Brownian motion first appeared in biology. Later Einstein, Smoluchowsk and other people used heat equation to derive some equations for Brownian motions. They viewed Brownian motion as some diffusion process. Later Wienner gave a rigorous foundation of Brownian motions, he constructed the Wienner measure on the space of the paths of the Brownian motions(in general situation is the space , consisting of continuous functions with value at time ). In economics, some economists proposed some models which turned out to be Brownian motion, too.
The story did not stop here. Another aspect came from quantum mechanics. Feynman developed his own formulation of quantum mechanics using the Feynman integral. Later Kac found that removing the factor in the Feynman integral gives a solution to the heat equations.
So Feynman integrals, Brownian motions are related by the heat equations, or diffusion processes.
In this post I will try to give several formulations of the Brownian motions from different aspects.
The first approche relies on Gaussian measures on Hilbert spaces. So, we need first say something on this.
On the real line , we have the standard Lebesgue measure with . Then we define the Gaussian measure to be where are two constants with .
For superior dimensions, we say that a probability measure on is Gaussian if there is a linear map such that the induced measure on , is Gaussian. We can show that the Fourier transform of a Gaussian measure is where is a vector in and is a positive operator on . In fact, there is a one-to-one correspondence between the set of Gaussian measures and the set of the pairs , so we can write .
We can continue this process to any dimension, even infinite dimension. That is to say, if is a real Hilbert space, then a probability measure on is said to be Gaussian if there is a continuous linear map, such that is a Gaussian measure on the real line. Just like the above statements, we can show that there exists a mean vector and a covariance operator (a vector is a mean vector if for all , assuming that given , the function is integrable. And a covariance operator is a symmetric, positive definite operator such that , assuming again some integrability). We can show that is a nuclear operator, which, roughly speaking, is so operator for which the trace can be defined and is finite, and is independent of the choice of basis. Conversely, we can show that for any such pair with a nuclear opeartor, there is a unique probability measure on which is Gaussian and . In the same way, we write .
To define Brownian motion, we set (equipped with the usual Lebesgue measure) and where is a symmetric, positive definite nuclear operator on . Thus we can define the square root of , which we choose to be positive, too(). For any , we write the characteristic function of the interval, . We can show that is a dense subspace of (what is more, we have that, , this is a remarkable fact). So, there is a sequence such that . Then we define .
Then I say that is a Brownian motion. This means that is a random variable on , and for , are independent and . These identities are not difficult to prove.
In the next post, I will try to establish Brownian motion from the point of view of diffusion process.