Diffusion constant $D$ as a function of step size and time step for random walk Consider the random walk in 1D with step size $\epsilon$ and time step $\delta t$. 
I've found two conflicting answers to my question. Wikipedia says 
$$ D = \dfrac{\epsilon^2}{2\delta t}\tag{1}$$
while Gardiner's Handbook of Stochastic Methods (3rd edition) says in section 3.8.2 on the Random Walk in 1D after eq. (3.8.45), "we see clearly the expression of $D$ as the mean square distance travelled per unit time" or
$$D = \dfrac{\epsilon^2}{\delta t}\tag{2}.$$
Which is correct? I suspect Wikipedia might have an error but I wanted to double check. 
 A: Universal constant factors do not really matter. You can always redefine the diffusion constant by a universal constant factor (independent of the system) and then update the formulae in which it appears to cancel out that constant factor. 
As for why these two definitions are both reasonable, the answer is related to a rather pervasive discrepancy between PDE and probability theory. A PDE person would want to see
$$u_t=D u_{xx}$$
with just one constant, instead of 
$$u_t=\frac{1}{2} D u_{xx}.$$
A probabilist on the other hand would like to see that the variance of the increments in a time interval of length $1$ is just $D$. These aren't compatible because of this factor of $2$. That is, a PDE person will tend to define the diffusion constant the first way, while a probabilist will tend to define it the second way. 
Two other places where this factor of $2$ shows up include the Ito formula (where you wind up seeing it either as a $1/2$ in a PDE or a $\sqrt{2}$ in a SDE) and the Gaussian density (where the denominator in the exponent is not $\sigma^2$ but $2\sigma^2$).
