# Discussion about Hairer's existence theory for stochastic differential equations

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The discussion started below this submission creation request, but it is better deserves a separate chat thread.

asked Aug 15, 2014 in Chat

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http://arxiv.org/abs/1303.5113 also http://arxiv.org/abs/1401.3014 , there are also great later articles by Hairer citing these. These are, to my mind, the biggest step in mathematical quantum field theory since Wilson.

answered Aug 15, 2014 by (7,550 points)

The first one is a book-sized paper. Could you please point to a particular result relevant for quantum YM? (The word ''Yang'' is in none of the two papers.)

The first (2013) work says on p.4 in Remark 1.1 that they obtain a nonperturbative Euclidean equivalent of superrenormalizable QFT. Thus in 4D this is relevant at best to scalar QED, which is (apart from its relatives) the only superrenormalizable theory in 4D.

It won't contribute to the solution of the YM problem.

''The details of the method requires superrenormalizability for the time being, but this is just a technical limitation'' The same was claimed for the methods of constructive field theory, which could construct $\Phi^4_3$ long, long ago but then got stuck. The technical limitations are enormous! I believe true progress only when someone actually overcomes that barrier.

Nevertheless, the treatise is good, and valuable for the analysis of true SDE (rather than QFTs).

@RonMaimon, for the record Hairer just got Fields prize yesterday for this regularity stuff.

Oh wow! He got recognized so quickly? Great! congrats to him.

But this is Euclidean $\Phi^4_3$, which is unrelated to its Minkowski version unless $O(4)$ invariance and reflection positivity are shown.

O(4) invariance is really automatic, as it is present in O(4) invariant regulators (his regulator is a smooth function convolved with the noise, you can make it a rotationally invariant bump, he always does in his specific examples anyway), and the convergence was established to be  independent of regulator (this is one of the great advances of his formalism, the form of the regulator clearly doesn't matter in taking the limit). The thing that he constructs is an SPDE solution, with a good (distributional) continuum limit, whose long-time solution, if you take a constant time slice, has statistics that eventually converges to a random pick from $\phi^4_3$ path integral. These are already known to be reflection positive from other work, and it should be easy to show it directly, you asked about it, let an answer come. I believe (not sure, haven't thought at all) that with appropriate reflection positive statistics on the initial conditions, it is extremely easy to show reflection positivity  for all time slices and for all regulators, not just asymptotically at the large t limit.

There is no way reflection positivity is going to stop the program (unlike, say, the superrenormalizability business which requires a real idea advancing this stuff). The program constructs the stochastic quantized field theory in what can only be called 'the right way'.

Any difficulty in turning the SPDE construction into a field theory construction can only lie in establishing that the long-time limit is converging (but it is true that the long time limit will be converging to the appropriate thing). You could, for example, define the SPDE for $\phi^3$ field theory in 4d, and it should be ok as an SPDE, the theory is superrenormalizable, but it would not converge at large times to anything, the solutions would run away to $\phi\rightarrow -\infty$.

It might also have a finite time limit to the SPDE evolution, there are no results that guarantee global solutions, as sometimes they won't exist. The results he proves show local existence of SPDE solutions, and you need a further estimate for global solutions. This one of the places where he focuses his current research, connecting the SPDE to the stationary distribution at long times. Again, he is interested in SPDEs, but the formalism is general enough to apply to any (bosonic/real-Euclidean-action) QFT.

I don't claim that Hairer solved all the problems of constructing QFTs at one stroke, he just solved the main one--- defining a rigorous renormalization procedure that is completely non-perturbative, easy to work with and prove things about, is general enough that it should work precisely in those cases when traditional renormalization does, and corresponds exactly to what physicists know about renormalization (it's essentially a framework for producing convergence from a rigorous version of the OPE--- the OPE defines the possible renormalization terms in products of regularized distributions, and then the analysis parts just show the statistical convergence when the regulator is relaxed).

Because he is doing stochastic quantization, his renormalization for the stochastic time evolution is completely separately from constructing a vacuum state, and it works even when there is no vacuum state. The two problems are decoupled, and this allows you to think about short-distance renormalization without considering long-distance properties, like confinement. This separation property is central, it's what allowed the Glimm/Jaffe program to succeed too, but there the separation was intrinsically due to the superrenormalizability, it wasn't something that could be generalized. Here it can be generalized to any nonlinear SPDE even stochastically quantized Yang-Mills theory.

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