TY - JOUR
T1 - The Stratonovich heat equation: A continuity result and weak approximations
AU - Deya, Aurélien
AU - Quer-Sardanyonsz, Lluís
AU - Jolis Gimenez, Maria
PY - 2013/2/8
Y1 - 2013/2/8
N2 - 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.
AB - 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.
KW - Convergence in law
KW - Convolutional rough paths theory
KW - Stochastic heat equation
KW - Stratonovich integral
U2 - 10.1214/EJP.v18-2004
DO - 10.1214/EJP.v18-2004
M3 - Article
SN - 1083-6489
VL - 18
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
ER -