I'm interesting to the stochastic PDE
$$\left\{\begin{array}{l}\dfrac{\partial u}{\partial t}(t,x)=\Delta_{\mathbf{\alpha }}u(t,x) + {\dot{W}}(t,x), \\u(0,x)=u_{0}(x),\,\,\,\, x\in\mathbb{R}.\end{array}\right.$$
where $\Delta_{\alpha}= -\left(-\Delta\right)^{\frac{\alpha}{2}}$ with $0 < \alpha \leq 2$. My problem is that I dont know some properties of this operator, especially
- Its domain ?
- If there exists a ONB of $L^{2}(\mathbb{R})$ of eigenvectors of this operator ?
- If there exists $\delta\in (0,1)\,$ such that $(\Delta_{\alpha})^{-1 + \delta}\,$ is a trace class operator.
- If there exists a reference where I can find a study of this operator ? Thank you for your help