mathematical theory results for instability for nonlinear fractional diffusion
July 16, 2015 by zulfahmed
Zulfikar Ahmed <zulfikar.moinuddin.ahmed@gmail.com>


7:03 AM (0 minutes ago)




Dear Professor Donoho,
We are very lucky and there is a very recent paper from 2015 in arxiv by Roquejoffre and Tarfulea that give theory results for phase transition to turbulence in nonlinear fractional diffusions in the sense that the level sets of the solution with compactly supported initial value has exponential radius in time. This is extremely welcome for our project of showing that turbulence phase transition for global financial volatility is a quantitative feature of continuoustime finance without having to repeat very good work already done to show this in numerical simulations (see snapshot) and
This result uses fractional power of Laplacian, and perhaps could have extensions to timefractional case as well. With this result we reduce our work greatly for the FisherKPP reaction nonlinearity. This does not necessarily cover all nonlinearities but for nonfractional diffusions, it is known that the normal lines from the level sets pass through the convex hull of initial compact support by
[9] C.K.R.T. Jones, Asymptotic behavior of a reactiondiffusion equation in higher space dimensions, Rocky Mountain J. Math. 13, (1983), pp. 355364.
The numerical simulations refer to the Ph.D. thesis of AC Coulon.
[7] A.C. Coulon, Fast propagation in reactiondiffusion equations with fractional diffusion, PhD thesis (Universit´e Paul Sabatier and Universitat Politecnica de Catalunya), 2014.
This is extremely interesting and gives us concrete phase transition to turbulence phenomena for KPP nonlinearity which is a function of u only and not its gradient as in NavierStokes. So our work on movement toward simulations is still valuable and in fact we can be guided by this result to ask the same question in R^d with nonlinearities that can be detected from data.
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Zulfikar Ahmed <zulfikar.moinuddin.ahmed@gmail.com>






holding off on doing all practical things till clarity regarding volatility turbulence phase transitions occurs
The BIG questions of S4 physics and the mind are too much for me to handle right now.
Onsager 1940s must guide the path
We now know with mathematical results that NavierStokes is not necessary for turbulent phase transition which is fantastic! Nonlinear fractional FisherKPP does it.
We know we must pythoncode the MATLAB code which is boring but has to be done
So let’s just think through this:
Consider forcing term classes: we are on R^d and say we show that some class F of forcing terms produce turbulent phase transitions just like FisherKPP, which we expect to be pretty broad but we don’t want to do all the work to show these because we want to keep focus on the goal of quelling volatility storms and not get distracted by ANYTHING.
Then we want to understand what models of continuoustime volatility can match data, i.e. find the F from data on a complex network rather than on R^d.
Let’s assume that any nonlinear F arises. At this point we have challenged the fundamental models of quant finance of a century but have not hit the prize of phase transition to turbulence. Of course it may be more messy that we have range of F that arise and some of these F lead to phase transitions to turbulence and others don’t.
So let’s be conservative and suppose that this a messier problem of this type. Then actually there is a less clean approach: STATISTICAL MODELS for estimating such things from data. This may be the obvious escape route from more sleepless nights.
Well maybe we have not clinched victory yet but we’ve made giant strides in this problem in terms of thought in uncharted territory, which is not as beautiful as the explanation of potential of Laplace geometrically — and you are a sweetheart for being there for that too.
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