Robert Adler: What do I do?
For the last three and a half decades most of my work
has concentrated on the study and
application of stochastic processes which, in one form or another, have a
strong spatial component. (As opposed to the usual study of temporal
processes.) These are called random fields . In particular, I have been interested in the geometrical
properties of various structures generated by these processes.
While my work is primarily theoretical, to my most pleasant surprise,
much of it has found application.
For example,
my work on random geometry, the basis of which began in the mid 1970's and has been
extensively broadened over the years, has been used in areas as diverse as
astrophysics and
mapping the structure of the brain.
The term `random fields' means different things to different people.
For example, people working in interacting particle systems use the term to
refer to processes indexed by lattices in R^N, N>1, exhibiting some
kind of multi-parameter Markov properties. (In fact, some people will even use
the term to describe processes on the integers satisfying a sort of two-sided
Markov property.)
For others, and that included myself when,
in 1981, I published a Wiley monograph under the tile The Geometry of
Random Fields,
a random field was
any (generally continuous) stochastic process defined over some (usually simply
connected) subset of Euclidean space; i.e.\ some sort of random surface.
Twenty five years have passed, and much in the area has
changed. In particular, the last few years have seen an explosion of
material on random fields on manifolds. This work, which began
with the PhD thesis of Jonathan Taylor ,
was originally
motivated by specific applications in brain imaging, but has gone
well beyond this and has revolutionised the way we now think about
most of the geometric properties of random fields on parameter spaces
as simple as the unit interval; i.e. for simple random processes in time.
Consequently, Jonathan and I spent a good part of 2003-2006
on producing a Springer monograph
Random Fields and Geometry which describes this theory. You can
read the preface and introductions
to the three parts of this book on my
publications page. Since this was a book aimed primarily at mathematicians, and
there are many applications to this theory, we are now working
on another book, this time together with
Keith Worsley, which will have an
easy introduction to the theory and also cover lots of these applications. Early
chapters can, at this point, also be found on my publications page.
The
initial data is actually directional, as it represents radiation coming into
a point from the surrounding universe. As such, it is actually a random field
on a two dimensional sphere. The maps shown here are projections of
the full sky in Galactic coordinates, with the plane of the Milky Way
placed horizontally in the middle of the
map and with the Galactic center at the center.
Very briefly,
astronomers are interested in the apparent randomness in the patterns in
maps like these, and in determining whether or not the
image can be considered as the realisation of a Gaussian (or other) random
field.
As with the astrophsyics example, the main problem of interest lies
in determining whether these pictures are consistent with a noise model,
or whether regions of activity really contain information
about which part of the brain handles a specific task.
You can get more information on this from Keith Worsley's
home page, which is from where
the figure also comes.
These are the applications, but they are what others have done with my theory,
and are not what I do my myself. I find that doing theory is quite hard enough
for me. Actually handling real life problems is beyond my technical capabilities.
One very rich class of random fields is provided by Gaussian processes. These
are far more natural to study in the multi-parameter scenario than, for
example, Markov processes or martingales, the theories of both of which are
closely tied to the total order of their parameter space - the real line.
As a result, I have been interested in Gaussian processes over the years, and
in 1990 published a set of lecture notes
An Introduction to Continuity, Extrema,
and Related Topics for General Gaussian Processes
with the IMS, much of which also now reappears in the first part of
Random Fields and Geometry
Here is part of the Introduction the the notes:
"... on what these notes are meant to be, and what
they are not meant to be.
They are meant to be an introduction to what I call
the ``modern'' theory of sample path properties of Gaussian
processes, where by ``modern'' I mean a theory based on concepts
such as entropy and majorising measures. They are directed at
an audience that has a reasonable probability background, at
the level of any of the standard texts (Billingsley, Breiman,
Chung, etc.). It also helps if the reader already knows something
about Gaussian processes, since the modern treatment is very
general and thus rather abstract, and it is a substantial help
to one's understanding to have some concrete examples to hang
the theory on. To help the novice get a feel for what we are talking
about, Chapter 1 has a goodly collection of examples.
