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I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole proof, on the next page only the formula (3.10) is repeated.

a proof

My problem is, I do not see how to get equation (3.10) from (3.11). I did some computations, to no avail. I have checked the original work of McKean ([44]), but there explanation is even more concise. We may write (3.11) in the form $$ u(t,x) = e^{-t} H_t f(x) + \int _0 ^t e^{-s} H_s u^2 (t-s, x ) ds, $$ where $H_t g(x) = \frac{1}{\sqrt{2 \pi t} } \int e^{\frac{-z^2}{2t}} g(x-z) dz $.

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    $\begingroup$ Search for "mild formulation" of semilinear PDEs or "Duhamel formula". (Performing the substitution $s \mapsto t-s$ in the second term might help you to see that (3.11) is precisely such a mild formulation.) $\endgroup$
    – Martin Hairer
    Commented Jul 4, 2014 at 13:43

1 Answer 1

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The answer given by Martin Hairer is perfect. But if you are not familiar with the not familiar with pdes, just differentiate the right hand side wrt to t. Note that when you differentiate the integrand in the second line wrt to, that is the same as differentiating wrt to s and multiplying by minus one. Then do a partial integration to through the derivative on e^{-s}times the heat kernel. Finally use the differential equation that the heat kernel satisfies. Then use partial integration for the z-derivatives again and observe that then acting on the u, they are the same as x-derivatives. If you collect all terms carefully, the KPP equation pops up. To get into it, make sure you understood how Lemma 3.2 and 3.3 are proven.

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