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2,500 | Coupling and Harnack inequalities for Sierpinski carpets | math.PR | Uniform Harnack inequalities for harmonic functions on the pre- and graphical
Sierpinski carpets are proved using a probabilistic coupling argument. Various
results follow from this, including the construction of Brownian motion on
Sierpinski carpets embedded in $\R^d$, $d\geq 3$, estimates on the fundamental
solution ... | math |
2,501 | A general decomposition theory for random cascades | math.PR | This announcement describes a probabilistic approach to cascades which, in
addition to providing an entirely probabilistic proof of the Kahane-Peyri\`ere
theorem for independent cascades, readily applies to general dependent
cascades. Moreover, this unifies various seemingly disparate cascade
decompositions, including ... | math |
2,502 | The dimension of the Brownian frontier is greater than 1 | math.PR | Consider a planar Brownian motion run for finite time. The frontier or
``outer boundary'' of the path is the boundary of the unbounded component of
the complement. Burdzy (1989) showed that the frontier has infinite length. We
improve this by showing that the Hausdorff dimension of the frontier is
strictly greater than... | math |
2,503 | No directed fractal percolation in zero area | math.PR | We show that fractal (or "Mandelbrot") percolation in two dimensions produces
a set containing no directed paths, when the set produced has zero area. This
improves a similar result by the first author in the case of constant retention
probabilities to the case of retention probabilities approaching 1. | math |
2,504 | Markov chains in a field of traps | math.PR | A general criterion is given for when a Markov chain trapped with probability
p(x) in state x will be almost surely trapped. The quenched (state x is a trap
forever with probability p(x)) and annealed (state x traps with probability
p(x) on each visit) problems are shown to be equivalent. | math |
2,505 | Vertex-reinfoced random walk on Z visits finitely many states | math.PR | Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a
random walk that is biased to visit sites it has already visited a lot. We show
that this reinforcement scheme, in contrast to the scheme of
edge-reinforcement, causes random walk on a line to get trapped in a finite
set. | math |
2,506 | Sets avoided by Brownian motion | math.PR | Any fixed cylinder is hit almost surely by a 3-dimensional Brownian motion,
but is there a random cylinder that is in the complement? We answer this for
cylinders, and then replacing a cylinder with a more general set. | math |
2,507 | First passage percolation and a model for competing spatial growth | math.PR | We generalize Richardson's model by starting with two sites of different
colors and giving each new site the color of the site that spawned it. We show
that co-existence is possible. | math |
2,508 | Paths with exponential intersection tails and oriented percolation | math.PR | We show that oriented percolation occurs whenever a condition is satisfied
called "exponential intersection tails". This condition says that a measure on
paths exists for which the probability of two independent paths intersecting in
more than k sites is exponentially small in k. | math |
2,509 | The probability that Brownian motion almost covers a line | math.PR | Lower and upper estimates are given for the probability that the
epsilon-enlargement of planar Brownian motion to time 1 (the epsilon sausage)
contains a unit line segment. The estimates imply that Brownian motion to time
1 itself contains no line segment. | math |
2,510 | On minimal parabolic functions and time-homogeneous parabolic h-transforms | math.PR | Does a minimal harmonic function $h$ remain minimal when it is viewed as a
parabolic function? The question is answered for a class of long thin
semi-infinite tubes $D\subset \R^d$ of variable width and minimal harmonic
functions $h$ corresponding to the boundary point of $D$ ``at infinity.''
