Hitting questions play a central role in the theory of Markov processes.  For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions.  For a general Markov process, we can determine whether the process hits a given set in terms of potential theory.  There has also been a huge amount of work on the related question of when a process has multiple points. 

For SPDE, much less is known, but there have been a growing number of papers on the topic in recent years.  Potential theory provides an answer in principle.  But unfortunately, solutions to SPDE are infinite dimensional processes, and the potential theory is intractible.  As usual, the critical case is the most difficult. 

We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case