
10:22 PM (0 minutes ago)



Dear Professor Donoho,
As I learn more about fractional partial differential equations, I become aware of work of the partial differential equations in the R^d case that is applicable, which are fractional diffusionreaction equations which have recent development. My intuitive and simplistic use of (Laplacian)^beta is made precise in the R^d case by the RieszFeller fractional derivatives which can be defined as a pseudodifferential equation using Fourier transform on R^d as
the operator D^{beta, theta}_x such that it’s Fourier transform is multiplication by
psi_{beta,theta}( xi ) = – abs( xi )^beta exp( i * sign(xi) theta* pi/2 )
This behaves as expected when beta=2 and theta=0 when it’s abs(xi)^2 for (d^2/dx^2). This, we are told (snapshot) is the logarithm of the characteristic of the generic Levy stable probability density in the parametrization of Feller ([1] Feller, W. (1952). On a generalization of Marcel Riesz’ potentials and the semigroups generated by them. Meddelanden Lunds Universitets M a t e m a t i s k a S e m i n a r i u m , ( C o m m . S ́e m . M a t h ́e m . U n i v e r s i t ́e d e Lund), Tome suppl. d ́edi ́e a M. Riesz, 7381.) The attached 2014 paper studies reactiondiffusion equations fractionalized to find rigorous R^d theory of the equations I have been calling ‘NavierStokes type’. http://arxiv.org/pdf/1409.1817.pdf
This paper is perfect in terms of making rigorous the types of FPDE that interest us for applications to volatility modeling modulo the issue of phase transition to turbulent dynamics and I will study this paper in more detail. Our interest is to carry over to complex networks these types of equations from space R^d.
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