Probability Research
Probability theory borders on many fields both of pure and applied nature, namely, ergodic theory, statistics, mathematical analysis, game theory, physics, geometry, combinatorics, etc.
Combining the idea of randomness with the idea of time or/and space, we arrive at the idea of a random process. Say, the simple one-dimensional random walk is a discrete-time random process. Its scaling limit (when step sizes in space and time tend to zero in an appropriate way) is the famous Brownian motion (discovered by botanist Brown, understood by physicist Einstein, and idealized by mathematician Wiener), the most important continuous-time random process.
The usual (linear) Brownian motion describes a point that moves chaotically but does not rotate. Imagine now a rigid body that rotates chaotically, while its center does not move. Such a "Brownian rotation" is a Brownian motion in the Lie group of all rotations of our three-dimensional space. It is related to the linear 3-dim Brownian motion by some stochastic differential equations.
Instead of a rigid body, imagine now a liquid, or gas, or dust, filling the space, in a chaotic motion. We get a stochastic flow. It may be non-singular, in which case it can be described by a countable system of stochastic differential equations. However, it may also be singular; such singularities as coalescence, stickness, and splitting are under study nowadays. Here we get Brownian motions in "undergroups" beyond topological (semi) groups. They generate "black noises", inherently nonlinear counterparts of the classical white noise generated by the usual Brownian motion.
Ergodic theory grows in response to a challenge of mechanistic determinism of classical physics: the future is uniquely determined by the past, therefore, nothing is random, and nobody has a free will.
Here is a very simple example of a discrete-time deterministic evolution:
| time | n | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| state | xn | .846 | .214 | .582 | .950 | .318 | .685 |
The state xn at time n is a real number of (0,1), and the deterministic evolution is just xn - xn = a (mod 1), where a = e-1=0.3678... is an irrational constant.+1
Imagine that we know the first decimal digit of xn for n = 0, 1, ..., 999 . Can we predict the first decimal digit of x1000 ? It appears that we can do it with a high probability of success (assuming that x0 is chosen at random, uniformly).
Another simple example of a deterministic evolution (still in discrete time, but states are now two-dimensional):
| time | n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| state | xn | .243 | .834 | .259 | .943 | .570 |
| yn | .346 | .590 | .424 | .684 | .627 |
Here xn, yn are real numbers of (0,1), and xn = 2xn + yn , yn = xn + yn (mod 1).+1+1
Imagine that we know the first decimal digit of xn, as well as yn, for n = 0, 1, ..., 999 . Can we predict the first decimal digit of x1000 ? It appears that we cannot.
Each example deals with an ergodic measure-preserving invertible transformation. However, the transformation of the first example is of zero entropy, while the transformation of the second example is of positive entropy.
Do you think that it sheds some light on weather forecasting?
An Example From Mathematical Analysis
Two lands can have a common boundary, but three lands can meet at a point. Does it mean that points of trilateral contact are rare due to some general geometric argument? The question is not simple, since three branching domains of fractal nature (like the circulatory system, branching from artery to capillary vessels) can penetrate a spatious zone of trilateral contact. A right formulation of the problem was found in mathematical analysis (potential theory); however, the best result achieved was that points of 11-lateral contact must be rare. The problem was solved recently by probabilistic means (stochastic analysis plus hypercontractivity).
An auction (namely, a single-unit first-price sealed-bid symmetric auction) is a game with incomplete information. Each player wants to guess bids of others, however, he only knows their (common) probability distribution. Striving to maximize his expected profit (or, more generally, expected utility), he adheres to his optimal strategy computed by solving a differential equation. That is the theory. Does it conform to reality? That can be verified (to some extent) by a statistical test, even though the only empirically observable values are bids.
