Nonlinear diffusions and the KPP equation
July 12, 2015 by zulfahmed
Dear Professor Donoho,
I learned that the form in which I had been seeking for volatility dynamics is equation in one dimension studied by Kolmogorov-Petrovsky-Piskunov in the 1930s and in multidimensions by Aronson-Weinberger and many others
(d/dt) u = (d/dx)^2 u + f(u)
It is clear that I must study the research more carefully in order to understand when there are phase transitions to turbulence precisely for f(u); we know this is the case when f(u) = kappa*u*du/dx as in Navier-Stokes but for f(u) = kappa*u^gamma, gamma>0 I have more study to do with the attached papers for guidance. I thank you for patience, sir, as I am self-studied and did not know the nonlinear parabolic equation literature.
 Aronson, D. G.; Weinberger, H. F. Multidimensional nonlinear diffusions arising in population genetics. Adv. in Math. 30 (1978), no. 1, 33–76.
 Kolmogorov, A. N.; Petrovsky, I. G.; Piskunov, N. S. Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d’Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Série Internationale, Section A 1 (1937), 1–26.