We study the nonlinear stochastic heat equation driven by space–time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized stochastic convolution via Young-type inequalities.
"Weak nonmild solutions to some SPDEs." Illinois J. Math. 54 (4) 1329 - 1341, Winter 2010. https://doi.org/10.1215/ijm/1348505531