_{\delta\to
0}u_\delta^1(x,y)=u^1(x,y). Then \[ :G^{2n}\mu_x:\st\lim_{\delta\to
0}\int :G_{y,\de}^{2n}:\,d\mu_x(y) \] where the limit is taken in
L^2. (:G_{y,\delta}^{2n}: is the 2n-th Wick power of G_{y,\delta}, that is, a
normalized Hermite polynomial of degree 2n in G_{y,\delta}).
This process has a natural metric \begin{eqnarray}&&
d(x,y)\st\frac1{(2n)!}\(E(:G^{2n}\mu_x:-:G^{2n}\mu_y:)^2\)^{1/2}\nn\\
&&\quad=\(\int\!\! \int
\(u^1(u,v)\)^{2n} \left( d(\mu_x(u)-\mu_y(u)) \right) \left(
d(\mu_x(v)-\mu_y(v))
\right)\)^{1/2}\nn.
\end{eqnarray} A well known metric entropy condition with
respect to d gives a sufficient condition for the continuity of
\{:G^{2n}\mu_x:,\,x\in R^m\} and hence for
\{\gamma_{n}(\mu_x),\,x\in R^m\}.
Different extentions of an isomorphism theorem of Dynkin are developed and are used
to study two distinct but related families of functionals of L\'evy processes;
n-fold ``near-intersections'' of a single
L\'evy
process, which is also referred to as a self-intersection local time, and continuous additive functionals of several independent L\'evy
processes. Intersection local times for n independent L\'evy
processes are also studied. They are related to both of the above families. In all three cases
sufficient conditions are obtained for the almost sure continuity of these
functionals in terms of the almost sure continuity of associated Gaussian chaos
processes. Concrete sufficient conditions are given for the
almost sure continuity of these functionals of L\'evy processes.
Let u(x), x\in R^q be a symmetric non-negative definite function which
is bounded
away from zero but which may have u(0)=\infty. Let p_{x,\de}(\cdot) be the density of an R^q valued canonical normal random variable with mean x and variance \delta and let \{G_{x,\delta};(x,\delta)\in R^q\times [0,1]\}
be the mean zero Gaussian process with covariance
\[
EG_{x,\delta}G_{y,\delta'}=\int\!\!\int u(s-t)p_{x,\delta}(s)p_{y,\delta'}(t)\,ds\, dt.
\]
A finite positive measure \mu on R^q is said to be in \cal{G}^r, with respect to u, if
\[
\int \int (u(x,y))^{r}\,d\mu(x)\,d\mu(y)<\infty.
\]
When \mu\in\cal{G}^{|\bar m|}, a multiple Wick product chaos \cal{H}_{\bar m,1,0,\mu}(\bar{x}) is defined to be the limit in L^2, as \delta\to 0, of
\cal{H}_{\bar m,1,0,\mu,\delta}(\bar{x}):=
\int
\prod_{j=1}^{k}:\prod_{p=1}^{m_j} G_{y+x_{j,p},\delta}:\,d\mu(y),
where \bar
m=(m_1,\ldots,m_k)\in Z_+^k, and with |\bar m|\st\sum_{j=1}^k
m_j,
\[
\bar{x}=(x_{1,1},\dots,x_{1,m_1},\dots, x_{k,1},\ldots,x_{k,m_k})\in
(R^q)^{|\bar m|}.
\],
and :\prod_{p=1}^{m_j}
G_{y+x_{j,p},\delta,N}: denotes the Wick product of the
m_j normal random variables \{G_{y+x_{j,p},\delta}\}_{p=1}^{m_j}.
Consider also the associated decoupled chaos processes \newline \cal{H}_{r,1,0,\mu}^{dec}(x_1,\ldots,x_r), r\le |\bar m| defined as the limit in L^2, as \delta\to 0, of
\[
\cal{H}_{r,1,0,\mu,\delta}^{dec}(x_1,\ldots,x_r):=
\int
\prod_{j=1}^{r}G_{y+x_{j},\delta}^{(j)}\,d\mu(y)
\]
where \{G_{x,\delta}^{(j)}\} are independent copies of G_{x,\delta}.
Define
S_{\bar{m},\epsilon}:=
\{\bar{x}:|x_{j,p}-x_{j',q}|>\epsilon,
\forall\, 1\le p\le m_j,1\le q\le m_{j'}, 1\le j,j'\le k, j\ne j'\}.
Note that a neighborhood of the diagonals of \bar{x} in
(R^q)^{|\bar m|} is excluded, except those points on the diagonal which originate in
the same Wick
product. Set S_{\bar{m},\ne}:=\cup_{\ep>0}S_{\bar{m},\epsilon}.
One of the main results of this paper is:
If \cal{H}_{r,1,0,\mu}^{dec}(x_1,\dots,x_{r}) is continuous on
(R^q)^r for all
r\le |\bar m|, then
\cal{H}_{\bar m,1,0,\mu}(\bar{x})
is continuous on S_{\bar{m},\ne}.
