Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A priori bounds for the $\Phi^4$ equation in the full sub-critical
  regime   We derive a priori bounds for the $\Phi^4$ equation in the full sub-critical
regime using Hairer's theory of regularity structures. The equation is formally
given by \begin{equation} \label{e}(\partial_t-\Delta)\phi = -\phi^3 + \infty
\phi +\xi, \tag{$\star$} \end{equation} where the term $+\infty \phi$
represents infinite terms that have to be removed in a renormalisation
procedure. We emulate fractional dimensions $d<4$ by adjusting the regularity
of the noise term $\xi$, choosing $\xi \in C^{-3+\delta}$. Our main result
states that if $\phi$ satisfies this equation on a space-time cylinder $P=
(0,1) \times \{ |x| \leq 1 \}$, then away from the boundary $\partial P$ the
solution $\phi$ can be bounded in terms of a finite number of explicit
polynomial expressions in $\xi$, and this bound holds uniformly over all
possible choices of boundary data for $\phi$. The derivation of this bound
makes full use of the super-linear damping effect of the non-linear term
$-\phi^3$. A key part of our analysis consists of an appropriate re-formulation
of the theory of regularity structures in the specific context of \eqref{e},
which allows to couple the small scale control one obtains from this theory
with a suitable large scale argument. Along the way we make several new
observations and simplifications. Instead of a model $(\Pi_x)_x$ and the family
of translation operators $(\Gamma_{x,y})_{x,y}$ we work with just a single
object $(\mathbb{X}_{x, y})$ which acts on itself for translations, very much
in the spirit of Gubinelli's theory of branched rough paths. Furthermore, we
show that in the specific context of \eqref{e} the hierarchy of continuity
conditions which constitute Hairer's definition of a \emph{modelled
distribution} can be reduced to the single continuity condition on the
"coefficient on the constant level".

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The categories of this paper are: bayes-an
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The categories of this paper are: eess
20
The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th