Singular KPZ Type Equations

Abstract : In this thesis, we investigate the existence and the uniqueness of the solution of the generalised KPZ equation. We use the recent theory of regularity structures inspired from the rough path and introduced by Martin Hairer in order to give a meaning to this singular equation. The procedure contains an algebraic part through the renormalisation group and a stochastic part with the computation of renormalised stochastic processes. One major improvement in the theory of the regularity structures is the definition of the renormalisation group using a Hopf algebra on some labelled trees. This new construction paves the way to simple formulas very useful for the renormalised stochastic processes. Then the convergence is obtained by an efficient treatment of some Feynman diagrams.
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Yvain Bruned. Singular KPZ Type Equations. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2015. English. ⟨NNT : 2015PA066517⟩. ⟨tel-01306427v2⟩

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