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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.