The main point of the modern theory is that the geometric structure of the
parameter space of a Gaussian process has very little to do with its basic
sample path properties. Thus, rather than having one literature treating
Gaussian processes on the real line, another for multiparameter processes, yet
another for function indexed processes, etc., there should be a way of treating
all these processes at once. That this is in fact the case was noted by Dudley
in the late sixties, and his development of the notion of entropy was meant to
provide the right tool to handle the general theory.
While the concept of entropy turned out to be very useful, and in the hands of
Dudley and Fernique lead to the development of necessary and sufficient
conditions for the sample path continuity of stationary Gaussian processes, the
general,
non-stationary case remained beyond its reach. This case was finally solved
when, in 1987, Talagrand showed how to use the notion of majorising measures to
fully characterise the continuity problem for general Gaussian processes.
All of this would have been a topic of interest only for specialists, had it
not been for the fact that on his way to solving the continuity problem
Talagrand also showed us how to use many of our old tools in more efficient
ways than we had been doing in the past.
It was in response to my desire to understand Talagrand's message clearly that
these notes started to take form. The necessary prompting came from
Holger
Rootzen and Georg Lindgren, who asked me to
deliver, in (a very cold) February 1988 in Lund, a short series
of lectures on Gaussian processes. On the basis that the best way
to understand something is to try to explain it to someone
else, I decided to lecture on Talagrand's results, and, after further
prompting, wrote the notes up. They have grown considerably over the
past two years, one revision being finished in (an even colder)
February in Ottawa, under the gracious hospitality of Don Dawson and
Co., and the last major revision during a much more pleasant summer in
Israel.
I rather hope that what is now before
you will provide not only a generally accessible introduction to majorising
measures and their ilk, but also
to the general theory of continuity, boundedness, and suprema distributions for
Gaussian processes.
Nevertheless,
what these notes are not meant to provide is an
encyclopaedic and overly scholarly treatment of Gaussian sample
path properties.
I have chosen material
on the basis of what interests me, in the hope that this will
make it easier to pass on my interest to the reader. The choice
of subject matter and of type of proof
is therefore highly subjective.
If what you want is an encyclopaedic treatment, then
you don't need notes with the title An Introduction to ...."
This is what I wrote in 1989. What I did not know at the time was that
Ledoux and Talagrand were also writing at the time, and they did
produce an "encyclopaedic treatment", published under the title
Probability in
Banach spaces. Isoperimetry and Processes
During the 1990's I got distracted from random fields and Gaussian
processes for a while, and took a trip into the Markovian-martingalian
world of stochastic partial differential equations, looking at what
became known as superprocesess. To describe these here are a few
paragraphs from a grant proposal I wrote during that time, which, to my
great surprise, was actually funded.
"Once upon a time, there was a random walk, which even undergraduate
probabilists understood. Then, as time went on, it became normalised
in space and time, and, eventually, it converged to the Brownian
motion. By then, only graduate students in probability theory could
really understand what it was, but they, and many others, soon
realised that Brownian motion was a good thing .
For a start, Brownian motion was related to the heat equation
u_t(x,t)=u_{xx}(x,t), and this was good because it helped
probabilists understand Brownian motion and because it helped analysts
understand the heat equation. However, beyond this, it turned out that
Brownian motion was at the heart of many other stochastic
processes. Using it, one could construct all continuous Markov
processes, or all stationary Gaussian processes, and it served as the
archetypical example for wide varieties of other processes.
More recently, there was a branching random walk, which, using their
experience from simple walks and simple Brownian motion, probabilists
immediately ran in speeded up time, and normalised in space and
time. The result, they (eventually) agreed to call super Brownian
motion . Immediately, the world of probabilists, led by the Markov
theorists, realised that super Brownian motion was also a good
thing .
Like its simpler counterpart, it turned out that super Brownian motion
was also related to the heat equation, but this time it was the non-linear
PDE u_t(x,t)=u_{xx}(x,t)- u^2(t,x).
Furthermore, it provided a very nice example of a positive
solution to a stochastic PDE, and of a general state space Markov
process, and had lots of interesting
properties. As a result, since the late 1980's there has been
great activity among probabilists in studying this new
phenomenon.
