Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients (Q2125633)

Let \(\alpha\in(1,2)\), \(\gamma\in(1/2,1)\) and \(\rho\in(0,\frac{\alpha+1}{2})\) and consider the space \[ C^+_\rho(\mathbb R)=\{g\in C(\mathbb R):\,g\ge 0,\,\sup_{x\in\mathbb R}(1+|x|)^\rho g(x)<\infty\}. \] If \(f\in C^+_\rho(\mathbb R)\) and \(T>0\) then the equation \begin{align*} \frac{\partial X}{\partial t}&=-(-\Delta)^{\alpha/2}X+X^\gamma\,\dot W \\ X(0)&=f \end{align*} driven by a space-time white noise \(\dot W\) on \([0,T]\times\mathbb R\) has a weak solution \(X\) (in the probabilistic sense) with continuous paths in \(C^+_\rho(\mathbb R)\).