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I have been learning about the theory of regularity structures, for which the common motivation is Taylor series. However, I keep seeing direct sums in the definition of a regularity structure, which makes me think that they are really about finite sums of basis functions / noise.

Question: Do regularity structures for common SPDEs provide infinite Taylor-like expansions of solutions to the SPDE? Or are they finite?

A reference would be really useful. Thank you!

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    $\begingroup$ You are right that regularity structures are set up for Taylor-like jets of finite order. The right starting analogy is describing Holder functions of order $\gamma \in (k, k+1)$ by their Taylor jet of up to order $k$ with a good estimate on the remainder (see e.g. Section 13.1 in this book of Friz and Hairer). In the same way that solving a smoothly driven PDE in some Holder space would only require Taylor jets of some finite order, solving a fixed singular SPDE in regularity structures only requires generalised Taylor jets of a fixed maximal order. $\endgroup$
    – Rhys Steele
    Commented Dec 1, 2023 at 10:49
  • $\begingroup$ Hi Rhys! I've thought about this for quite a while and I still don't understand. In the fixed point argument itself, infinite iterations are required and so we get an infinite series of iterated integrals. I understand that, depending on your level of regularity, you get a canonical rough path lift after declaring finitely many integrals. But for the actual solution itself, you can in infinite series arising from Picard iteration, right? That's why I don't understand how the regularity structure can be finite dimensional. $\endgroup$
    – NZK
    Commented Dec 21, 2023 at 15:13
  • $\begingroup$ The fixed point argument is run in a $D^\gamma$ space which is a space of jets in monomials which have degree less than $\gamma$. In particular, these are always finite jets so that whilst the Picard iteration has infinitely many steps, in each of these steps one is only changing the coefficients of a finite jet. It might seem like the different parts of the fixed point map can produce jets that contain monomials of higher order (i.e. when taking a product of two jets which have monomials of positive degree) but one always then truncates back to order $\gamma$. $\endgroup$
    – Rhys Steele
    Commented Dec 21, 2023 at 15:36
  • $\begingroup$ So for example, one really solves the (lift to modelled distributions of the) $\Phi_3^4$ equation in a space of jets of the form $f(x) = I(\Xi) + u_0(x) \mathbf{1} - I(I(\Xi)^3) - u_1(x) I(I(\Xi)^2) + v_i(x) X_i$ where the coefficients $u_0, u_1, v_i$ satisfy some continuity conditions that are collectively expressed in the definition of a modelled distribution. Any given step in the Picard iteration then involves just updating the values of the $u_0, u_1$ and $v_i$. $\endgroup$
    – Rhys Steele
    Commented Dec 21, 2023 at 15:47
  • $\begingroup$ Hmm I am almost there – So then what is the statement that says that the reconstruction operator applied to this truncated fixed point gives a "true" fixed point? $\endgroup$
    – NZK
    Commented Dec 21, 2023 at 19:10

1 Answer 1

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So one good start is "Rough Stochastic PDEs". Here Hairer builds a notion of an SPDE solution using rough paths

  • first building the invariant solution $\psi_{t}(x)$ and lifting to a rough path $\psi_{t}(\cdot)\mapsto \Psi_{t}(\cdot)$. This allowed, because we do this for each fixed $t$. So we get a different rough path lift for each $t$.
  • Here is where the "Taylor"-expansion comes in. In order to define a semigroup, we need to define the rough path integral in terms of $d\Psi_{t}(x)$ and for that we use the notion of Gubinelli-derivative in order to define integrals of the form $$\int Y d\Psi_{t}(x)$$, for $Y$ satisfying certain regularity (and existence of a process $Y'$ called Gubinelli derivative eg. see the book by Hairer-Friz) This is important because in the rough-path theory as developed by Lyons, the limitation was to integrating $\int f(X)dX$. The Taylor-expansion comes here because Gubinelli derivatives are required to satisfy $$Y_{s,t}=Y_{s}'X_{s,t}+Rem_{s,t}, (\text{ Gubinelli expansion})$$ and so in a sense we are expanding the process $Y$ with respect to the noise process $X$. Of course here $Y'$ is not a derivative process, but it is just thought of this way. It is yet another postulated process that happens to satisfy this relation for a remainder $Rem_{s,t}$ with better regularity.
  • This is a good foundation for regularity structures. In the above picture ,via the invariance measure, we simply needed to do one lift of the noise, but regularity structures allows us to do introduce multiple postulated objects and their interactions. How many? Well enough for the fixed point argument to go through (see the article "introduction to regularity structures). But as you can see from the first main regularity structures article, the vocabulary is always finite dimensional. So yes in that sense, you can say that the expansion is finite. The problem arises when we start studying combinations of them eg. triple product of lifted noise $(\Psi)*(\Psi)*(\Psi)$. These can grow very large. In particular, the complication comes with keeping track of the covariances of all those variations objects and their relations. It turned out that trees were a great bookkeeping mechanism to help identify the appropriate constants to subtract.
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    $\begingroup$ Thank you – this is incredibly helpful. $\endgroup$
    – NZK
    Commented Dec 3, 2023 at 3:39
  • $\begingroup$ I am still a bit confused – see my comment below the question. $\endgroup$
    – NZK
    Commented Dec 21, 2023 at 15:14
  • $\begingroup$ @NZK These are indeed very interesting questions about the fixed point. Can you open a new question with them? $\endgroup$
    – Thomas Kojar
    Commented Dec 21, 2023 at 19:28
  • $\begingroup$ Absolutely, thanks! $\endgroup$
    – NZK
    Commented Dec 21, 2023 at 19:34
  • $\begingroup$ Asked here: mathoverflow.net/questions/460811/… $\endgroup$
    – NZK
    Commented Dec 21, 2023 at 20:03

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