## A Stroll Through Dynamics (and some Random Walks too)

Dynamical Systems is a vast area of mathematics, and one in which I have some small experience with a small bit of the research community. I won’t go into the overall field and all of its various subfields. Rather, I will concentrate on providing the interested reader some idea of what my Masters work was about. Modern research in many fields contains elements from a variety of areas of mathematics, so I will try and link to relevant concepts where possible. The reader is assumed to have some familiarity with basic mathematical analysis and elementary differential equations.

The work I did was primarily concerned with the asymptotic (as time goes to infinity) dynamics, or behavior, of certain Partial Differential Equations. These equations were forced by nonautonomous terms, as well as stochastic terms that model noise. The fact that these are PDE has several important implications, a crucial one being that the phase space for the dynamics is infinite-dimensional. That is, initial data are considered to lie in a Banach space, and this type of functional analysis notation is found throughout the work. Much of modern PDE theory is done this way, because these spaces form a natural framework within which to deduce relatively general ideas about these type of equations. In fact, the study of PDE is what motivated many huge developments in modern analysis.

Another complication is introduced via the noise term. Stochastic differential equations are a very large area of modern interest. There is an entire subfield of real analysis and probability theory that has been concerned with formalizing the framework for such equations, and much of it is quite technical. The rest of my post (and my work itself) will be in a pathwise setting. So you can think of the noise term below as simply a continuous function. It should be noted that the particular function chosen will be dependent on the specific path one takes in the probability space. Further simple details can be found under this article on Wiener Processes, and any good book or lecture notes on stochastic calculus. If you are unfamiliar with this material, then please ignore some of the technical language.

To model this noise, let $(\Omega, \mathcal{F}, P)$ be the standard probability space where $\Omega = \{ \omega \in C(\mathbb{R},\mathbb{R} ): \omega(0) = 0 \}$, $\mathcal{F}$ is the Borel $\sigma$-algebra induced by the compact-open topology of $\Omega$ and $P$ is the Wiener measure on $(\Omega, \mathcal{F})$. Denote by $\{\theta_t\}_{t \in \mathbb{R}}$ the family of shift operators given by

$\theta_t \omega (\cdot) = \omega (\cdot +t) - \omega (t) \quad \mbox{ for all } \ \omega \in \Omega \ \mbox{ and } \ t \in \mathbb{R}.$

It is known that $(\Omega, \mathcal{F}, P, \{\theta_t\}_{t \in \mathbb{R}})$ is a metric dynamical system. Given $\tau \in\mathbb{R}$ and $\omega \in \Omega$, consider the following stochastic equation defined for $\mathbb{R} x \in {\mathbb{R}^n}$ and $t > \tau$,

${\frac {\partial u}{\partial t}} + \lambda u- {\rm div} \left (|\nabla u |^{p-2} \nabla u \right )=f(t,x,u )+g(t,x)+\alpha h(u,x) {\frac {dW}{dt}},$

with initial condition

$u( \tau, x ) = u_\tau (x), \quad x\in {\mathbb{R}^n},$

where $p\ge 2$, $\alpha>0$, $\lambda>0$, $\varepsilon>0$, $g \in L^2_{loc}(\mathbb{R}, L^2(\mathbb{R}^n))$, and $W$ is a two-sided real-valued Wiener process on $(\Omega, \mathcal{F}, P)$. The function $h$ is used to model either additive or multiplicative noise (e.g. $h = h(x)$ or $h = u$), although more interesting generalizations are possible. The function $f$ satisfies some dissipative conditions, which makes existence of solutions and other properties relatively straightforward.

There are many ways to analyze and study such equations. Existence and Uniqueness results are of course a major area of mathematical analysis, especially concerning stochastic PDEs. There are questions of bifurcation and other qualitative properties of solutions to equations like the above one. One major simplification of solution behavior can occur when we restrict ourselves to looking at the equation asymptotically; that is, what happens to the function $u$ as $t \to \infty$? This is a central question in dynamical systems, and has especially interesting and concrete results in the case of autonomous deterministic finite-dimensional systems, such as the Lorenz equations. In this case, the concept of an attractor is well-defined, and can be characterized and understood. There are still open questions of course, mostly relating to chaos or strange attractors, but the amount we know about these kinds of simple systems is quite significant.

