Two approaches two SPDEs not equivalent?

Question

I have arrived at needing SPDEs and encountered a strange thing. In the literature, two approaches are mentioned: One where the equation is thought of as an SDE in an infinite dimensional space; an other where the solution is thought of a random field which changes over time (?). Now, I have read that these two approaches are not translatable one to one. Does that mean that I can have existence of a solution for one approach but not for the other? Same question for uniqueness? How do I chose which approach best suits my problem?

Answer 1

Indeed in the reference "Stochastic integrals for spde’s: A comparison", they show that the two approaches

• Walsh random field approach
• the framework of the stochastic evolution in Hilbert spaces

are equivalent. However, as explained eg. "Invariant measures for the nonlinear stochastic heat equation with no drift term" for example there can be differences in the theorems conditions that are preferable depending on the context:

Here we emphasize that we study the invariant measure using the Walsh random field approach [27], whereas such studies are mostly carried out under the framework of the stochastic evolution in Hilbert spaces [13]. Even though both theories are equivalent (see [16]), the differences in many technical aspects are still substantial. As the random field approach often produces results that are more explicit, we try to use this approach to obtain more precise conditions for the existence of an invariant measure.