Feynman Path Integral and Diffusion Process --- similarity of mathematical tools in
quantum physics to those in stochastic processes
Chapter 4 of this book derives Schrodinger equation from Feynman path integral for a
particle moving in a field of potential. If we remove the imaginary "i", the mathematics can be carried over
to the study of diffusion process. This was done by Kac in 1949 on the Feynman-Kac formula.
When studying Markov jump process and diffusion process, we can discretize time into small time
intervals, just like a movie is actually discretized into frames. A key concept of such processes
is the so-called "generator" A. In jump process, the generator A can be interpreted as transition rate,
so that the one-frame transition matrix is K(dt) = I + A dt (here dt means \Delta t,
which is a finite but small time priod, and we omit higher order terms). Thus the long-term transition matrix is
K(t) = K(dt)^{t/dt} = (I + A dt)^{t/dt} = e^{At}. This is the so-called semi-group representation.
Such a symbolic representation is also used to represent the solution to Schrodinger equation,
|psi(t)> = e^{-iHt/h} |psi(0)>, so that in Heisenberg picture, O(t) = e^{iHt/h} O e^{-iHt/h}.
It is often easy to write down A for jump process, where the state space is discrete.
In diffusion process, the state space is continuous. We can further discretize the spatial domain,
just like each frame of a movie is discretized into pixels. Then we see that the transition rate A
is actually in the form of second derivative or Laplacian for Brownian motion. The generator A also
involves first derivative for more complex diffusion process described by stochastic differential
equations.
Another way to think about A is to interpret the transition matrix K(t) as a "verb", so that for
a function f, K(t) can act on f, so that g = K(t)f becomes another function, where
g(x) = (K(t)f)(x) = E[f(X(t))|X(0)=x]. The right hand side is a local average of f
around x, i.e., we let a population of random walkers start from x, and let them walk for a
time period t, and then we compute the average of f values of where they end up. Thus g = K(t)f is a smoothed
or massaged version of f. So the "verb" meaning of A is the smoothing rate or massaging rate.
The calculation of Af = (K(dt)f-f)/dt involves the second-order Taylor expansion because E[dB_t] = 0
and E[dB_t^2] = sigma^2 dt for Brownian motion B_t. It can be easily shown that A involves second
derivative operator, thus A is naturally a "verb".
From K(t) = e^{At}, we can write down the Kolmogorov forward and backward equations for the jump
process, dK/dt = KA and dK/dt = AK. For Brownian motion, the corresponding equations become heat
equations. Einstein derived the heat equation in the form of a forward equation in his study of
Brownian motion. Usually the forward equation is harder to derive than the backward equation,
because it involves transposing A or finding the adjoint of A using integral by part. One can always
establish the correspondence
between the stochastic differential equation that describes a single random walker
and the partial differential equation that describes the whole population of random walkers.
Sometimes it is easier to solve SDE, and sometimes it is easier to solve PDE. Of course, this
also depends on one's own background.
The Feynman-Kac formula involves a partial differential equation du/dt = (A-V)u, with u(0, x) = f(x).
This is similar to the backward equation dK/dt = AK, because if we let u = Kf, then du/dt = Au. So
u(t) = e^{At}f = K(t)f, and u(t, x) = (K(t)f)(x) = E[f(X(t))|X(0)=x]. That is, we solve PDE by first
solve SDE to obtain K(t), and then calculate conditional expectation. In Feynman-Kac, there is
an extra potential function V. But the solution of
u(t, x) is similar: u(t, x) = E[f(X(t))e^{\int_{0}^{t} V(X(s))ds|X(0)=x], where there is an extra term
e^{\int_s V(X(s))ds}. The Feynman-Kac formular can be proved,
again by second order Taylor expansion. It allows us to solve PDE by solving SDE and compute
the solution of PDE by expectation, i.e., the path integral or sum over histories.
We can map the above PDE to Schrodinger equation with Hamiltonian p^2/2m + V(x) =
-h^2/2m d^2/dx^2 + V(x), and the expectation formular
to Feynman path integral with Langrangian m(x^dot)^2/2 - V(x), where we
add back the imaginary "i" as well as Planck constant h.
The Feynman-Kac formular is also useful for deriving Black-Scholes formula for option pricing.
The equation du/dt = Au and the solution u(t) = e^{At}u(0) is essentially the same as the calculation
of long term interest from the interest rate. So we can play with the interest rate r and generator
A together and at some point replace e^{(T-t)A} f by K(t, T) f, which can be calculated by
conditional expectation, very much like the path integral formulation.
In Feynman's formulation, he went through the following steps. (1) discretize time into movie frames;
(2) calculate the integral I, which is a finite many fold integral; and (3) calculate the limit of
I as dt --> 0. It is unnesessary to let dt --> 0 in the second step, because then one has to definite
a measure (i.e., volume) over the infinite dimensional space of continuous paths.
I always feel that it makes life much easier to take
the limit in the third step for solving stochastic differential equation, where the first step
means to change stochastic differential equation to difference equation or time series.
In the second step, the Ito calculus means dB_t^2 = dt as far as integral or sum is concerned.
The rigorous definition of the measure as well as stochastic differential equation
is mathematically important, but it takes too many pages in the
textbooks, and it is not useful for calculation. In fact, it does not even make sense in reality. In
Brownian motion, at the time scale of 10^{-2} second, the dust particle is bombarded by a huge number of water molecules.
But at the time scale of 10^{-12} second, the picture changes completely where things become discrete, either there is a
collison or there is none. Also, in financial market, the transition of a trader is made in discrete time. In
both cases, we take the limit in the last step as an approximation to the discrete time reality.
Despite the similarity between Feynman and Kac (so to speak), Feynman is about probability amplitude
because of the imaginary "i", and Kac is about real transition probability. Feynman made this
difference clear in his thesis.
There are two other mathematical tools in quantum theory that are commonly used in probability and
statistics. One is of course the eigen decomposition of Hermitian matrices or operators, which is
the same mathematics underlying principal component analysis and spectral analysis of Markov process.
The other is the Gaussian integral in path integral treatment of free field, which is the same mathematics for multivariate
Gaussian distribution.