The equation above, however, is nonautonomous, stochastic, and infinite-dimensional. It also contains two nonlinearities, and is in an unbounded domain. I won’t try and give a detailed explanation of how the modern theory has found a way to overcome these difficulties, and extend some of the nice theory of attractors to this case, but I will try and give you some ideas. There is much work to be done in terms of understanding things, especially at a more detailed level, but the existence of some object we might call an attractor can be deduced (and was the primary goal of my work).

There are many ways to discuss attraction, and many of the technical details aren’t important to get some idea of the work. Usually we think of an attractor (more formally, the global attractor) as an object which every bounded set asymptotically converges towards as time goes to infinity under the flow of the differential equation. The flow is usually taken to be a solution operator of the equation, although other formalizations are possible. In the case of nonautonomous and random systems, however, there are reasons that one must separate “forward” and “backward” or pullback attraction. The justification is somewhat technical, but suffice to say that it is easier to understand a stochastic process by studying its history than it is to understand it moving forward. So instead of studying forward or global attractors, many researchers have begun investigating the idea of a pulback attractor, which has proven to be an important generalization of the autonomous deterministic case. It is in fact strictly a generalization, since in the autonomous deterministic case, these types of attractors are all the same object.

To study such an attractor, we need a notion of an Absorbing Set (in this case, a Pullback Absorbing Set), as well as certain compactness conditions. The former idea is a rough generalization of the concept that after a sufficient period of time, all dynamics are confined to a bounded subset of the phase space. The compactness conditions are important for a few reasons, but mostly to characterize important properties of the attractor itself (e.g. that it itself is compact). The very interested reader could consult the main theorem of this paper to understand the details in this particular case. It should be noted that here flows have been modelled using Cocycles of a particular form so as to take random and initial times into account.

Once existence results are established, one could study various properties of these attractors. An interesting one, especially in the case of chaotic and infinite dimensional systems, is the dimensionality of the attractor. This potentially has important implications to understanding the formation of turbulence and other complex behavior, and has been used successfully in those areas. It should be noted that, due to the still open question about regularity of the Navier-Stokes equations, these dynamical results only hold absolutely in the two dimensional setting. Some approaches have been made in the three dimensional case using various generalizations, but I want to emphasize that this is an area ripe for exploration.

In the case of our work, we also showed upper-semicontinuity of the attractor as the intensity of the noise, $\alpha \to 0$. For this, we showed the existence of attractors $\mathcal{A}_\alpha, \mathcal{A}_0$ corresponding to the stochastic case with given intensity, and the deterministic case respectively. This is defined in a technical sense using the Hausdorff semi-metric. Roughly speaking, this means that for every small $\epsilon$-distance of the deterministic attractor $\mathcal{A}_0$, we can find a small enough $\alpha$ such that the stochastic attractor, $\mathcal{A}_\alpha$ is contained inside of this distance of the deterministic attractor. This was shown for just the multiplicative noise case, but it should be easy to show for the additive noise case.

How is this dynamical information useful? It tells us that adding very small stochastic perturbations to our equations won’t cause the dynamics to grow very much in the norm of our metric space (in this case, the $L^2(\mathbb{R}^n)$ or “energy norm”). We can interpret this as saying the dynamics are stable to such stochastic perturbations, and that if all we are interested in is asymptotic behavior, the deterministic model would be sufficient, as long as the intensity of the noise is very small, and as long as we are ok with stochasticity possibly “collapsing” the dynamics (see below). This says little about transient behavior of these systems, but their dissipative character makes their long-time dynamics at least amenable to understanding in this way.

What about lower-semicontinuity? Is it possible for a small stochastic perturbation to collapse the stochastic attractor into a strictly smaller subset of the deterministic one? This question, it turns out, relies on a lot of information about the detailed structure of the attractor itself; this is information we cannot obtain for such a general family of equations. Nevertheless, it would be interesting to see someone try and investigate this phenomena

The full text of my thesis can be read here. The bulk of the work appeared in two papers, here, and here. If anyone is interested in the details and motivation for all of these ideas, I encourage you to investigate the references in the introductory chapter on dynamical systems and attractors in general, as well as some textbooks on nonautonomous attractor theory.

Finally, I would like to thank my advisor Bixiang Wang for all of his support during my time at New Mexico Tech. I should also mention that part of this work (the multiplicative noise case) was done jointly with a good friend, Michael Austin Lewis