# Stochastic Lattice Differential Equations and Applications (17frg671)

Arriving in Banff, Alberta Sunday, September 10 and departing Sunday September 17, 2017

## Organizers

Xiaoying Han (Auburn University)

Hakima Bessaih (Department of Mathematics, University of Wyoming)

Maria J. Garrido-Atienza (University of Sevilla)

Bjorn Schmalfuss (Institut for Mathematics)

## Objectives

\section{Objectives}

While classical noise perturbations used in natural sciences and engineering applications are usually

Classical random dynamical systems driven by white noise are investigated from the point of view of Markov processes. In particular one topic of research is the invariant measure with respect to the Markov transition property. This method fails in general when the noise is fractional. An alternative for the investigation of these object is the theory of random dynamical systems (RDS). The RDS theory has been developed by L. Arnold (see the monograph [Arnold]) and colleagues. It allows us to study stability behavior of differential equations containing a general type of noise, which can be characterized by several objects such as random attractors and their dimension, random fixed points, random inertial, stable or unstable manifolds, and Lyapunov exponents.

Finite dimensional It\^o equation with sufficiently smooth coefficients generate RDS. This assertion follows from the flow property generated by the It\^o equation, due to Kolmogorov's theorem for a H{\"o}lder continuous version of a random field with finitely many parameters in [Kunita]. However this method fails for infinite dimensional stochastic equations, i.e., for systems with infinitely many parameters, and in particular for stochastic lattice equations. To justify the flow property or the generation of an RDS by a lattice system, a special transform technique is used. Such a transform reformulates a stochastic differential equation to a path-wise random differential equation. The application of this technique to stochastic lattice systems is restricted to the case that the random perturbations of the lattice system are given by either an additive noise term in each component $c dB_i(t)$ or a simple multiplicative noise term $c u_i dB_i(t)$ at each node $i \in \mathbb Z$ (see [bates-lisei-lu, bates, Caraballo-Lu, HanShenZhou, han-review] and the references therein).

\section{Research Plan} The proposed activity for our Focused Research Group at BIRS will be mainly focusing on the study of random dynamical systems generated by stochastic lattice equations perturbed by fractional Brownian motion. In a first step we intend to prove the existence of solutions to general stochastic lattice systems with fBm and that the solutions generate RDSs. We will consider general situations where the intensity of the feedback of the noise to the system is determined by the component of the state variable. In other words, we will consider noise coefficients with the form $\sigma_i(u_i)dB_i^H$ where $B_i^H$ is a component of an infinite dimensional fBm and $\sigma_i$ are sufficiently regular functions. The solutions of these stochastic lattice systems are to generate an RDS.

In order to obtain an RDS for this generalized class of stochastic lattice systems we have to use modern techniques of Stochastic Analysis. When the Hurst parameter of the fBm is larger than 1/2, these techniques are based on a generalization of the Young--integration defined throughout fractional integrals. For $H\in (1/3,1/2]$ we will use the rough path theory or the theory of compensated fractional derivatives. All these theories provide a stochastic integral in the pathwise sense. Note that in comparison, the It\^o- or Stratonovich integral is given as a limit

During the stay at BIRS we will initiate the following projects:

\section{Justification for using BIRS} Every member of the team has a common interest in stochastic lattice systems with fractional Brownian motions and their applications. In addition, there had been collaborations among different team members on related topics. However, the team members are from three different countries, four different institutions, which makes collaborations not as effective as expected. BIRS provides a great opportunity for the team members to work together in person, and to focus on one specific topic during the specific period of time. BIRS also provides helpful supports such as working and living facilities. We truly believe these will greatly facilitate the progress of our major research project and promote collaborations among mathematicians from different parts of the world.

While classical noise perturbations used in natural sciences and engineering applications are usually

*white noise}, we would like to introduce a more general type of noise -- the {\em fractional Brownian motion} (fBm). In probability theory, an fBm (denoted by $B^H$) is a centered Gau{\ss*--process with a special covariance function determined by the Hurst parameter $H\in (0,1)$. For $H=1/2$, $B^H$ is the Brownian motion (denoted by $B$) where the generalized temporal derivative is the white noise. For $H\not =1/2$, $B^H$ does not enjoy the semi-martingale property and as a consequence classical techniques of Stochastic Analysis are not applicable. In particular the fBm with a Hurst parameter $H \in (1/2,1)$ enjoys the property of a long range memory, which roughly implies that the decay of stochastic dependence with respect to the past is only sub-exponentially slow. This long-range dependence property of fBm makes it a realistic choice of noise for problems with long memory in the applied sciences such as climate modeling, financial econometrics, internet traffic, DNA sequencing, and polymers (see e.g., [Mandelbrot, w1, w2]). \textit{The proposed research to be carried out by our Focused Research Group at BIRS is to initiate the study of stochastic lattice systems driven by a fractional Brownian motion.}Classical random dynamical systems driven by white noise are investigated from the point of view of Markov processes. In particular one topic of research is the invariant measure with respect to the Markov transition property. This method fails in general when the noise is fractional. An alternative for the investigation of these object is the theory of random dynamical systems (RDS). The RDS theory has been developed by L. Arnold (see the monograph [Arnold]) and colleagues. It allows us to study stability behavior of differential equations containing a general type of noise, which can be characterized by several objects such as random attractors and their dimension, random fixed points, random inertial, stable or unstable manifolds, and Lyapunov exponents.

