Quasi-likelihood estimation for stochastic fractional heat equation

By the quasi-likelihood method, in this note we consider parameter estimation of the fractional heat equation $\frac{\partial }{\partial t}u(t,x)={\Delta }_{\alpha }u(t,x)dt+\sigma \dot{W}(t,x),t\ge 0,x\in R$with initial condition $u(0,x)=0$, where $\dot{W}(t,x)$ is a space–time white noise and ${\Delta }_{\alpha }=-{(-\Delta )}^{\alpha /2}$ is the fractional Laplacian with $\alpha \in (1,2]$. By using the quasi-likelihood method we obtain the estimator of ${\sigma }^2$ and give the asymptotic behaviors of the estimator provided that the spatial process $x\mapsto u(t,x)$ can be observed at some discrete points $\{x_j=jh,j=0,1,2,\dots ,n\}$ with $h=h\left(n\right)\to 0$, $n{h}^{1+\gamma }\to R\ne 0$ for some $0\le \gamma <1$, as $n\to \infty$.