Weak convergence for quasilinear stochastic heat equation driven by a fractional noise with Hurst parameter \(H \in (\frac{1}{2},1)\) (Q2841323)

The authors prove a convergence result for the quasi-linear stochastic heat equation on \([0,1]\) NEWLINE\[NEWLINE \frac{\partial U}{\partial t} - \frac{\partial^2 U}{\partial x^2} = b(U) + \frac{\partial W^H}{\partial t \partial x}, NEWLINE\]NEWLINE with Dirichlet boundary conditions NEWLINE\[NEWLINE U(t,0) = U(t,1) = 0, \quad t \in [0,T], NEWLINE\]NEWLINE where \(W^H\) is a fractional noise with Hurst parameter \(H \in (\frac{1}{2},1)\).NEWLINENEWLINEThe solutions to this equation are understood to be mild solutions; that is, a continuous random field \(U = \{ U(t,x): (t,x) \in [0,T] \times [0,1] \}\) is a solution if it satisfies NEWLINE\[NEWLINE\begin{multlined} U(t,x) = \int_0^1 G_t(x,y)u_0(y)dy + \\ \int_0^t \int_0^1 G_{t-s}(x,y)b(U(s,y))dy ds + \int_0^t \int_0^1 G_{t-s}(x,y)W^H(ds,dy),\end{multlined} NEWLINE\]NEWLINE where \(G\) denotes the Green function associated to the heat equation.NEWLINENEWLINEAssuming that \(u_0 : [0,1] \rightarrow \mathbb{R}\) is continuous and \(b : \mathbb{R} \rightarrow \mathbb{R}\) is Lipschitz, the authors' main result states that the mild solution can be approximated in law in the space \(C([0,T] \times [0,1])\) by the processes NEWLINE\[NEWLINE\begin{multlined} U_n(t,x) = \int_0^1 G_t(x,y)u_0(y)dy + \\ \int_0^t \int_0^1 G_{t-s}(x,y)b(U_n(s,y))dy ds + \int_0^t \int_0^1 K_H^* G_{t-s}(x,y) \theta_n(s,y) dy ds,\end{multlined} NEWLINE\]NEWLINE where \(\{ \theta_n(t,x), (t,x) \in [0,T] \times [0,1] \}\), \(n \in \mathbb{N}\), stand for the Kac-Stroock process in the plane.