Questions tagged [regularity-structures]
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19 questions
0 votes
0 answers
87 views
Piecewise linear approximation: time-independent renormalization constant
We are interested in the piecewise linear approximation of the $\Phi^4_d,d=2,3$ model, interpreted in the mild sense: $(\partial-\Delta)\phi=\phi^3+\xi.$ for this kind of approximation, how to define ...
- 607
1 vote
0 answers
110 views
Lattice and piecewise approximation of SPDE
This question concerns lattice & piecewise linear approximation (version of Wong-zakai theorem). Regularity structures allowed to tackle these topics for several SPDE, for example: https://arxiv....
- 607
0 votes
0 answers
128 views
Nelson estimate for $:\phi^2:$ in where $\phi$ is the $3d$ Gaussian free field
If $\phi$ is the Gaussian free field on the $2$-torus then the Nelson estimate says that $e^{-\langle :\phi^n:,z\rangle}\in L^p$ for all $p\geq 1$, smooth $z$, and where $:\phi^n:$ is the Wick power. ...
- 2,221
9 votes
0 answers
356 views
The "easy to show" direction of Hairer's $\Phi_3^4$ is Orthogonal to GFF
I was reading the note https://hairer.org/Phi4.pdf where Hairer shows that the $\Phi_3^4$ measure is orthogonal to the GFF. He defines the following set $$A_\psi:=\{\Phi:\lim_{n\to\infty} e^{-3n/4}\...
- 2,221
4 votes
1 answer
408 views
Renormalization of powers of the Gaussian free field
Let $h$ denote the Gaussian free field on $\Omega\subseteq \mathbb R^d$ for $d\geq 2$. As $h$ is distribution valued, powers of $h$ are not well defined. However in dimension $d=2$ we know that we can ...
- 2,221
6 votes
2 answers
408 views
Is there any real connection between homology and the sewing lemma/boundary operator in rough paths theory?
This question is concerned with the sewing lemma and "boundary operator" arising in rough paths theory. I give some background here. Let $\Delta_1^T, \Delta_2^T,\Delta_3^T$ denote the $1,2,3$...
- 2,221
4 votes
0 answers
206 views
Quasi-invariance of $\Phi_3^4$ under translation by nonsmooth shifts
In https://hairer.org/Phi4.pdf Hairer shows that the $\Phi_3^4$ measure is mutually singular with respect to any nonzero smooth shift. Is it also mutually singular with respect to any nonzero ...
- 2,221
5 votes
0 answers
513 views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 607
2 votes
0 answers
129 views
$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE $$\partial_t u=\Delta u-u^3+\xi,$$ where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
- 2,221
4 votes
1 answer
322 views
Truncated fixed point and regularity structures
This question arose via the helpful comments on this earlier question. In Hairer's theory of regularity structures, fixed point problems are first solved in certain spaces $D^\gamma$ which consist of ...
- 507
3 votes
1 answer
365 views
Do regularity structures involve infinite "Taylor" series?
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 ...
- 507
1 vote
0 answers
106 views
Can we extend regularity structure to matrix?
I have been trying to understand if we can apply regularity structure to solve differential equations related to Gorini–Kossakowski–Sudarshan–Lindblad or GKSL equations. This is also known as the ...
- 495
8 votes
0 answers
392 views
Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps
Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...
- 91
5 votes
1 answer
658 views
How to compare pathwise convergence and convergence in probability
This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here. Motivation: It appears pathwise convergence can ...
- 495
0 votes
1 answer
258 views
When and why do we require the condition that :"a subset bounded from below and has no accumulation points?"
I have been tyring to understand the first condition given in the link https://en.wikipedia.org/wiki/Regularity_structure for quite some time now, at least a year. I have posted a similar question in ...
- 495
2 votes
0 answers
234 views
Algebraic normalisation of regularity structures: can there be a explicit expression of g?
This is related to Bruned, Hairer, and Zambotti - Algebraic renormalisation of regularity structures. In the method of re-normalization the functional $g$ shown in page 6 plays a major role. However, ...
- 21
2 votes
1 answer
274 views
Why does the correct scaled dimension for SPDEs count time as two dimensions?
In this video, Felix Otto says that the correct way to count dimensions for parabolic equations is $2+\text{number of space dimensions}$. He said nothing about this. In the accompanying notes it is ...
user69208
8 votes
0 answers
386 views
What are morphisms between regularity structures?
In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here: Is there any way of extending this to morphisms ...
user69208
10 votes
1 answer
2k views
Is there any reason to use paracontrolled calculus over regularity structures?
Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
user69208