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Nonlinear fractional diffusions with forcing term $F=z^2-z$ are addressed by http://arxiv.org/pdf/1202.6072.pdf.  Analytic properties of Feller semigroups for the linear problem are used here which are more generally useful for more general nonlinearities.  One of the key points is that the KPP nonlinearity has Lipschitz bounds for the actual problems which are not available more generally for $F$.  We need to study the properties of Feller semigroups for $2\alpha$-stable Levy jump processes more carefully to understand instabilities in financial markets.