arXiv subjects

Yingtong Hou

Publications and source records attributed to Yingtong Hou.

5 recordsLinked to original sources

arXiv2026

Banach fixed point and flow approach for rough analysis

In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.

math.PR↗

arXiv2026

Flows driven by multi-indices Rough Paths

In this work, we introduce a solution theory for scalar-valued rough differential equations driven by multi-indices rough paths. To achieve this task, we will show how the flow approach using the log-ODE method introduced by Bailleul fits perfectly in this setting. In addition, we also describe the action of the translation of multi-indices rough paths at the level of rough differential equations.

math.PR↗

arXiv2025

Renormalising Feynman diagrams with multi-indices

In this work, we provide a method to obtain the renormalised measure in quantum field theory directly from the renormalisation of the expansion of the original measure. Our approach is based on BPHZ renormalisation via multi-indices, a combinatorial structure extremely successful for describing scalar-valued singular SPDEs. We propose the multi-indices counterpart to the Hopf algebraic program initiated by Connes and Kreimer for the renormalisation of Feynman diagrams. This new Hopf algebra also bridges the gap between the analysis of "pre-Feynman diagrams" and traditional diagrammatic methods. The construction relies on a well-chosen extraction-contraction coproduct of multi-indices equipped with a correct symmetry factor. We illustrate our method by the $ Φ^4 $ measure example.

math-ph↗

arXiv2024

Multi-indice B-series

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 Butcher's B-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi-indices B-series uniquely characterize the Taylor expansion of one-dimensional local and affine equivariant maps.

math.NA↗

arXiv2024

Multi-indices coproducts from ODEs to singular SPDEs

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 Butcher-Connes-Kreimer and extraction/contraction coproducts with multi-indices. The main idea is to use the fact that these coproducts are the adjoints of dual products for which one has explicit simple formulae. We are able to derive the explicit formulae via an inner product defined from a symmetry factor easily computable for multi-indices.

math.PR↗