We propose a novel way to study numerical methods for ordinary differential equations in
one dimension via the notion of multi‐indice. The main idea is to replace rooted trees in …

Malliavin calculus provides a characterization of the centered model in regularity structures
that is stable under removing the small-scale cut-off. In conjunction with a spectral gap …

In this work, we study the two main symmetries for the one-dimensional generalised KPZ
equation (gKPZ): the chain rule and the It\^ o Isometry. We consider the equation in the full …

We derive the renormalised equation for the quasi-generalised KPZ equation with space-
time white noise. We complement the program initiated by Gerencs\'er and Hairer for solving …

In this work, we introduce explicit formulae for the coproducts at play for multi-indices in
ODEs and in singular SPDEs. The two coproducts described correspond to versions of the …

We characterise the chain rule symmetry for the geometric stochastic heat equations in the
full subcritical regime for Gaussian and non-Gaussian noises. We show that the …

Any commutative connected graded Hopf algebra gives rise to a specific notion of rough
path. This article provides a review of the different Hopf algebras, providing the various kinds …

We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-
indices which recently appeared in the literature, which can be briefly described as the …

Algebraic structures with multiple copies of a given type of operations interrelated by various
compatibility conditions have long being studied in mathematics and mathematical physics …

This thesis delves into the convergence properties of the generalised KPZ equation and
variants thereof. The first part of this thesis extends the solution theory for discrete SPDEs to …