Ito's- and Tanaka's-type formulae for the stochastic heat equation: The linear case

Mihai Gradinaru (1) , Ivan Nourdin (1, 2) , Samy Tindel (1)

1 IECN - Institut Élie Cartan de Nancy
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution X as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô's- and Tanaka's-type formulae associated to X.

Keywords: Stochastic heat equation; Malliavin calculus; Itô's formula; Tanaka's formula; Chaos decomposition

Domains: Probability [math.PR]; Analysis of PDEs [math.AP]

Mihai Gradinaru, Ivan Nourdin, Samy Tindel. Ito's- and Tanaka's-type formulae for the stochastic heat equation: The linear case. Journal of Functional Analysis, 2005, 228, pp.114-143. ⟨10.1016/j.jfa.2005.02.008⟩. ⟨hal-00091290⟩