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Previous set of notes: Notes 3.
Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.
In these notes, random variables will be denoted in boldface.
Definition 1 A real random variable
is said to be normally distributed with mean
and variance
if one has
for all test functions
. Similarly, a complex random variable
is said to be normally distributed with mean
and variance
if one has
for all test functions
, where
is the area element on
.
A real Brownian motion with base pointis a random, almost surely continuous function
(using the locally uniform topology on continuous functions) with the property that (almost surely)
, and for any sequence of times
, the increments
for
are independent real random variables that are normally distributed with mean zero and variance
. Similarly, a complex Brownian motion with base point
is a random, almost surely continuous function
with the property that
and for any sequence of times
, the increments
for
are independent complex random variables that are normally distributed with mean zero and variance
.
Remark 2 Thanks to the central limit theorem, the hypothesis that the increments
be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).
Real and complex Brownian motions exist from any base point or
; see e.g. this previous blog post for a construction. We have the following simple invariances:
Exercise 3
- (i) (Translation invariance) If
is a real Brownian motion with base point
, and
, show that
is a real Brownian motion with base point
. Similarly, if
is a complex Brownian motion with base point
, and
, show that
is a complex Brownian motion with base point
.
- (ii) (Dilation invariance) If
is a real Brownian motion with base point
, and
is non-zero, show that
is also a real Brownian motion with base point
. Similarly, if
is a complex Brownian motion with base point
, and
is non-zero, show that
is also a complex Brownian motion with base point
.
- (iii) (Real and imaginary parts) If
is a complex Brownian motion with base point
, show that
and
are independent real Brownian motions with base point
. Conversely, if
are independent real Brownian motions of base point
, show that
is a complex Brownian motion with base point
.
The next lemma is a special case of the optional stopping theorem.
Lemma 4 (Optional stopping identities)
- (i) (Real case) Let
be a real Brownian motion with base point
. Let
be a bounded stopping time – a bounded random variable with the property that for any time
, the event that
is determined by the values of the trajectory
for times up to
(or more precisely, this event is measurable with respect to the
algebra generated by this proprtion of the trajectory). Then
and
and
- (ii) (Complex case) Let
be a real Brownian motion with base point
. Let
be a bounded stopping time – a bounded random variable with the property that for any time
, the event that
is determined by the values of the trajectory
for times up to
. Then
Proof: (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take . Let
be an upper bound for
. Since
is a real normally distributed variable with mean zero and variance
, we have
and
and
By the law of total expectation, we thus have
and
and
where the inner conditional expectations are with respect to the event that attains a particular point in
. However, from the independent increment nature of Brownian motion, once one conditions
to a fixed point
, the random variable
becomes a real normally distributed variable with mean
and variance
. Thus we have
and
and
which give the first two claims, and (after some algebra) the identity
which then also gives the third claim.
Exercise 5 Prove the second part of Lemma 4.
One theme in this course will be the central nature played by the gaussian random variables . Gaussians have an incredibly rich algebraic structure, and many results about general random variables can be established by first using this structure to verify the result for gaussians, and then using universality techniques (such as the Lindeberg exchange strategy) to extend the results to more general variables.
One way to exploit this algebraic structure is to continuously deform the variance from an initial variance of zero (so that the random variable is deterministic) to some final level
. We would like to use this to give a continuous family
of random variables
as
(viewed as a “time” parameter) runs from
to
.
At present, we have not completely specified what should be, because we have only described the individual distribution
of each
, and not the joint distribution. However, there is a very natural way to specify a joint distribution of this type, known as Brownian motion. In these notes we lay the necessary probability theory foundations to set up this motion, and indicate its connection with the heat equation, the central limit theorem, and the Ornstein-Uhlenbeck process. This is the beginning of stochastic calculus, which we will not develop fully here.
We will begin with one-dimensional Brownian motion, but it is a simple matter to extend the process to higher dimensions. In particular, we can define Brownian motion on vector spaces of matrices, such as the space of Hermitian matrices. This process is equivariant with respect to conjugation by unitary matrices, and so we can quotient out by this conjugation and obtain a new process on the quotient space, or in other words on the spectrum of
Hermitian matrices. This process is called Dyson Brownian motion, and turns out to have a simple description in terms of ordinary Brownian motion; it will play a key role in several of the subsequent notes in this course.
I am very saddened (and stunned) to learn that Oded Schramm, who made fundamental contributions to conformal geometry, probability theory, and mathematical physics, died in a hiking accident this Monday, aged 46. (I knew him as a fellow editor of the Journal of the American Mathematical Society, as well as for his mathematical research, of course.) It is a loss of both a great mathematician and a great person.
One of Schramm’s most fundamental contributions to mathematics is the introduction of the stochastic Loewner equation (now sometimes called the Schramm-Loewner equation in his honour), together with his subsequent development of the theory of this equation with Greg Lawler and Wendelin Werner. (This work has been recognised by a number of awards, including the Fields Medal in 2006 to Wendelin.) This equation (which I state after the jump) describes, for each choice of a parameter , a random (fractal) curve
in the plane; this random curve can be viewed as a nonlinear variant of Brownian motion, although the SLE curves tend to cross themselves much less frequently than Brownian paths do. By the nature of their construction, the
curves are conformally invariant: any conformal transformation of an
curve (respecting the boundary) gives another curve which has the same distribution as the original curve. (Brownian motion is also conformally invariant; given the close connections between Brownian motion and harmonic functions, it is not surprising that this fact is closely related to the fact that the property of a function being harmonic in the plane is preserved under conformal transformations.) Conversely, one can show that any conformally invariant random curve distribution which obeys some additional regularity and locality axioms must be of the form
for some
.
The amazing fact is that many other natural processes for generating random curves in the plane – e.g. loop-erased random walk, the boundary of Brownian motion (also known as the “Brownian frontier”), or the limit of percolation on the triangular lattice – are known or conjectured to be distributed according to for some specific
(in the above three examples,
is 2, 8/3, and 6 respectively). In particular, this implies that the highly non-trivial fact that such distributions are conformally invariant, a phenomenon that had been conjectured by physicists but which only obtained rigorous mathematical proof following the work of Schramm and his coauthors.
[Update, Sep 6: A memorial blog to Oded has been set up by his Microsoft Research group here. See also these posts by Gil Kalai, Yuval Peres, and Luca Trevisan.]
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