Global unique solution to the perturbation of the Burgers' equation forced by derivatives of space-time white noise

We consider the one-dimensional Burgers’ equation forced by fractional derivative of order $\frac{1}{2}$ applied on space-time white noise. Relying on the approaches of Anderson Hamiltonian from Allez and Chouk (2015, arXiv:1511.02718 [math.PR]) and two-dimensional Navier-Stokes equations forced by space-time white noise from Hairer and Rosati (2024, Annals of PDE, $\mathbf{\text{10}}$, pp. 1–46), we prove the global-in-time existence and uniqueness of its mild and weak solutions.