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On a smooth manifold $M$ with altas $\mathcal A=\{\phi:U_\phi\subseteq M\to \Bbb R^n\}_\phi$, we can define a tensor $T$ on $M$ as the collection $\{T_\phi\}_{\phi\in\cal A}$ such that whenever $U_\...
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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 ...
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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....
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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. ...
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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 $$A_\psi:=\{\Phi:\lim_{n\to\infty} e^{-3n/4}\...
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Let $h$ denote the Gaussian free field on $\Omega\subseteq \mathbb R^d$ for $d\geq 2$. As $h$ is distribution valued, powers of $h$ are not well defined. However in dimension $d=2$ we know that we can ...
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This question is concerned with the sewing lemma and "boundary operator" arising in rough paths theory. I give some background here. Let $\Delta_1^T, \Delta_2^T,\Delta_3^T$ denote the $1,2,3$...
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In https://hairer.org/Phi4.pdf Hairer shows that the $\Phi_3^4$ measure is mutually singular with respect to any nonzero smooth shift. Is it also mutually singular with respect to any nonzero ...
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Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
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One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE $$\partial_t u=\Delta u-u^3+\xi,$$ where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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