Suppose $f(u)$ is the widt... | math |
2,511 | Completely regular multivariate stationary process and the Muckenhoupt condition | math.PR | We give necessary and sufficient conditions for a multivariate stationary
stochastic process to be completely regular. We also give the answer to a
question of V.V. Peller concerning the spectral measure characterization of
such processes. | math |
2,512 | A support property for infinite dimensional interacting diffusion processes | math.PR | The Dirichlet form associated with the intrinsic gradient on Poisson space is
known to be quasi-regular on the complete metric space $\ddot\Gamma=$
$\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions,
the set $\ddot\Gamma\setminus\Gamma$ is $\e$-exceptional, where $\Gamma$ is the
space of loc... | math |
2,513 | Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization | math.PR | Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on
rigged Hilbert spaces are studied. An analytic approach based on a--priori
estimates is used. The extension of the problem to the $L^p$-setting is
discussed. As a direct application essential self--adjointness and strong
uniqueness in $L^p$ is pr... | math |
2,514 | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent | math.PR | Sufficient conditions are given for existence and uniqueness in
Smoluchowski's coagulation equation, for a wide class of coagulation kernels
and initial mass distributions. An example of non-uniqueness is constructed.
The stochastic coalescent is shown to converge weakly to the solution of
Smoluchowski's equation. | math |
2,515 | Stochastic bifurcation models | math.PR | We study an ordinary differential equation controlled by a stochastic
process. We present results on existence and uniqueness of solutions, on
associated local times (Trotter and Ray-Knight theorems), and on time and
direction of bifurcation. A relationship with Lipschitz approximations to
Brownian paths is also discus... | math |
2,516 | Limit Theorems for Sums of p-Adic Random Variables | math.PR | We study p-adic counterparts of stable distributions, that is limit
distributions for sequences of normalized sums of independent identically
distributed p-adic-valued random variables. In contrast to the classical case,
non-degenerate limit distributions can be obtained only under certain
assumptions on the asymptotic... | math |
2,517 | Existence and regularity for a class of infinite-measure $(ξ,ψ,K)$-superprocesses | math.PR | We extend the class of $(\xi,\psi,K)$-superprocesses known so far by applying
a simple transformation induced by a \lq\lq weight function\rq\rq\ for the
one-particle motion. These transformed superprocesses may exist under weak
conditions on the branching parameters, and their state space automatically
extends to a cer... | math |
2,518 | Rademacher's theorem on configuration spaces and applications | math.PR | We consider an $L^2$-Wasserstein type distance $\rho$ on the configuration
space $\Gamma_X$ over a Riemannian manifold $X$, and we prove that
$\rho$-Lipschitz functions are contained in a Dirichlet space associated with a
measure on $\Gamma_X$ satisfying some general assumptions. These assumptions
are in particular ful... | math |
2,519 | A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions | math.PR | The growth exponent $\alpha$ for loop-erased or Laplacian random walk on the
integer lattice is defined by saying that the expected time to reach the sphere
of radius $n$ is of order $n^\alpha$. We prove that in two dimensions, the
growth exponent is strictly greater than one. The proof uses a known estimate
on the thi... | math |
2,520 | On increasing subsequences of iid samples | math.PR | We study the fluctuations, in the large deviations regime, of the longest
increasing subsequence of a random i.i.d. sample on the unit square. In
particular, our results yield the precise upper and lower exponential tails for
the length of the longest increasing subsequence of a random permutation. | math |
2,521 | Markov Processes with Identical Bridges | math.PR | Let X and Y be time-homogeneous Markov processes with common state space E,
and assume that the transition kernels of X and Y admit densities with respect
to suitable reference measures. We show that if there is a time t>0 such that,
for each x\in E, the conditional distribution of (X_s)_{0 < s < t}, given X_0 =
x = X_... | math |
2,522 | A Simple Path to Biggins' Martingale Convergence for Branching Random Walk | math.PR | We give a simple non-analytic proof of Biggins' theorem on martingale
convergence for branching random walks. | math |
2,523 | The Stable Manifold Theorem for Stochastic Differential Equations | math.PR | We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic
differential equations (sde's) that are driven by spatial Kunita-type
semimartingales with stationary ergodic increments. Both Stratonovich and
It\^o-type equations are treated. Starting with the existence of a stochastic
flow for a sde, we i... | math |
2,524 | Stochastic analysis on configuration spaces: basic ideas and recent results | math.PR | The purpose of this paper is to provide a both comprehensive and summarizing
account on recent results about analysis and geometry on configuration spaces
$\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to a
complete description of the so--called ``lifting--procedure'', Markov resp.