Here is a very brief description of how to construct a super Brownian motion
via simple, discrete, approximations.
We start with a parameter N>0 that will eventually become large,
and a collection of N initial particles which,
at time zero, are independently distributed in R^d, d\geq 1, according
to some finite measure m.
Each of these N particles follows the path of independent copies of a
Markov process B, with infinitesimal generator A, until time t=1/N.
At time 1/N each particle, independently of the others, either dies or
splits into two, with probability 1/2 for each event.
The individual particles in the new population then follow
independent copies of B, starting at their place of birth, in the interval
[1/N,2/N), and the pattern
of alternating critical branching and spatial spreading continues until,
with probability one, there are no particles left alive.
Consider the measure valued Markov process
Under very mild conditions on B the sequence
{X^N\}_{N\geq 1} converges, in distribution, to
a measure valued process which is called the superprocess for B.
If B is a Brownian motion, then the limit is super Brownian motion."
I left the area of superprocesses after only a few years, since it became
far too technical and difficult for either my tastes or my abilities.
Nevertheless, I did make one very major major contribution to the area,
by giving some initial training to
Leonid Mytnik
in the area and not doing any serious damage to his native abilities in
the process.
For more serious information on superprocesses, the places to look are Dawson's
encyclopedic review, Dynkin's Wald Lectures
(Annals of Probability, 1993)
the two monographs by Dynkin
I and
II , and those by
LeGall.
or Etheridge. Furthermore,
Ed Perkins'
Saint Flour Lecture Notes are highly recommended.
For some pictures, try
Superprocesses: The Movie or some
simulations of simple
superprocesses in
one , two , or three dimensions. (In each one of these simulations you will first
see a simulation of many particles following a simple random walk, with total mass spreading out according to the
discrete heat equation with small random perturbation, followed by branching
random walks meant to simulate superprocesses.)
Here should go all the "other" things I have worked on over the years, but
you can see what they are from my publication list. So let me just list
two recent things that I am proud of. (I can allow myself some pride, since
the real work was done by others.)
One is a collection of well written, and carefully edited (mainly thanks
to Raisa Feldman and Murad Taqqu's hard work)
papers on heavy tailed distributions and processes.
(
A Users
Guide to Heavy Tails: Statistical Techniques for Analysing Heavy
Tailed Distributions and Processes )
This collection gives, I think, a very nice entry point into what is a
rather
important area, both mathematically and from the point of view of real, and
important, applications. I am not going to write out another Introduction: You can get basic information about the book from its website.
A second collection of papers, done together with
Peter Muller and Boris Rozovskii, is about
Stochastic Modelling in Physical Oceanography .
I think this is one of the most important areas that we have, not only
for the application of random field theory, but, perhaps more importantly,
for the motivation to develop new and exciting models.
RANDOM FIELDS
TWO APPLICATIONS
In a
Nobel Prize winning
astrophysics application, random field geometry was used to help analyse the
COBE abd Wmap astrophysical data being used to study the structure
of the premieval universe that followed the supposed `Big Bang'.
The idea here is that 99.97% of the
radiant energy of the Universe was released within the first year after the
Big Bang, and much of the structure of that time is still measurable in
terms of today's background microwave radiation.
Theories that attempt to explain the origin of large scale structure
seen in the Universe today must
therefore conform to the constraints imposed by these observations.
The Figures at the left, taken from a
NASA site show the anomolies in the cosmic
microwave background (CMB) radiation, divided by their standard
deviation. This was the first evidence of
anomolies in the CMB radiation, a sort of signature left over
from the creation of the universe.
A second application of random field geometry is in the area of bio-statistics.
Using either PET or fMRI technologies, images of brain activity
are taken from a number of individuals at rest,
and then when performing a task, such as the silent reading of words projected
on a screen. The underlying principle is that those parts of the brain
involved in performing the task require oxygen, and this surplus of
oxygen can also be detected and imaged. The difference
between the rest and task images, averaged over the subjects, and
standardised at each point by the standard deviation of the sample, is shown
in the figure on the left, where the red regions are regions of
statistically significant
activity.
GAUSSIAN PROCESSES
SUPERPROCESSES
POTPOURRI
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