Suppose I simulate a random walk with a random step of mean size $\sigma$ every time interval $\Delta t$. The step may be of fixed size, or it could be drawn from a distribution (with $\sigma$ representing the average step size), e.g. a normal distribution with variance $\sigma^2$, and it could be one- or two-dimensional.
Now I want to change the temporal discretization, i.e., change the size of $\Delta t$. I will have to change the size of $\sigma$ accordingly to get a random walk of similar behavior. In fact, I will have to keep $$\nu = \frac{\sigma^2}{\Delta t}$$ constant in order to get a similar behavior. What I am missing is a "name" for this quantity $\nu$. One could call it "variance growth rate"... but I guess that out there in "random walk world" there is surely a standard name for this quantity. It describes a kind of "speed" of the random walk (increasing $\nu$ gets the walker walk faster), but its units are not $m/s$ but $m^2/s$. Kinematic viscosity shares this unit... but I don't see any connection.
In the physics world I think $\nu$ is the diffusion coefficient but I may have a factor of two missing.