ERRATA
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Note from a reader:


Hi Andrew,

I happened upon your notes while looking for a clean derivation of the FPE, with
particular attention to Ito vs Strat. I found them useful, but your last section
4 is not really correct. You've calculated M1 and M2 after the Kramers-Moyal
expansion in terms of Ito. If you had started with an SDE dx = \mu dt + b dW,
and used strat. rules, you DO get the diffusivity inside one derivative only.
The probability flux you (and Risken and Gardiner) have, J = M1*P - (M2 P)'
really has to be defined under Ito.

The real one physicists use is J = M1*P - M2*P', which jives with linear
response theory and the convection-diffusion equation (which has to reduce to
the FPE for noninteracting particles).  One does get this starting from an SDE
(or langevin Eq) defined under strat.

Your notes, along with others actually helped me resolve all this....which took
me years!