Finite dimensional It\^o equation with sufficiently smooth coefficients generate RDS. This assertion follows from the flow property generated by the It\^o equation, due to Kolmogorov's theorem for a H{\"o}lder continuous version of a random field with finitely many parameters in [Kunita]. However this method fails for infinite dimensional stochastic equations, i.e., for systems with infinitely many parameters, and in particular for stochastic lattice equations. To justify the flow property or the generation of an RDS by a lattice system, a special transform technique is used. Such a transform reformulates a stochastic differential equation to a path-wise random differential equation. The application of this technique to stochastic lattice systems is restricted to the case that the random perturbations of the lattice system are given by either an additive noise term in each component $c dB_i(t)$ or a simple multiplicative noise term $c u_i dB_i(t)$ at each node $i \in \mathbb Z$ (see [bates-lisei-lu, bates, Caraballo-Lu, HanShenZhou, han-review] and the references therein).

\section{Research Plan} The proposed activity for our Focused Research Group at BIRS will be mainly focusing on the study of random dynamical systems generated by stochastic lattice equations perturbed by fractional Brownian motion. In a first step we intend to prove the existence of solutions to general stochastic lattice systems with fBm and that the solutions generate RDSs. We will consider general situations where the intensity of the feedback of the noise to the system is determined by the component of the state variable. In other words, we will consider noise coefficients with the form $\sigma_i(u_i)dB_i^H$ where $B_i^H$ is a component of an infinite dimensional fBm and $\sigma_i$ are sufficiently regular functions. The solutions of these stochastic lattice systems are to generate an RDS.

In order to obtain an RDS for this generalized class of stochastic lattice systems we have to use modern techniques of Stochastic Analysis. When the Hurst parameter of the fBm is larger than 1/2, these techniques are based on a generalization of the Young--integration defined throughout fractional integrals. For $H\in (1/3,1/2]$ we will use the rough path theory or the theory of compensated fractional derivatives. All these theories provide a stochastic integral in the pathwise sense. Note that in comparison, the It\^o- or Stratonovich integral is given as a limit

*in probability*of Darboux sums of adapted integrands, and hence is not pathwise. The advantage with respect to the method to obtain random (semi-)flows presented in [Kunita] lies in that it is relatively easy to control the exceptional sets by the pathwise character of the integral. Thus, we do not need Kolmogorov's theorem for a continuous version of a random field. This allows us to consider stochastic differential equations with an infinite dimensional state space. Therefore we expect to obtain random dynamical systems generated by the general class of stochastic lattice systems given above.During the stay at BIRS we will initiate the following projects:

- To study the RDS generated by stochastic lattice systems with state-dependent multiplicative fBm, and in particular the cell model \eqref{cell1}-\eqref{cell2} and the shell model \eqref{shell} perturbed by state-dependent multiplicative fBm.
- To analyze stability questions of the RDS generated by these stochastic lattice equations with general diffusion part. In particular, assuming that such a lattice system possesses the trivial solution we will formulate assumptions ensuring that this solution is locally exponentially stable (exponentially attracting).
- In addition, we are also interested in investigating the existence and geometric structures of global random attractors for the RDS generated by the above fBm-driven stochastic lattice systems, as well as fBm-driven stochastic lattice systems with delays (if time permits).

\section{Justification for using BIRS} Every member of the team has a common interest in stochastic lattice systems with fractional Brownian motions and their applications. In addition, there had been collaborations among different team members on related topics. However, the team members are from three different countries, four different institutions, which makes collaborations not as effective as expected. BIRS provides a great opportunity for the team members to work together in person, and to focus on one specific topic during the specific period of time. BIRS also provides helpful supports such as working and living facilities. We truly believe these will greatly facilitate the progress of our major research project and promote collaborations among mathematicians from different parts of the world.