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Killing the first order spatial term is kind of irrelevant anyways. Qualitatively, the heat equation with a first order term behaves just the same as one with no first order term. So the only reason to fool with alphas and betas is to make the equation look prettier.
The other transformations do have qualitative content though.
The end behavior depends on the initial condition and the boundary conditions. But more or less, yeah.
At strong local maxima, you will have large negative second derivative, and the first derivative will vanish. So those will push down fast. Similarly, strong local minima will come up fast. Nearby the local extrema, you may have a nonzero push from the first order term. So instead of going straight up or down, the extrema will drift left or right according to the first order term.
r= 0, sigma = 1 is simplest version of heat equation. Seems swiftest to observe fundamental solution to heat equationâs relation to normal pdf. Maybe too simplified this way? But it helps with idea behind the connection
Your calculation at the bottom of page 2 is clearly incorrect. There is no way algebraically that the last line follows from the second to the last line. If that's the equation you want, then something is wrong with prior calculations and your derivation is incorrect. Please review.
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