strong resp. ... | math |
2,525 | Measures on contour, polymer or animal models. A probabilistic approach | math.PR | We present a new approach to study measures on ensembles of contours,
polymers or other objects interacting by some sort of exclusion condition. For
concreteness we develop it here for the case of Peierls contours. Unlike
existing methods, which are based on cluster-expansion formalisms and/or
complex analysis, our met... | math |
2,526 | Cheeger's inequalities for general symmetric forms and existence criteria for spectral gap | math.PR | In this paper, some new forms of the Cheeger's inequalities are established
for general (maybe unbounded) symmetric forms, the resulting estimates improve
and extend the ones obtained by Lawler and Sokal (1988) for bounded jump
processes. Furthermore, some existence criteria for spectral gap of general
symmetric forms ... | math |
2,527 | On the conditioned exit measures of super-Brownian motion | math.PR | In this paper we present a martingale related to the exit measures of
super-Brownian motion. By changing measure with this martingale in the
canonical way we have a new process associated with the conditioned exit
measure. This measure is shown to be identical to a measure generated by a
non-homogeneous branching parti... | math |
2,528 | Concrete representation of martingales | math.PR | Let (f_n) be a mean zero vector valued martingale sequence. Then there exist
vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n)
dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the
same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . Similar results for tangent
sequen... | math |
2,529 | Unitary Brownian motions are linearizable | math.PR | Brownian motions in the infinite-dimensional group of all unitary operators
are studied under strong continuity assumption rather than norm continuity.
Every such motion can be described in terms of a countable collection of
independent one-dimensional Brownian motions. The proof involves continuous
tensor products and... | math |
2,530 | Some properties of the range of super-Brownian motion | math.PR | We consider a super-Brownian motion $X$. Its canonical measures can be
studied through the path-valued process called the Brownian snake. We obtain
the limiting behavior of the volume of the $\epsilon$-neighborhood for the
range of the Brownian snake, and as a consequence we derive the analogous
result for the range of... | math |
2,531 | Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations | math.PR | \def\R{\mathbb R}
We give a characterization of G-regularity for super-Brownian motion and the
Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times
\R^d$, which is not invariant by translation. We then prove that the hitting
probability of a Borel set $A\subset E$ for the graph of the Brownian ... | math |
2,532 | Non-degenerate conditionings of the exit measures of super-Brownian motion | math.PR | We introduce several martingale changes of measure of the law of the exit
measure of super Brownian motion. These changes of measure include and
generalize one arising by conditioning the exit measures to charge a point on
the boun dary of a 2-dimensional domain. In the case we discuss this is a
non-degenerate conditio... | math |
2,533 | The Repeated Solicitation Model | math.PR | This paper presents a probabilistic analysis of what we call the "repeated
solicitation model". To give a specific context, suppose B is a direct
marketing company with a list of S sales prospects. At epoch 1, B sends a
solicitation to every prospect on the list, and elicits X(1) replies. The
company deletes the respon... | math |
2,534 | Finite time extinction of super-Brownian motions with catalysts | math.PR | Consider a catalytic super-Brownian motion $X=X^\Gamma$ with finite variance
branching. Here `catalytic' means that branching of the reactant $X$ is only
possible in the presence of some catalyst. Our intrinsic example of a catalyst
is a stable random measure $\Gamma $ on $R$ of index $0< gamma <1$.
