Let ξ denote space-time white noise, and consider the following stochastic partial differential equations: (i)u = 1 2 u ′′ + uξ, started identically at one; and (ii)Ż = 1 2 Z ′′ + ξ, started identically at zero. It is well known 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 R+ × R d with d 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 R are multifractal.Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3,4]. We expand on aspects of the Barlow-Taylor theory, as well.

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In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on R with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller's comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at ±∞. These results generalize a recent work by Moreno Flores [25], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [6].MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.

In this paper, we study the stochastic heat equation in the spatial domain R d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. As a first application of these moments bounds, we find the necessary and sufficient conditions for the solution to have phase transition for the second moment Lyapunov exponents. As another application, we prove a localization result for the intermittency fronts.