# Differential equations driven by fractional Brownian motion as random dynamical systems: qualitative properties (08frg140)

## Organizers

David Nualart (University of Kansas)

Björn Schmalfuß (University of Paderborn)

Frederi Viens (Purdue University)

## Objectives

Background and relevance: stochastic integration and fractional

Brownian motion

A central mathematical object in Stochastics/Stochastic Processes is the Ito

integral. It plays an important role in many areas of pure and applied

mathematics including: mathematical finance, population dynamics, fluid

dynamics, statistics, signal processing, control, particle systems, to name a

few. The integrator of such an integral is often chosen to be Brownian motion

(the Wiener process) or its semimartingale generalizations. These random

functions are of unbounded total variation, so that their Stieltjes integrals

do not exist. Special properties of the integrators and the integrands are

necessary to generalize the definition of the Stieltjes integral to the Ito

integral, and enable the definition of solutions of differential equations

driven by Brownian motion.

A property of paramount importance to this effect for Brownian motion is the

independence of its increments. To move beyond integrals and processes

constructed using this property is one of the most important tasks in the

theory of Stochastics. We are most interested in using the fractional Brownian

motion (fBm) process $B^{H}$ where $Hin(0,1)$ is fixed. It is a type of

stochastic process which deviates significantly from Brownian motion and

semimartingales, and others classically used in probability theory. As a

centered Gaussian process, it is characterized by the stationarity of its

increments and a medium- or long-memory property which is in sharp contrast

with martingales and Markov processes. Specifically, $B^{H}left( 0right)

=0$ and $Varleft[ B^{H}left( tright) -B^{H}left( sright) right]

=leftvert t-srightvert ^{2H}$. It also exhibits power scaling and path

regularity properties with Holder parameter $H$, which are very distinct from

Brownian motion. In fact, this single textquotedblleft

Hursttextquotedblright parameter $Hin(0,1)$ is also responsible for the low

decorrelation speed of fBm: for increments that are $n$ time units apart, the

correlation is precisely $c_{H}n^{2H-2}$ where the constant $c_{H}=H(2H-1)$.

Note that the standard Brownian motion is included in this family of models:

it is fBm with $H=1/2$, since $c_{H}$ is indeed null in this case. FBm has

become a popular choice of late for applications where classical processes

cannot model these non-trivial properties; for instance long memory, which is

also known as persistence, and corresponds to the case $Hin(1/2,1)$, is of

fundamental importance for financial data and in internet traffic: see

cite{Mandelbrot}, cite{w1}, cite{w2} . Our research group will have these

applications in mind in all discussions.

The mathematical theory of fBm is currently being developed vigorously by a

number of stochastic analysts, in various directions, using complementary and

sometimes competing tools. Ever since the pioneering works of Z{"{a}}hle

cite{Zah98}, Decreusefond and "{U}st"{u}nel cite{DecUst99}, and Lyons

cite{L}, the main thrust has been to understand how to perform stochastic

integration with respect to fBm in a way which is consistent with some

properties of the classical Ito theory for Brownian motion. In the case of

higher regularity ($H>1/2$), simple trajectorial methods, labelled as

emph{pathwise}, can be used which make it easy to translate one integration

theory into another, as emph{fractional derivatives} allow a pathwise

estimate of the integrals in terms of integrand and integrator using special

norms. Pathwise integrals historically gave the first cases where adequate

solutions to stochastic differential equations (sde) were established, see

Nualart and Rascanu cite{NR02}; infinite-dimensional equations have been

treated with the same success as finite-dimensional ones, e.g. Nualart and

Maslowski cite{MN}, Viens et al. cite{TTV}.

Timeliness, motivation, and importance: stochastic differential

equations driven by fractional Brownian motion, from general theory to random

dynamical systems

Solving sde can be considered a benchmark for testing the adequacy of an

integration theory. Among those methods which allow the use of more irregular

fBm ($H<1/2$), the so-called rough path theory, which uses non-probabilistic

constructions even for very irregular signals, is more efficient at defining

solutions of non-linear sde driven by fBm; it also has the advantage of being

applicable to processes that share some regularity properties with fBm, but

are otherwise very different; its main disadvantage is to not allow the same

ease of calculations as for Ito equations for Brownian motion. The Skorohod

(divergence) integral theory, based on stochastic analysis and Malliavin

calculus, is better at exploiting the Gaussian property of fBm, for instance

to generalize the Ito integral and formula with ease, resulting in mean-zero

stochastic integrals, and other convenient constructions: see Cheridito and

Nualart cite{CN05}, Mocioalca and Viens cite{MV05}. But Skorohod integration

has not produced a way to define solutions to fully non-linear equations; this

is one of the longest-standing open problems in stochastic analysis.medskip

Many mathematicians have yet to move decisively beyond the basic "existence-uniqueness" theory of sde driven by fBm. While some problems of this type, such as for Skorohod sde, remain open and are worthy of study, in this focused research group, we will also delve deeper into the qualitative properties of fBm-driven equations. In particular, we will investigate the equations' asymptotics (e.g. in large time). The two most popular theories dealing with the asymptotic qualitative behavior for general sde are: the theory of emph{random dynamical systems} (RDS) and the theory of existence and uniqueness of emph{invariant measures} for the associated Markov semigroup. However, similarly to fBm itself, equations driven by fBm do not generate a Markov process. This precludes the study of invariant measures using classical tools for fBm-driven systems. It motivates our plan to concentrate on the investigation of fBm-driven sde as RDS, the interface between the two being at the heart of our focused research group.

