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I have a lot of numerical data which I'm looking to characterise as a (possibly continuous) random walk with variable (in space) step size, for example, along $x$ between $-1$ and $1$ with a step size of $1-x^2$.

In particular I'm interested in the diffusion coefficient $(\Delta x)^2/\Delta t$ as a function of $\Delta t$.

My feeling is that I can change variables from $x$ in order to make the step size constant, and then I can just get the variance and transform back. Unfortunately all the literature a quick search finds is in a financial context which was inscrutable.

(I think this question might be simple, but I am a physicist, and not even a very good one, and so I am at the stage where I don't even know the names of what to look for.)

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