Consequently, here ... | math |
2,535 | Fractional Brownian motion and the Markov Property | math.PR | Fractional Brownian motion belongs to a class of long memory Gaussian
processes that can be represented as linear functionals of an infinite
dimensional Markov process. This representation leads naturally to: - An
efficient algorithm to approximate the process. - An infinite dimensional
ergodic theorem which applies to... | math |
2,536 | Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes | math.PR | Martin boundaries and integral representations of positive functions which
are harmonic in a bounded domain $D$ with respect to Brownian motion are well
understood. Unlike the Brownian case, there are two different kinds of
harmonicity with respect to a discontinuous symmetric stable process. One kind
are functions har... | math |
2,537 | Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains | math.PR | For a symmetric $\alpha$-stable process $X$ on $\RR^n$ with $0<\alpha <2$,
$n\geq 2$ and a domain $D \subset \RR^n$, let $L^D$ be the infinitesimal
generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class
function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a
H\"older domain $D$ o... | math |
2,538 | Free probability for probabilists | math.PR | This is an introduction to some of the most probabilistic aspects of free
probability theory. | math |
2,539 | Some function spaces related to the Brownian motion on simple nested fractals | math.PR | In this paper we identify the domain of the Dirichlet form associated with
the Brownian motion on simple nested fractals with an integral Lipschitz space.
This result generalizes such an identification on the Sierpi\'nski gasket,
carried on by Jonsson. | math |
2,540 | On the thermodynamic limit for a one-dimensional sandpile process | math.PR | Considering the standard abelian sandpile model in one dimension, we
construct an infinite volume Markov process corresponding to its thermodynamic
(infinite volume) limit. The main difficulty we overcome is the strong
non-locality of the dynamics. However, using similar ideas as in recent
extensions of the standard Gi... | math |
2,541 | The restriction of the Ising model to a layer | math.PR | We discuss the status of recent Gibbsian descriptions of the restriction
(projection) of the Ising phases to a layer. We concentrate on the projection
of the two-dimensional low temperature Ising phases for which we prove a
variational principle. | math |
2,542 | A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums of independent identically distributed random variables | math.PR | Let X_1,X_2,... be a sequence of independent and identically distributed
random variables, and put S_n=X_1+...+X_n. Under some conditions on the
positive sequence tau_n and the positive increasing sequence a_n, we give
necessary and sufficient conditions for the convergence of sum_{n=1}^infty
tau_n P(|S_n|>t a_n) for a... | math |
2,543 | Continuum-sites stepping-stone models, coalescing exhcangeable partitions, and random trees | math.PR | Analogues of stepping--stone models are considered where the site--space is
continuous, the migration process is a general Markov process, and the
type--space is infinite. Such processes were defined in previous work of the
second author by specifying a Feller transition semigroup in terms of
expectations of suitable f... | math |
2,544 | A comparison inequality for sums of independent random variables | math.PR | We give a comparison inequality that allows one to estimate the tail
probabilities of sums of independent Banach space valued random variables in
terms of those of independent identically distributed random variables. More
precisely, let X_1,...,X_n be independent Banach-valued random variables. Let I
be a random varia... | math |
2,545 | On random sections of the cube | math.PR | Let $f(j,k,n)$ denote the expected number of $j$-faces of a random
$k$-section of the $n$-cube. A formula for $f(0,k,n)$ is presented, and for
$j\geq 1$, a lower bound for $f(j,k,n)$ is derived, which implies a precise
asymptotic formula for $f(n-m,n-l,n)$ when $1\leq l<m$ are fixed integers and
$n\to\8$. | math |
2,546 | An embedding for the Kesten-Spitzer random walk in random scenery | math.PR | For one-dimensional simple random walk in a general i.i.d. scenery and its
limiting process we construct a coupling with explicit rate of approximation
extending a recent result for Gaussian sceneries due to Khoshnevisan and Lewis.
Furthermore we explicity identify the constant in the law of iterated
logarithm. | math |
2,547 | Necessary and Sufficient Conditions for the Strong Law of Large Numbers for U-statistics | math.PR | Under some mild regularity on the normalizing sequence, we obtain necessary
and sufficient conditions for the Strong Law of Large Numbers for (symmetrized)
U-statistics. We also obtain nasc's for the a.s. convergence of series of an
analogous form. | math |
2,548 | Trees, not cubes: hypercontractivity, cosiness, and noise stability | math.PR | Noise sensitivity of functions on the leaves of a binary tree is studied, and
a hypercontractive inequality is obtained. We deduce that the spider walk is
not noise stable. | math |
2,549 | A pattern theorem for lattice clusters | math.PR | We consider general classes of lattice clusters, including various kinds of
animals and trees on different lattices. We prove that if a given local
configuration ("pattern") of sites and bonds can occur in large clusters, then
it occurs at least cN times in most clusters of size n, for some constant c>0.
