Related papers: Paracontrolled Distributions and the 3-dimensional…

The $\Phi^4_3$ equation is a singular stochastic PDE with important applications in mathematical physics. Its solution usually requires advanced mathematical theories like regularity structures or paracontrolled distributions, and even…

We construct the $\Phi^4_3$ measure on an arbitrary 3-dimensional compact Riemannian manifold without boundary as an invariant probability measure of a singular stochastic partial differential equation. Proving the nontriviality and the…

We prove the local wellposedness of the (renormalized) parabolic $\Phi^4_3$ model associated with the harmonic oscillator on $\mathbb{R}^3$, that is, the equation formally written as \begin{equation*} \partial_t X + HX= -X^3+\infty\cdot X +…

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a…

We study the fractional $\Phi^4_3$-measure (with order $\alpha > 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity,…

We consider time-dependent singular stochastic partial differential equations on the three-dimensional torus. These equations are only well-posed after one adds renormalization terms. In order to construct a well-defined notion of solution,…

We present a construction of the fractional $\Phi^4$ Euclidean quantum field theory on $\mathbb{R}^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for…

We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that…

In [C. Giannotti, A. Spiro, M. Zoppello, {\it Distributions and controllability problems (I)}, preprint posted on ArXiv (2024)], we introduced a new approach to the real analytic non-linear control systems of the form $\dot q^i = f^i(t, q,…

Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in H\"older and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by…

We prove the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on $\mathbb{R}^3$ to the dynamical $\Phi^4_3$ model by paracontrolled distributions on weighted Besov space. Our approach depends on…

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $\Phi^3_3$-measure and complete the program on the…

We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of…

We establish the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on a three dimensional periodic domain to the dynamic $\Phi^4_3$ model within the framework of paracontrolled distributions. Our…

We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a…

Following Parisi \& Wu's paradigm of stochastic quantization, we constructed in \cite{BDFT} a $\Phi^4$ measure on an arbitrary closed, compact Riemannian manifold of dimension $3$ as an invariant measure of a singular stochastic partial…

We study the lattice approximations to the dynamical $\Phi^4_3$ model by paracontrolled distributions proposed in [GIP13]. We prove that the solutions to the lattice systems converge to the solution to the dynamical $\Phi_3^4$ model in…

We review recent results on the analysis of singular stochastic partial differential equations in the language of paracontrolled distributions.

Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional…

We consider the continuum parabolic Anderson model (PAM) and the dynamical $\Phi^4$ equation on the $3$-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin…