An Itô type formula for the additive stochastic heat equation

Carlo Bellingeri (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We use the theory of regularity structures to develop an Itô formula for u, the solution of the one-dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular, for any smooth enough function φ we can express the random distribution (∂t−∂xx)φ(u) and the random field φ(u) in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of Malliavin calculus.

Mots clés: stochastic partial differential equations; Itô formula; Malliavin calculus; regularity structures

Domaines: Probabilités [math.PR]

Carlo Bellingeri. An Itô type formula for the additive stochastic heat equation. Electronic Journal of Probability, 2020, 25, pp.60 - 75. ⟨10.1214/19-EJP404⟩. ⟨hal-01725871v3⟩