In the progression of ideas about nature in diffusion theory, from the first formulation in 1855 after 1822 invention of Joseph Fourier, it had done well in the linear form with Schroedinger. Schroedinger came up with the heat equation with a potential that’s homogenous. This is the linearity assumption of quantum mechanics. The rest of the infrastructure besides the introduction of Dirac operators whose geometry was unearthed by Michael Atiyah who deserved credit for having defined them and proved wonderful things about them in geometry. Navier-Stokes and nonlinear partial differential equations without Schroedinger type linear term but rather more interesting nonlinearities are the bread and butter of anomalous diffusions. To understand these things at all gives one — or at least me–a giant headache. I can understand things when they are clear and simple. For me, here there is a long weaning off western science from linear partial differential equations not simply as a curiosity for fun (which is really what I like) but because the foundations of science need a clean overhaul once in a while and these transitions are always anarchic as Paul Feyerabend will tell you in no uncertain terms in Against Method. The general idea that the observations dictate science and there is a coherent structure of the edifice is a total illusion. We have here anarchic crisis and the old guard don’t want to be the next Aristotle when Descartes comes along. Quantum mechanics and gravitation never fit for very good reason that quantum mechanics is a damn linear approximation of something else, but what can it be? Well since we are not beholden to any physics dogma, let’s be completely crazy and claim boldly and totally intuitively that the solution lies in the residuals of the power law Poisson process on volatility data, where you get a pretty damn good idea of what the power law approaches to modeling natural phenomena that Mandelbrot pushed correctly in my view does: it leaves out a clear exact something or other that looks like this:

So this could be quite wrong but this is a tiny residual in volatility for power law on data for 1900 stocks integrated shock distribution of the entire complex network of the financial markets. I’ve worked in finance since 1995 on and off mostly not paying too much attention to too many things, but this sort of connection when one looks at anomalous diffusion which is a real science — check out metzler-klafter-anomalous-diffusion with an excellent presentation of the literature and history. To the extent that this entire movement toward ‘fractional’ proceeds based on the dogma of everything is power law in physics which is the rage it seems, this residual tells a different story. Obviously one needs to move to pseudodifferential operators with nonpolynomial symbols but which ones are good enough. This residual graph suggests that we want mixtures of powers and exponents of powers.

Of course exact formulas are harder and so on but that’s not so important as to get the right nonlinearities in some exact theory too. This is volatility showing a parametric form in power law residuals, not hard science data. This is an opportunity to get some information about where Mandelbrot did not manage to get to.

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