Let SUBEXP is the complexity class of all problems solvable in sub-exponential time in the length of the input. What are the known properties of this class? Is it known to be contained in PSPACE, if so, is it known to be contained in NP? Is it thought to lie outside of PSPACE? My intuition says there should be problems that are Solvable in at least Sub-expoential time and Sub-exponential space, but not be in PSPACE.
SUBEXP is neither known or widely believed to lie in PSPACE (and -- contrary to one of the comments -- this is not known to have any connection to SETH). It is not known whether the containment "SUBEXP in PSPACE" would imply that SUBEXP is in NP; there are oracles relative to which P=NP and PSPACE = EXP, and thus there are oracles relative to which this implication fails.