1,342 research outputs found
Schoenberg's Theorem via the law of large numbers
A classical theorem of S. Bochner states that a function
is the Fourier transform of a finite Borel measure if and only
if is positive definite. In 1938, I. Schoenberg found a beautiful
complement to Bochner's theorem.
We present a non-technical derivation of of Schoenberg's theorem that relies
chiefly on the de Finneti theorem and the law of large numbers of classical
probability theory.Comment: Some errors and misprints corrected; new references have been adde
Slices of Brownian Sheet: New Results, and Open Problems
We can view Brownian sheet as a sequence of interacting Brownian motions or
slices. Here we present a number of results about the slices of the sheet. A
common feature of our results is that they exhibit phase transition. In
addition, a number of open problems are presented
Brownian Sheet Images and Bessel-Riesz Capacity
We show that the image of a 2-dimensional set under d-dimensional,
2-parameter Brownian sheet can have positive Lebesgue measure, if and only if
the set in question has positive (d/2)-dimensional Bessel-Riesz capacity. Our
methods solve a problem of J.-P. Kahane.Comment: 19 page
Talagrand Concentration Inequalities for Stochastic Partial Differential Equations
One way to define the concentration of measure phenomenon is via Talagrand
inequalities, also called transportation-information inequalities. That is, a
comparison of the Wasserstein distance from the given measure to any other
absolutely continuous measure with finite relative entropy. Such
transportation-information inequalities were recently established for some
stochastic differential equations. Here, we develop a similar theory for some
stochastic partial differential equations.Comment: 19 pages. Keywords: Stochastic partial differential equations,
stochastic heat equation, stochastic fractional heat equation, concentration
of measure, transportation-information inequality, relative entropy,
Wasserstein distanc
Analysis of a Stratified Kraichnan Flow
We consider the stochastic convection-diffusion equation
for and , subject to
being a nice initial profile. Here, the velocity field is assumed to be
centered Gaussian with covariance structure
where is a continuous and bounded positive-definite function on
.
We prove a quite general existence/uniqueness/regularity theorem, together
with a probabilistic representation of the solution that represents as an
expectation functional of an exogenous infinite-dimensional Brownian motion. We
use that probabilistic representation in order to study the It\^o/Walsh
solution, when it exists, and relate it to the Stratonovich solution which is
shown to exist for all .
Our a priori estimates imply the physically-natural fact that, quite
generally, the solution dissipates. In fact, very often,
\begin{equation}
P\left\{\sup_{|x_1|\leq m}\sup_{x_2\in\mathbb{R}} |u(t\,,{\bf x})|
= O\left(\frac{1}{\sqrt t}\right)\qquad\text{as }
\right\}=1\qquad\text{for all },
\end{equation} and the rate is shown to be unimproveable.
Our probabilistic representation is malleable enough to allow us to analyze
the solution in two physically-relevant regimes: As and as . Among other things, our analysis leads to a "macroscopic multifractal
analysis" of the rate of decay in the above equation in terms of the reciprocal
of the Prandtl (or Schmidt) number, valid in a number of simple though still
physically-relevant cases
Analytical Solution For Navier-Stokes Equations In Two Dimensions For Laminar Incompressible Flow
The Navier-Stokes equations describing laminar flow of an incompressible
fluid will be solved. Different group of general solutions for Navier stokes
equations governing Laminar incompressible fluids will be derived.Comment: 10 page
On the Stochastic Heat Equation with Spatially-Colored Random forcing
We consider the stochastic heat equation of the following form
\frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) +
\sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where \sL is the
generator of a L\'evy process and is a spatially-colored, temporally
white, gaussian noise. We will be concerned mainly with the long-term behavior
of the mild solution to this stochastic PDE. For the most part, we work under
the assumptions that the initial data is a bounded and measurable
function and is nonconstant and Lipschitz continuous. In this case, we
find conditions under which the preceding stochastic PDE admits a unique
solution which is also \emph{weakly intermittent}. In addition, we study the
same equation in the case that is replaced by its
massive/dispersive analogue where .
Furthermore, we extend our analysis to the case that the initial data is
a measure rather than a function. As it turns out, the stochastic PDE in
question does not have a mild solution in this case. We circumvent this problem
by introducing a new concept of a solution that we call a \emph{temperate
solution}, and proceed to investigate the existence and uniqueness of a
temperate solution. We are able to also give partial insight into the long-time
behavior of the temperate solution when it exists and is unique. Finally, we
look at the linearized version of our stochastic PDE, that is the case when
is identically equal to one [any other constant works also].In this
case, we study not only the existence and uniqueness of a solution, but also
the regularity of the solution when it exists and is unique.Comment: +100 page
Zeros of a two-parameter random walk
We prove that the number gamma(N) of the zeros of a two-parameter simple
random walk in its first N-by-N time steps is almost surely equal to N to the
power 1+o(1) as N goes to infinity. This is in contrast with our earlier joint
effort with Z. Shi [4]; that work shows that the number of zero crossings in
the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity.
We prove also that the number of zeros on the diagonal in the first N time
steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi.Comment: 14 page
Intermittence and nonlinear parabolic stochastic partial differential equations
We consider nonlinear parabolic SPDEs of the form \partial_t u=\sL u +
\sigma(u)\dot w, where denotes space-time white noise,
is [globally] Lipschitz continuous, and \sL is the
-generator of a L\'evy process. We present precise criteria for existence
as well as uniqueness of solutions. More significantly, we prove that these
solutions grow in time with at most a precise exponential rate. We establish
also that when is globally Lipschitz and asymptotically sublinear, the
solution to the nonlinear heat equation is ``weakly intermittent,'' provided
that the symmetrization of \sL is recurrent and the initial data is
sufficiently large.
Among other things, our results lead to general formulas for the upper
second-moment Liapounov exponent of the parabolic Anderson model for \sL in
dimension . When \sL=\kappa\partial_{xx} for , these
formulas agree with the earlier results of statistical physics
\cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the
two exactly-solvable cases where and
Images of the Brownian Sheet
An N-parameter Brownian sheet in R^d maps a non-random compact set F in R^N_+
to the random compact set B(F) in \R^d. We prove two results on the image-set
B(F):
(1) It has positive d-dimensional Lebesgue measure if and only if F has
positive (d/2)-dimensional capacity. This generalizes greatly the earlier works
of J. Hawkes (1977), J.-P. Kahane (1985a; 1985b), and one of the present
authors (1999).
(2) If the Hausdorff dimension of F is strictly greater than (d/2), then with
probability one, we can find a finite number of points \zeta_1,...,\zeta_m such
that for any rotation matrix \theta that leaves F in B(\theta F), one of the
\zeta_i's is interior to B(\theta F). In particular, B(F) has interior-points
a.s. This verifies a conjecture of T. S. Mountford (1989).
This paper contains two novel ideas: To prove (1), we introduce and analyze a
family of bridged sheets. Item (2) is proved by developing a notion of
``sectorial local-non-determinism (LND).'' Both ideas may be of independent
interest.
We showcase sectorial LND further by exhibiting some arithmetic properties of
standard Brownian motion; this completes the work initiated by Mountford
(1988).Comment: 27 pages, submitted for publicatio
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