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.
