Schoenberg's Theorem via the law of large numbers

A classical theorem of S. Bochner states that a function $f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only if $f$ 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

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

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

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 $$\partial_t u(t\,,{\bf x}) =\nu\Delta u(t\,,{\bf x}) + V(t\,,x_1)\partial_{x_2}u(t\,,{\bf x}),$$ for $t>0$ and ${\bf x}=(x_1\,,x_2)\in\mathbb{R}^2$, subject to $\theta_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure $$\text{Cov}[V(t\,,a)\,,V(s\,,b)]= \delta_0(t-s)\rho(a-b)\qquad\text{for all $s,t\ge0$ and $a,b\in\mathbb{R}$},$$ where $\rho$ is a continuous and bounded positive-definite function on $\mathbb{R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $u$ 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 $\nu>0$. 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 $t\to\infty$} \right\}=1\qquad\text{for all $m>0$}, \end{equation} and the $O(1/\sqrt t)$ 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 $t\to\infty$ and as $\nu\to 0$. 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

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 $\dot{F}$ 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 $u_0$ is a bounded and measurable function and $\sigma$ 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 $\mathcal{L}u$ is replaced by its massive/dispersive analogue $\mathcal{L}u-\lambda u$ where $\lambda\in\R$. Furthermore, we extend our analysis to the case that the initial data $u_0$ 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 $\sigma$ 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

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

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 $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and \sL is the $L^2$-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 $\sigma$ 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 $(1+1)$. When \sL=\kappa\partial_{xx} for $\kappa>0$, 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 $u_0=\delta_0$ and $u_0\equiv 1$

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