# Some intermittency problems for parabolic SPDE (17rit683)

Arriving in Banff, Alberta Sunday, August 20 and departing Sunday August 27, 2017

## Organizers

Kunwoo Kim (Pohang University of Science and Technology, Korea)

Davar Khoshnevisan (University of Utah)

Carl Mueller (University of Rochester)

## Objectives

The proposers have an ongoing project to study the rate of decay of solutions to the parabolic Anderson in the case where the spatial variable $x$ does not lie in $\mathbf{R}$, but on the unit circle. We previously discussed the high peaks which occur when $x\in\mathbf{R}$; but these peaks are widely spaced across the real line. When $x$ lies in the unit circle, we expect very different behavior, in which the entire solution should decay to 0 exponentially, and at a precise rate. The study of this exponential behavior is the first goal of our project. Even when $x\in\mathbf{R}$, we expect the solution to decay between the widely spaced peaks, so exponent decay should really be the typical behavior at a given location.

The intermittency exhibited by solutions to the parabolic Anderson model should be typical of many models, although most will not be as tractable. Firstly, we propose to study a variation on the parabolic Anderson model with nonlinear noise coefficient: \(\partial_tu=\Delta u+g(u)\dot{W}.\) Here again, $X$ lies on the unit circle, and the noise coefficient $g(u)$ has close to linear behavior. One of us, Khoshnevisan, has written an extensive series of papers exploring intermittency in the presence of such a noise coefficient. We expect to carry over many of these ideas to the case where $x$ lies on the unit circle, and to show that, in contrast to the parabolic Anderson on the line, the solution to the parabolic Anderson model on the circle has a sharp exponential rate of decrease. Such an undertaking hinges on underlying quantitative bounds which have other consequences. Next we describe one such possibility: Consider the stochastic Allen--Cahn equation $\partial_t u = \Delta u + u - u^3 + \lambda u\dot{W}$ on the circle, where $\lambda>0$ describes the level of the noise $\dot{W}$ in the system. Our intermittency analysis of the parabolic Anderson model, if successful, is likely to imply the existence of a unique $\lambda_c\in(0\,,\infty)$ such that: (i) If $\lambda<\lambda_c$, then the solution to the stochastic Allen--Cahn equation goes to zero exponentially fast; and (ii) if $\lambda>\lambda_c$, then the solution does not go to zero as $t\to\infty$. In particular, this shows that there exists a phase transition --- in the sense of nonequilibrium statistical mechanics --- only above the noise threshold $\lambda_c$ and not below $\lambda_c$. In particular, this will give a rigorous proof of a phenomenon that was predicted earlier, using physical and simulation arguments, by physicists Zimmerman et al (2000;

The intermittency exhibited by solutions to the parabolic Anderson model should be typical of many models, although most will not be as tractable. Firstly, we propose to study a variation on the parabolic Anderson model with nonlinear noise coefficient: \(\partial_tu=\Delta u+g(u)\dot{W}.\) Here again, $X$ lies on the unit circle, and the noise coefficient $g(u)$ has close to linear behavior. One of us, Khoshnevisan, has written an extensive series of papers exploring intermittency in the presence of such a noise coefficient. We expect to carry over many of these ideas to the case where $x$ lies on the unit circle, and to show that, in contrast to the parabolic Anderson on the line, the solution to the parabolic Anderson model on the circle has a sharp exponential rate of decrease. Such an undertaking hinges on underlying quantitative bounds which have other consequences. Next we describe one such possibility: Consider the stochastic Allen--Cahn equation $\partial_t u = \Delta u + u - u^3 + \lambda u\dot{W}$ on the circle, where $\lambda>0$ describes the level of the noise $\dot{W}$ in the system. Our intermittency analysis of the parabolic Anderson model, if successful, is likely to imply the existence of a unique $\lambda_c\in(0\,,\infty)$ such that: (i) If $\lambda<\lambda_c$, then the solution to the stochastic Allen--Cahn equation goes to zero exponentially fast; and (ii) if $\lambda>\lambda_c$, then the solution does not go to zero as $t\to\infty$. In particular, this shows that there exists a phase transition --- in the sense of nonequilibrium statistical mechanics --- only above the noise threshold $\lambda_c$ and not below $\lambda_c$. In particular, this will give a rigorous proof of a phenomenon that was predicted earlier, using physical and simulation arguments, by physicists Zimmerman et al (2000;

*PRL}***85}(17):3612--3615). Going further afield, we would like to study intermittency and decay properties for other physically motivated equations, many of which have often been much less studied. For example, Andrew Majda (1993; ${\it Phys.\ Fluids^MA****$ {\bf 5**:1963--1970) has studied shear flows with a particular random forcing term. These flows are associated to the equation \(\partial_tT+v\cdot\nabla T=\kappa\Delta T\), with the extra condition that div $v=0$. He goes on to remark that observations support the existence of strong intermittency for this model. Although space is three dimensional in this model, Majda assumes that random fluctuations only affect one of the three dimensions. With this simplification, Majda is able to derive detailed properties of the model, always within the context of finding explicit formulas for the moments of the solution. He states that intermittency manifests itself through nonGaussian probability distributions. Using our ideas about the parabolic Anderson model, we believe we can go further to study the intermittent behavior of solutions in the original context of the formulation of high peaks. We also wish to study the exponential decay of solutions between the peaks.