I'm currently reading through a book on the Fokker-Planck equation:
H. Risken, The Fokker–Planck Equation, 2nd ed., Springer Series in Synergetics, Springer, Berlin, 1996.
$\dots$ and I'm having some confusion over the dimensions of a certain function.
From the book it describes a natural framework for modelling the dynamics of complex systems, in the form of the stochastic differential equation:
$$ \frac{dx(t)}{dt} = D^{(1)}(x,t) + \sqrt{D^{(2)}(x,t)}\Gamma(t) $$
$\dots$ where $x$ is some state variable, $t$ is time, and the two functions that characterize the system are $D^{(1)}(x,t)$ and $D^{(2)}(x,t)$ are the drift and diffusion functions. Also $\Gamma(t)$ is a Gaussian white noise source. This is interpreted in terms of Itô's definition of stochastic integro-differential equations.
My question is about the dimensions of $D^{(2)}(x,t)$. Since $\Gamma(t)$ is just a random variable and has no dimensions, therefore $D^{(2)}(x,t)$ must have dimensions:
$$ \Big[D^{(2)}(x,t)\Big] = X^2T^{-2}. $$
However later when going over the subject of the Kramers-Moyal expansion and estimation of parameters, the following equation is stated regarding the diffusion function:
$$ M^2(x,t,\tau)/2=D^{(2)}(x,t)\tau + \mathcal{O}(\tau^2) $$
$\dots$ with the second moment
$$ M^2(x,t,\tau)=\langle [x(t+\tau)-x(t)]^2\rangle\rfloor_{x(t)=x} $$
$\dots$ so in the limit of $\tau\rightarrow 0$
$$D^{(2)}(x,t) \approx \frac{1}{2\tau} \langle [x(t+\tau)-x(t)]^2\rangle\rfloor_{x(t)=x}.$$
But this indicates that $D^{(2)}(x,t)$ has dimensions $X^2T^{-1}$. Does anyone know why there is this difference, or am I making an error somewhere? Thank you
It has dimensions $X^2T^{-1}$, which comes out of the second calculation. Your error in the first argument was to assume that $\Gamma(t)$ has no units. But the covariance function of $\Gamma(t)$ is $\delta(t-t')$, which has units of $T^{-1}$ (since when it is integrated, it has to become dimensionless). This means that $\Gamma(t)$ itself must have units of $T^{-1/2}$.
Another way of seeing this is to write the equation as an Itô stochastic differential equation (which is mathematically more rigorous), which takes the form
$$dX_t=D^{(1)}(X_t,t)dt+\sqrt{D^{(2)}(X_t,t)}dW_t,$$
where $W_t$ is a Wiener process, which has variance $t$. Thus, the increment $dW_t$ has variance $dt$, which means that $dW_t$ is of the order $dt^{1/2}$. Therefore, the prefactor has to have units $XT^{-1/2}$ to match the units of $dX_t$, which are $X$.