When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of \cal{H}_{r,1,0,\mu}^{dec}(x_1,\dots,x_{r}) on
(R^q)^r. Also several variants are considered and related to different types of decoupled processes. These results have applications in the study of intersections of L\'evy process and continuous additive functionals of several L\'evy processes.
We derive a large deviation principle for the occupation time
functional, acting on functions with 0 Lebesgue integral, for both super Brownian
motion and
critical branching Brownian motion in three dimensions. Our technique, based on
a moment formula of Dynkin, allows us to compute the exact rate
functions, which differ for the
two processes. Obtaining the exact rate function for the super Brownian motion solves a
conjecture of Lee and Remillard . We also show the corresponding CLT and
obtain similar results for the superprocesses and critical branching process built over
the symmetric stable process of index \beta in
R^d, with d<2\beta<2+d.
We prove functional laws of the iterated logarithm for L^0_n, the number
of returns
to the origin, up to step n, of recurrent random walks on Z^2 with
slowly varying
partial Green's function. We find two distinct functional laws of the
iterated logarithm
depending on the scaling used. In the special case of finite variance
random walks, we
obtain one limit set for L^0_{n^x}/ (\log n \log_3 n)\,;\hspace{.1in}0\leq
x\leq1,
and a different limit set for L^0_{xn}/ (\log n \log_3
n)\,;\hspace{.1in}0\leq x\leq1.
In both cases the limit sets are classes of distribution functions, with
convergence in
the weak topology.
Let X=\{X_n,\,n\geq 1\}, X'=\{X'_n,\,n\geq 1\} be two independent copies of a
symmetric random walk in
Z^4 with finite third moment. In this paper we study the asymptotics of I_n, the number of
intersections up to step n of the paths of X and X' as n\rar \infty. Our main result is
\[
\limsup_{n\rar \ff}{I_{n}\over \log(n) \log_{3}(n)}={1\over
2\pi^2 |Q|^{1/2}}
\]
a.s. where Q denotes the covariance matrix of X_1. A similar result holds for J_n, the number
of points in the intersection of the ranges of X and X' up to step n.
Let X=\{X_n,\,n\geq 1\}, X'=\{X'_n,\,n\geq 1\} and X''=\{X''_n,\,n\geq 1\} be three
independent copies of a symmetric random walk in
Z^3 with E(|X_{1}|^{2}\log_+ |X_1|)<\infty. In this paper we study the asymptotics of I_, the
number of triple intersections up to step n of the paths of X, X' and X'' as n\rar \infty. Our
main result is
\[
\limsup_{n\rar \ff}{I_{n}\over \log(n) \log_{3}(n)}={1\over
\pi |Q|}
\]
a.s. where Q denotes the covariance matrix of X_1. A similar result holds for J_n, the number
of points in the triple intersection of the ranges of X, X' and X'' up to step n.
Capacitary moduli for Levy processes and intersections.
ps(1.2 M)
Abstract:
Gaussian processes and the local times of symmetric Levy processes.
(with M. Marcus)
ps(1.1 M)
Abstract:
Dirichlet processes and an intrinsic characterization of renormalized intersection local times,
ps(1.1 M)
Abstract:
Thick Points for Planar Brownian Motion and
the Erdos-Taylor Conjecture on Random Walk,
(with A. Dembo, Y. Peres and O. Zeitouni).
ps(1.7 M)
Abstract:
Thick points for spatial brownian motion: multifractal analysis of occupation measure,
(with A. Dembo, Y. Peres and O. Zeitouni).
ps(1.7 M)
ps.Z (924 K)
Abstract:
Renormalized self-intersection local times and Wick power chaos processes,
(with Michael B. Marcus).
dvi.Z (182 K)
ps.Z (924 K)
Abstract:
Additive functionals of several L\'evy processes and self-intersection
local times,
(with Michael B. Marcus).
ps (1.3 M)
ps.Z (710 K)
Abstract:
Multiple Wick Product Chaos Processes,
(with Michael B. Marcus).
ps (1.3 M)
ps.Z (710 K)
Abstract:
Occupation Time Large Deviations for Critical Branching Brownian Motion, Super
Brownian Motion and
Related Processes,
(with Jean-Dominique Deuschel).
Annals of Probability.
dvi file (165 K)
ps.Z (726 K)
Abstract:
Functional laws of the iterated logarithm for local times of
recurrent random walks on Z^2.
(with Endre Cs\'{a}ki and P\'{a}l R\'{e}v\'{e}sz )
Ann. Inst. Henri Poincare. dvi file (66 K),
ps.Z (545 K)
Abstract:
Laws of the Iterated Logarithm for Intersections of Random Walks on Z^4.
(with M. Marcus),
Ann. Inst. Henri Poincare , to appear. dvi file (89 K),
postscript file (1.1 M)
ps.Z (611 K)
Abstract:
Laws of the Iterated Logarithm for Triple Intersections of Random Walks on Z^3.
dvi file (50 K),
ps.Z (594 K)
Abstract: