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6 1 Introduction
In view of these developments, we believe that it is an opportune time to try to
summarise some of the main results of the theory in a way that is as elementary as
possible, yet sufficiently precise to provide a technical working knowledge of the
theory. We therefore include elementary but essentially complete proofs of several
of the main results, including the continuity and definition of the It
ˆ
o–Lyons map,
the lifting of a class of Gaussian processes to the space of rough paths, etc. In
contrast to the available textbook literature [
LQ02
,
LCL07
,
FV10b
], we emphasize
Gubinelli’s view on rough integration [
Gub04
,
Gub10
] which allows to linearise
many considerations and to simplify the exposition. That said, the resulting theory
of rough differential equations is (immediately) seen to be equivalent to Davie’s
definition [
Dav08
] and, generally, we have tried to give a good idea what other
perspectives one can take on what amounts to essentially the same objects.
1.2 Analogies with other branches of mathematics
As we have just seen, the main idea of the theory of rough paths is to “enhance”
a path
X
with some additional data
X
, namely the integral of
X
against itself, in
order to restore continuity of the It
ˆ
o map. The general idea of building a larger
object containing additional information in order to restore the continuity of some
nonlinear transformation is of course very old and there are several other theories
that have a similar “flavour” to the theory of rough paths, one of them being the
theory of Young measures (see for example the notes [
Bal00
]) where the value of
a function is replaced by a probability measure, thus allowing to describe limits of
highly oscillatory functions.
Nevertheless, when first confronted with some of the notions just outlined, the
first reaction of the reader might be that simply postulating the values for the right
hand side of (1.8) makes no sense. Indeed, if
X
is smooth, then we “know” that there
is only one “reasonable” choice for the integral
X
of
X
against itself, and this is the
Riemann integral. How could this be replaced by something else and how can one
expect to still get a consistent theory with a natural interpretation? These questions
will of course be fully answered in these notes.
For the moment, let us draw an analogy with a very well established branch of
geometric measure theory, namely the theory of varifolds [Alm66, LY02].
Varifolds arise as natural extensions of submanifolds in the context of certain
variational problems. We are not going into details here, but loosely speaking a
k
-dimensional varifold in
R
n
is a (Radon) measure
v
on
R
n
× G(k, n)
, where
G(k, n)
denotes the space of all
k
-dimensional subspaces of
R
n
. Here, one should
interpret
G(k, n)
as the space of all possible tangent spaces at any given point for
a
k
-dimensional submanifold of
R
n
. The projection of
v
onto
R
n
should then be
interpreted as a generalisation of the natural “surface measure” of a submanifold,
while the conditional (probability) measure on
G(k, n)
induced at almost every point
by disintegration should be interpreted as selecting a (possibly random) tangent
space at each point. Why is this a reasonable extension of the notion of submanifold?