The theory of RDS, developed by L. Arnold and coworkers (see cite{Arn98}),

can be used to describe the asymptotic and qualitative behavior of systems of

random and stochastic differential/difference equation in terms of stability,

Lyapunov exponents, invariant manifolds, and attractors. A RDS consists of two

parts. The first part is a model for the noise path $omega$, leading to a

emph{metric dynamical system}. In particular, it is known that the fractional

Brownian motion forms an emph{ergodic} metric dynamical system, see Maslowski

and Schmalfu{ss } cite{MS04}. The second part of a RDS is the dynamics of an

sde: it is given by the solution mapping $phi(t,omega,x)$, which describes a

solution at time $t$ starting at time zero with initial condition $x$ in some

phase space $E$ which is driven be a noise path $omega$. This mapping $phi$

satisfied a generalized (semi)-group property, called emph{cocycle}%

--property, see Arnold cite{Arn98}. In other contexts, some answers have

already been given to basic questions such as stationarity and ergodicity

using fBm: see for instance Hairer cite{Hm05}, Hairer and Ohashi cite{HO07},

where conditions are given so that results familiar to the Brownian case also

hold for fBm. In Maslowski and Schmalfu{ss } cite{MS04},

infinite-dimensional RDS driven by fBm are exhibited as stochastic evolution

systems, and are shown to have unique, exponentially attracting fixed points.

Specific objectives for fractional-Brownian-driven random dynamical

systems: stability, infinite systems, invariant manifolds, Lyapunov

exponents

For many infinite dimensional Brownian-driven sde with non-trivial diffusion

coefficients, it is not known whether these equations generate a RDS. The

reason is that typically stochastic differential equations are only defined

$omega$-almost surely because the exceptional set is related to the

definition of an Ito integral as a limit of random variables emph{in

probability}. However, such a family of exceptional sets does not allow to use

the theory of RDS's full power. One advantage of sde driven by an fBm with

$H>1/2$ is that one can consider pathwise integrals, which avoid exceptional

sets. The case of $H>1/2$ is also interesting from the physical standpoint

since it is the case where path memory (persistence) is the longest; in that

sense, it is most different from standard Brownian motion.

On the other hand, when $H<1/2$, as we alluded to above in the relation

between pathwise and Skorohod integration, there does not exist a universally

accepted way of looking at stochastic integration or sde's, even finite

dimensional, and the problem of solving sde's in the Skorohod sense is

entirely open. Our focused research group will address the issue by trying to

understand whether any synergy can exist between the various approaches, which

have thus far been largely disjoint. In particular, it is possible that the

difficulties being experienced by stochastic analysts are related to the

non-existence of a RDS interpretation of sdes driven by irregular fBm. Since

non-linear equations driven by irregular fBm can be solved via the rough path

method (see recent progress by Nualart and Hu cite{NuaHu06}, and also the

original works of Lyons and Qian cite{LioQui98} or cite{LioQia02}), finding

even a single counter-example where such a solution occurs in a semi-explicit

way, but does not allow a RDS to be defined, would help understand the open

problem.

A key concept describing the dynamics of RDS generated by fBm-driven sde is

the so-called global attractor, which is an invariant random set attracting

other bounded random sets; it is a hallmark of the concept of emph{stability}%

. The essential dynamics take place in a neighborhood of the attractor. Even

for infinite dynamical systems the attractor often has a finite (fractal)

dimension which allows to describe the dynamics by finitely many parameters.

To show that there is an attractor, the main point is to show the existence of

an absorbing set (see Chueshov and Schmalfu{ss } cite{r-8}). In the

classical theory, it can usually can be constructed via Gronwall's lemma. This

leads to the question of finding a Gronwall lemma with respect to the special

fractional norms used for estimating stochastic integrals of fBm; this forces

one to try and modify the phase space of the RDS accordingly. Some ideas for

this kind of Gronwall lemma can be found for finite dimensional systems in

Garrido, Maslowski and Schmalfu{ss } cite{GarMasSchm08}. Our focused

research group will discuss these findings, their scope, and their extensions

to infinite systems driven by fBm.

Our plan to discuss stability for fBm-driven sde will continue with the

existence of stable and unstable manifolds and Lyapunov exponents, see Lu and

Schmalfu{ss } cite{r-9}. Such smooth manifolds are invariant under the

dynamics of the systems, and on them, the states are attracted or repelled by

a steady state. Standard methods to obtain these manifolds are the

Lyapunov-Perron method, based on fixed point arguments. We plan to discuss how

to obtain these fixed points with respect to the special fractional norms for

fBm integration, which will again force us to use an appropriate phase

space.

