Consider the stochastic heat equation \(\begin{aligned} \partial _t u_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in (0,T],\, x\in D, \end{aligned}\) where b is a generalized function, D is either [0, 1] or \(\mathbb {R}\) , and \(\dot{W}\) is space-time white noise on \(\mathbb {R}_+\times D\) . If the drift b is a sufficiently regular function, then it is well-known that any analytically weak solution to this equation is also analytically mild, and vice versa . We extend this result to drifts that are generalized functions, with an appropriate adaptation of the notions of mild and weak solutions. As a corollary of our results, we show that for \(b\in L_p(\mathbb {R})\) , \(p\geqslant 1\) , this equation has a unique analytically weak and mild solution, thus extending the classical results of Gyöngy and Pardoux (Probab Theory Related Fields 94(4):413–425, 1993).