An analogous t... | math |
2,550 | Fourier-Walsh coefficients for a coalescing flow (discrete time) | math.PR | A two-dimensional array of independent random signs produces coalescing
random walks. The position of the walk, starting at the origin, after N steps
is a highly nonlinear, noise sensitive function of the signs. A typical term of
its Fourier-Walsh expansion involves the product of about square roof of N
signs. | math |
2,551 | Extinction for two parabolic stochastic PDE's on the lattice | math.PR | It is well known that, starting with finite mass, the super-Brownian motion
dies out in finite time. The goal of this article is to show that with some
additional work, one can prove finite time die-out for two types of systems of
stochastic differential equations on the lattice Z^d. Our first system involves
the heat ... | math |
2,552 | Scaling limit of Fourier-Walsh coefficients (a framework) | math.PR | Independent random signs can govern various discrete models that converge to
non-isomorphic continuous limits. Convergence of Fourier-Walsh spectra is
established under appropriate conditions. | math |
2,553 | The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations | math.PR | Theorem 1 is a formula expressing the mean number of real roots of a random
multihomogeneous system of polynomial equations as a multiple of the mean
absolute value of the determinant of a random matrix. Theorem 2 derives closed
form expressions for the mean in special cases that include earlier results of
Shub and Sma... | math |
2,554 | Sample path large deviations for a class of Markov chains related to disordered mean field models | math.PR | We prove a large deviation principle on path space for a class of discrete
time Markov processes whose state space is the intersection of a regular domain
$\L\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$
are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The
transitio... | math |
2,555 | The LIL for canonical U-statistics of order 2 | math.PR | Let X,X_1,X_2,... be independent identically distributed random variables and
let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that
the bounded law of the iterated logarithm, $\limsup_n (n\log\log
n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the
following three condi... | math |
2,556 | Stationary Measures for Random Walks in a Random Environment with Random Scenery | math.PR | Let $\Gamma$ act on a countable set V with only finitely many orbits. Given a
$\Gamma$-invariant random environment for a Markov chain on V and a random
scenery, we exhibit, under certain conditions, an equivalent stationary measure
for the environment and scenery from the viewpoint of the random walker. Such
theorems ... | math |
2,557 | Lattice trees, percolation and super-Brownian motion | math.PR | This paper surveys the results of recent collaborations with Eric Derbez and
with Takashi Hara, which show that intergrated super-Brownian excursion (ISE)
arises as the scaling limit of both lattice trees and the incipient infinite
percolation cluster, in high dimensions. A potential extension to oriented
percolation i... | math |
2,558 | Monotonicity property for a class of semilinear partial differential equations | math.PR | We establish a monotonicity property in the space variable for the solutions
of an initial boundary value problem concerned with the parabolic partial
differential equation connected with super-Brownian motion. | math |
2,559 | A variational coupling for a totally asymmetric exclusion process with long jumps but no passing | math.PR | We prove a weak law of large numbers for a tagged particle in a totally
asymmetric exclusion process on the one-dimensional lattice. The particles are
allowed to take long jumps but not pass each other. The object of the paper is
to illustrate a special technique for proving such theorems. The method uses a
coupling th... | math |
2,560 | Noise sensitivity on continuous products: an answer to an old question of J. Feldman | math.PR | A relation between sigma-additivity and linearizability, conjectured by Jacob
Feldman in 1971 for continuous products of probability spaces, is established
by relating both notions to a recent idea of noise stability/sensitivity. | math |
2,561 | Lévy Processes on $U_q(g)$ as Infinitely Divisible Representations | math.PR | L\'evy processes on bialgebras are families of infinitely divisible
representations. We classify the generators of L\'evy processes on the compact
forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then
we show how the processes themselves can be reconstructed from their generators
and study sev... | math |
2,562 | Vertex-reinforced random walk on arbitrary graphs | math.PR | Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988a), is a
random process in a continuously changing environment which is more likely to
visit states it has visited before. We consider VRRW on arbitrary graphs and
show that on almost all of them, VRRW visits only finitely many vertices with a
positive prob... | math |
2,563 | The Propagation of Molecular Chaos by Markov Transitions | math.PR | We establish a necessary and sufficient condition for the propagation of
chaos by a family of many-particle Markov processes, if the particles live in a
Polish space: a sequence of n-particle Markov transition functions propagates
chaos if and only if it propagates chaos for pure initial states. | math |
2,564 | Singularity of Some Random Continued Fractions | math.PR | We prove singularity of some distributions of random continued fractions that
correspond to iterated function systems with overlap and a parabolic point.