Our focused research groups also intends to deal with the interesting question

of stability expressed in terms of pointwise Lyapunov exponents. The recent

work Viens and Zhang cite{VZ} considers the stochastic heat equation with

multiplicative infinite-dimensional noise which is fBm in time. If the spatial

behavior of the noise is non trivial, it is shown that when $Hleq1/2$, the

main Lyapunov exponent is finite and positive, but when $H>1/2$, it is

infinite, and in fact the solution explodes no slower than $expleft(

ct^{2H}/log tright) $ and no faster than $expleft( ct^{2H}/log

tright) $. Surprisingly, the case of $Hleq1/2$ is the easiest one to

analyze, and leads to familiar results. In the case of long memory ($H>1/2$),

the behavior of the solution is yet unclear. Is the memory length the main

factor in this difficulty for $H>1/2$? Or are there scaling arguments which

would help us show that the magnitude of the noise is the main driving force?

We will discuss these questions which could have important consequences in

statistical physics.

Specific objectives for long memory: memory preservation and

non-linear time series, effect on climate models, delay equations, and

statistical estimation

Further physically intriguing results regarding memory length are provided to

us by engineering and sociological practitioners. In Jumarie cite{Jg02},

arguments are given indicating that the introduction of random decision times

can lead to fractal behavior in dynamical systems, specifically with long

memory. In the opposite direction, numerical analysis on some linear dynamical

systems driven by fBm indicate they may lose the long memory property:

Grigoriu cite{Gm07}. The mathematical issue here is whether fBm's long memory

is preserved when passed through a RDS. As a first step to elucidate the

truths behind the numerical indications, we will discuss the use of

discrete-time RDS driven by long-memory time series. Indeed, it has been known

for a few years that some linear and nonlinear quadratic ARCH($infty$) time

series exhibit precisely fractional Brownian memory: Giraitis et al.

cite{GLRS04}. Here we will focus the discussion on how long memory may effect

products of non-independent matrices; an ARCH($infty$) model may be

considered as a product of infinite-dimensional matrices based on IID noise

terms, but it might be more efficient to model the discrete version of an

fBm-driven system as a product of $2times2$ matrices whose components are

correlated over long ranges. It is possible that for such a problem, using

continuous time and fractional stochastic calculus may provide a distinct

avenue of attack.

FBm is a good candidate to model random long-time influences in climate

systems, see Palmer et al. cite{Palmer2005}, cite{Palmer}. Here, the first

issues to address are existence questions for sde of from fluid dynamics when

they are driven by an fBm. For instance, although several authors appear

interested by the issue of driving a Navier-Stokes equation by fBm, no

definitive publications have yet appeared. The stochastic heat equation

mentioned above can be considered a toy model for this problem, and could

help, via Feynman-Kac representation as in cite{hv} for the corresponding

vorticity equation, understand the full non-linear equation driven by fBm.

After the existence question is better understood, our group will discuss the

dynamics of larger climate systems with long memory.

Stochastic differential delay equations and their asymptotic behavior have

received much attention in recent years (see for instance work of Caraballo

cite{Ca90}, Garrido-Atienza and collaborators cite{CaGaRe03}, cite{GaRe03},

Lisei cite{Lisei01}, Taniguchi et al. cite{TaLiuTru02}): such delayed

problems often appears in applications in physics, biology, engineering,

finance, etc. They offer an alternative to assuming that the driving noise is

fBm, while preserving some of fBm's memory features; but by combining the

effect of delay with the long memory of fBm, more realistic models will

emerge; to our knowledge these have not been considered. During our stay at

BIRS, we plan to discuss whether such stochastic delay equations with long

memory generate RDS, and if so, whether there exist random fixed points.

Judging for instance by the work of Garrido-Atienza et al. cite{CGS}

(stationary solutions for delayed sde driven by Brownian motion), our sde

coefficients will have to satisfy particular conditions.

Stochastic analysis has recently proved useful in tackling statistical

estimation issues for fBm. $H$ can be estimated consistently, and in some

cases with asymptotic normality, using simple power variation statistics, see

Tudor and Viens cite{TV07}. Similar constructs with weights are capable of

giving rise to various Ito integral limits, see for instance Nourdin et al.

cite{NNT07}. The statistics results have also motivated the study of

generalizations of fBm, such as the Rosenblatt process, and higher-order

Hermite processes, which share fBm's covariance structure, but can be

arbitrarily highly non-Gaussian. For simple fBm-driven sde with non-linear

drift scale parameter $a$, Gaussian and Malliavin calculus have yielded the

strong consistency and asymptotic normality of $a$'s maximum likelihood

estimator based on discrete observations: cite{TVstat}. Much remains to be

understood in estimating parameters for RDS driven by fBm; our group will

discuss the future research directions which are most likely to be of interest

to statisticians and other practitioners who need to know how to determine and

test their long-memory models.

Conclusion

Stochastic differential equations with fractional Brownian motion have the potential to provide a wealth of new models in many applied areas; our plan to use ideas from random dynamical systems to study their properties will give them the new perspective they deserve.

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