The Stratonovich heat equation : a continuity result and weak approximations

Aurélien Deya (1) , Maria Jolis (2) , Lluís Quer-Sardanyons (2)

1 IECL - Institut Élie Cartan de Lorraine
2 UAB - Departament de Matemàtiques [Barcelona]

We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques are used to provide an almost sure continuity result for the solution with respect to the solution of the 'smooth' equation obtained by replacing the noise with an absolutely continuous process. This continuity result is then exploited to prove weak convergence results based on Donsker and Kac-Stroock type approximations of the noise.

Mots clés: stochastic heat equation; convergence in law; Stratonovich integral; convolutional rough paths theory

Domaines: Probabilités [math.PR]

Aurélien Deya, Maria Jolis, Lluís Quer-Sardanyons. The Stratonovich heat equation : a continuity result and weak approximations. Electronic Journal of Probability, 2013, 18 (3), ⟨10.1214/EJP.v18-2004⟩. ⟨hal-00687262v2⟩