These arose while studying the conductance of Galton-Watson trees. | math |
2,565 | Loss of tension in an infinite membrane with holes distributed by Poisson law | math.PR | If one randomly punches holes in an infinite tensed membrane, when does the
tension cease to exist? This problem was introduced by R. Connelly in
connection with applications of rigidity theory to natural sciences. We outline
a mathematical theory of tension based on graph rigidity theory and introduce
several probabil... | math |
2,566 | Quasi-invariance and reversibility in the Fleming-Viot process | math.PR | Reversible measures of the Fleming-Viot process are shown to be characterized
as quasi-invariant measures with a cocycle given in terms of the mutation
operator. As applications, we give certain integral characterization of
Poisson-Dirichlet distributions and a proof that the stationary measure of the
step-wise mutatio... | math |
2,567 | Phase transition and percolation in Gibbsian particle models | math.PR | We discuss the interrelation between phase transitions in interacting lattice
or continuum models, and the existence of infinite clusters in suitable
random-graph models. In particular, we describe a random-geometric approach to
the phase transition in the continuum Ising model of two species of particles
with soft or ... | math |
2,568 | How to Couple from the Past Using a Read-Once Source of Randomness | math.PR | We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PA... | math |
2,569 | Splitting: Tanaka's SDE revisited | math.PR | The weak solution of Tanaka's SDE is not a function of the driving Brownian
motion, and therefore it has no Wiener chaos expansion. However in some sense
explained here it has a generalised chaos expansion involving infinite products
of stochastic differentials accumulating at the minimum of the Brownian path.
This is ... | math |
2,570 | Coalescence of skew Brownian motions | math.PR | We prove that two skew Brownian motions with the same skewness parameter
(different from 0) and driven by the same Brownian motion coalesce a.s. | math |
2,571 | Layered Multishift Coupling for use in Perfect Sampling Algorithms (with a primer on CFTP) | math.PR | In this article we describe a new coupling technique which is useful in a
variety of perfect sampling algorithms. A multishift coupler generates a random
function f() so that for each real x, f(x)-x is governed by the same fixed
probability distribution, such as a normal distribution. We develop the class
of layered mu... | math |
2,572 | Diffeomorphic flows driven by Levy processes | math.PR | We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x)
+ \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u},
Y_{s,s}(x)=x\in\R^d. $$ driven by a L\'evy process whose paths have finite
p-variation almost surely for some $p\in[1,2)$ defines a flow of locally
C^1-diffeomorphisms provided the vector field f is ... | math |
2,573 | Path-wise solutions of SDE's driven by Levy processes | math.PR | In this paper we show that a path-wise solution to the following integral
equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under
the assumption that X_t is a L\'evy process of finite p-variation for some $p
\geq1$ and that f is an $\alpha$-Lipschitz function for some alpha>p. There are
two types o... | math |
2,574 | Random Walks and Electric Networks | math.PR | A popular account of the connection between random walks and electric
networks. | math |
2,575 | Markov Transitions and the Propagation of Chaos | math.PR | The propagation of chaos is a central concept of kinetic theory that serves
to relate the equations of Boltzmann and Vlasov to the dynamics of
many-particle systems. Propagation of chaos means that molecular chaos, i.e.,
the stochastic independence of two random particles in a many-particle system,
persists in time, as... | math |
2,576 | q-probability: I. Basic discrete distributions | math.PR | For basic discrete probability distributions, $-$ Bernoulli, Pascal, Poisson,
hypergeometric, contagious, and uniform, $-$ $q$-analogs are proposed. | math |
2,577 | The supremum of Brownian local times on Holder curves | math.PR | For $f: [0,1]\to \R$, we consider $L^f_t$, the local time of space-time
Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions
