Multiplicative stochastic heat equations on the whole space

Understand

We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains.

• The two main equations our result covers are, on the one hand the parabolic Anderson model on $\mathbf{R}^3$, and on the other hand the KPZ equation on $\mathbf{R}$ via the Cole-Hopf transform.
• To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces.
• One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.