Abstract
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition.
Highlights
We give a few examples of singular SPDEs to which the framework developed in this article can be applied
Remark 1.10 At first sight, this may appear to contradict the results of [1] where the authors consider the three-dimensional parabolic Anderson model in a rather general setting which covers that of domains with boundary
Remark 1.11 The recent result [8] is consistent with our result in the sense that it shows that the “natural” notion of solution to (1.4) with homogeneous Neumann boundary condition (i.e. c± = 0) does not coincide with the Hopf–Cole solution with homogeneous boundary data
Summary
We give a few examples of singular SPDEs to which the framework developed in this article can be applied. Remark 1.10 At first sight, this may appear to contradict the results of [1] where the authors consider the three-dimensional parabolic Anderson model in a rather general setting which covers that of domains with boundary Since this scales in exactly the same way as the KPZ equation (after applying the Hopf–Cole transform), one would expect to observe a similar “boundary renormalisation” in this case. Remark 1.11 The recent result [8] is consistent with our result in the sense that it shows that the “natural” notion of solution to (1.4) with homogeneous Neumann boundary condition (i.e. c± = 0) does not coincide with the Hopf–Cole solution with homogeneous boundary data In this particular case, one possible interpretation is that, for any fixed time, the solution to the KPZ equation is a forward/backwards semimartingale (in its own filtration) near the right/left boundary point.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