whose H\"older norm of order $\al$ is less than or equal to 1. We show that the
supremum of $L^f_1$ over $f$ in $\sS_\al$ is finite is $\al>\frac12$ and
infinite if $\al<... | math |
2,578 | On the cover time of planar graphs | math.PR | The cover time of a finite connected graph is the expected number of steps
needed for a simple random walk on the graph to visit all the vertices. It is
known that the cover time on any n-vertex, connected graph is at least (1+o(1))
n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for
bounded-degree plana... | math |
2,579 | Some measure-preserving point transformations on the Wiener space and their ergodicity | math.PR | Suppose that T is a map of the Wiener space into itself, of the following
type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also
that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work
we prove that if T preserves the Wiener measure, then necessarily u is in the
first Wie... | math |
2,580 | Precise Propagation of Upper and Lower Probability Bounds in System P | math.PR | In this paper we consider the inference rules of System P in the framework of
coherent imprecise probabilistic assessments. Exploiting our algorithms, we
propagate the lower and upper probability bounds associated with the
conditional assertions of a given knowledge base, automatically obtaining the
precise probability... | math |
2,581 | Super-Brownian motion with reflecting historical paths | math.PR | We consider super-Brownian motion whose historical paths reflect from each
other, unlike those of the usual historical super-Brownian motion. We prove
tightness for the family of distributions corresponding to a sequence of
discrete approximations but we leave the problem of uniqueness of the limit
open. We prove a few... | math |
2,582 | Exponential and moment inequalities for U-statistics | math.PR | A Bernstein-type exponential inequality for (generalized) canonical
U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-J{\o}rgensen
inequalities for sums of independent random variables are extended to
(generalized) U-statistics of any order whose kernels are either nonnegative or
canonical | math |
2,583 | Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes | math.PR | A derivation operator and a divergence operator are defined on the algebra of
bounded operators on the symmetric Fock space over the complexification of a
real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar
properties as the derivation and divergence operator on the Wiener space over
$\eufrak{h}$.... | math |
2,584 | No more than three favourite sites for simple random walk | math.PR | We prove that, with probability one, eventually there are no more than three
favourite (i.e. most visited) sites of simple random walk. This partially
answers a relatively long standing question of Pal Erdos and Pal Revesz. | math |
2,585 | The identification capacity and resolvability of channels with input cost constraint | math.PR | Given a general channel, we first formulate the idetification capacity
problem as well as the resolvability problem with input cost constraint in as
the general form as possible, along with relevant fundamental theorems. Next,
we establish some mild sufficient condition for the key lemma linking the
identification capa... | math |
2,586 | On the critical exponents of random k-SAT | math.PR | There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from sat... | math |
2,587 | Occupation Time Fluctuations in Branching Systems | math.PR | We consider particle systems in locally compact Abelian groups with particles
moving according to a process with symmetric stationary independent increments
and undergoing one and two levels of critical branching. We obtain long time
fluctuation limits for the occupation time process of the one-and two-level
systems. W... | math |
2,588 | Mixing times for Markov chains on wreath products and related homogeneous spaces | math.PR | We develop a method for analyzing the mixing times for a quite general class
of Markov chains on the complete monomial group G \wr S_n (the wreath product
of a group G with the permutation group S_n) and a quite general class of
Markov chains on the homogeneous space (G \wr S_n) / (S_r \times S_{n - r}).
