Let $xi$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $dot{u}=frac{1}{2} u + uxi$, started identically at one; and (ii) $dot{Z}=frac12 Z + xi$, started identically at zero. It is well kn
own that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $mathbf{R}_+timesmathbf{R}^d$ with $dge 2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $mathbf{R}$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.
Given a field ${B(x)}_{xinmathbf{Z}^d}$ of independent standard Brownian motions, indexed by $mathbf{Z}^d$, the generator of a suitable Markov process on $mathbf{Z}^d,,,mathcal{G},$ and sufficiently nice function $sigma:[0,infty)to[0,infty),$ we cons
ider the influence of the parameter $lambda$ on the behavior of the system, begin{align*} rm{d} u_t(x) = & (mathcal{G}u_t)(x),rm{d} t + lambdasigma(u_t(x))rm{d} B_t(x) qquad[t>0, xinmathbf{Z}^d], &u_0(x)=c_0delta_0(x). end{align*} We show that for any $lambda>0$ in dimensions one and two the total mass $sum_{xinmathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for dimensions greater than two there is a phase transition point $lambda_cin(0,infty)$ such that for $lambda>lambda_c,, sum_{mathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for $lambda<lambda_c,,sum_{mathbf{Z}^d}u_t(x) otto 0$ as $ttoinfty.$
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).
The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses
of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.
Consider an infinite system [partial_tu_t(x)=(mathscr{L}u_t)(x)+ sigmabigl(u_t(x)bigr)partial_tB_t(x)] of interacting It^{o} diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solutio
n under standard regularity assumptions on the nonlinearity $sigma$. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the $k$th moment Lyapunov exponent is frequently of sharp order $k^2$, in contrast to the continuous-space stochastic heat equation whose $k$th moment Lyapunov exponent can be of sharp order $k^3$. When the underlying walk is transient and the noise level is sufficiently low, we prove also that the solution is a.s. uniformly dissipative provided that the initial profile is in $ell^1(mathbf {Z}^d)$.
We prove that if $f:mathbb{R}tomathbb{R}$ is Lipschitz continuous, then for every $Hin(0,1/4]$ there exists a probability space on which we can construct a fractional Brownian motion $X$ with Hurst parameter $H$, together with a process $Y$ that: (i)
is Holder-continuous with Holder exponent $gamma$ for any $gammain(0,H)$; and (ii) solves the differential equation $dY_t = f(Y_t) dX_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.
Consider the stochastic heat equation $partial_tu=mathscr{L}u+lambdasigma(u)xi$, where $mathscr{L}$ denotes the generator of a L{e}vy process on a locally compact Hausdorff Abelian group $G$, $sigma:mathbf{R}tomathbf{R}$ is Lipschitz continuous, $lam
bdagg1$ is a large parameter, and $xi$ denotes space-time white noise on $mathbf{R}_+times G$. The main result of this paper contains a near-dichotomy for the (expected squared) energy $mathrm{E}(|u_t|_{L^2(G)}^2)$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $u$ is intermittent, the energy of the solution behaves generically as $exp{operatorname {const}cdot,lambda^2}$ when $G$ is discrete and $geexp{operatorname {const}cdot,lambda^4}$ when $G$ is connected.
We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which have only been carefully studied in some particular cases so far. Th
en, we establish a comparison principle for the solution, following the ideas of Mueller. We think it is of particular interest to obtain such a result for a hyperbolic equation. Finally, using the results mentioned above, we aim to show that the solution exhibits a chaotic behavior, in a similar way as was established by Conus, Joseph, and Khoshnevisan for the heat equation. We study the two cases where 1. the initial conditions have compact support, where the global maximum of the solution remains bounded and 2. the initial conditions are bounded away from 0, where the global maximum is almost surely infinite. Interesting estimates are also provided on the behavior of the global maximum of the solution.
Consider the stochastic heat equation $partial_t u = (frac{varkappa}{2})Delta u+sigma(u)dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)in (0, infty)timesR^d$, and $dot{F}$ is a centered Gaussian noise that is white in time and has spatia
lly-correlated coordinates. We analyze the large-$|x|$ fixed-$t$ behavior of the solution $u$ in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function $f$ of the noise is of Riesz type, that is $f(x)propto |x|^{-alpha}$, then the fluctuation exponents of the solution are $psi$ for the spatial variable and $2psi-1$ for the time variable, where $psi:=2/(4-alpha)$. Moreover, these exponent relations hold as long as $alphain(0, dwedge 2)$; that is precisely when Dalangs theory implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.
We consider a family of nonlinear stochastic heat equations of the form $partial_t u=mathcal{L}u + sigma(u)dot{W}$, where $dot{W}$ denotes space-time white noise, $mathcal{L}$ the generator of a symmetric Levy process on $R$, and $sigma$ is Lipschitz
continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $mathcal{L}f=cf$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.