We derive an... | math |
2,589 | Random polynomials having few or no real zeros | math.PR | Consider a polynomial of large degree n whose coefficients are independent,
identically distributed, nondegenerate random variables having zero mean and
finite moments of all orders. We show that such a polynomial has exactly k real
zeros with probability n^{-b+o(1)}$ as n --> infinity through integers of the
same pari... | math |
2,590 | Random walks on wreath products of groups | math.PR | We bound the rate of convergence to uniformity for certain random walks on
the complete monomial groups G \wr S_n for any group G. These results provide
rates of convergence for random walks on a number of groups of interest: the
hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n,
and S_m \w... | math |
2,591 | A signed generalization of the Bernoulli-Laplace diffusion model | math.PR | We bound the rate of convergence to stationarity for a signed generalization
of the Bernoulli-Laplace diffusion model; this signed generalization is a
Markov chain on the homogeneous space (Z_2 \wr S_n) / (S_r \times S_{n-r}).
Specifically, for r not too far from n/2, we determine that, to first order in
n, 1/4 n \log ... | math |
2,592 | Favourite sites of simple random walk | math.PR | We survey the current status of the list of questions related to the
favourite (or: most visited) sites of simple random walk on Z, raised by Pal
Erdos and Pal Revesz in the early eighties. | math |
2,593 | Stationary Markov chains with linear regressions | math.PR | In a previous paper we determined one dimensional distributions of a
stationary field with linear regressions and quadratic conditional variances
under a linear constraint on the coefficients of the quadratic expression. In
this paper we show that for stationary Markov chains with linear regressions
and quadratic condi... | math |
2,594 | Critical exponents, conformal invariance and planar Brownian motion | math.PR | In this review paper, we first discuss some open problems related to
two-dimensional self-avoiding paths and critical percolation. We then review
some closely related results (joint work with Greg Lawler and Oded Schramm) on
critical exponents for two-dimensional simple random walks, Brownian motions
and other conforma... | math |
2,595 | Gaussian Random Matrix Models for q-deformed Gaussian Random Variables | math.PR | We construct a family of random matrix models for the q-deformed Gaussian
random variables G_\mu=a_\mu+a^\star_\mu where the annihilation operators a_\mu
and creation operators a^\star_\nu fulfil the q-deformed commutation relation
a_\mu a^\star_\nu-q a^\star_\nu a_\mu=\Gamma_{\mu\nu}, \Gamma_{\mu\nu} is the
covariance... | math |
2,596 | Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K-exclusion processes | math.PR | We study aspects of the hydrodynamics of one-dimensional totally asymmetric
K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove
that the weak solution chosen by the particle system is the unique one with
maximal current past any fixed location. A uniqueness result is needed because
we can pr... | math |
2,597 | The Randomness Recycler: A new technique for perfect sampling | math.PR | For many probability distributions of interest, it is quite difficult to
obtain samples efficiently. Often, Markov chains are employed to obtain
approximately random samples from these distributions. The primary drawback to
traditional Markov chain methods is that the mixing time of the chain is
usually unknown, which ... | math |
2,598 | A signal-recovery system: asymptotic properties and construction of an infinite volume limit | math.PR | We consider a linear sequence of `nodes', each of which can be in state 0
(`off') or 1 (`on'). Signals from outside are sent to the rightmost node and
travel instantaneously as far as possible to the left along nodes which are
`on'. These nodes are immediately switched off, and become on again after a
recovery time. Th... | math |
2,599 | Microscopic shape of shocks in a domain growth model | math.PR | Considering the hydrodynamical limit of some interacting particle systems
leads to hyperbolic differential equation for the conserved quantities, e.g.
the inviscid Burgers equation for the simple exclusion process. The physical
solutions of these partial differential equations develop discontinuities,
called shocks. Th... | math |
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