Invariant measure theory for SPDEs with distributional solutions (Hilbert versus Polish)

Question

SPDEs such as the stochastic heat equation for $d\geq 2$ with space-time white noise and the stochastic quantization equation have distributional solutions and we still try to make sense of their asymptotic distribution.

Q: what are the approaches of making sense of invariant measures for spdes with distributional solutions? Do the standard invariant measure results from Hilbert space carry through eg. Bismu-Elworth-Li formula? What are some results that work for dual Hilbert spaces but fail for Polish spaces?

I understand that in the case the distributional solution $u\in H^{-\alpha}$ for some Hilbert space H, then we can identify that space with H via dual pairing and use the standard invariant measure results as listed say in DaPrato and Zabczyk.

Another approach is regularizing ,say by Wick powers, and studying their behaviour.

But what if $u\in C^{-\alpha}$, where $C^{\alpha}$ is the $\alpha$-Hölder space? Do some Hilbert space results fail to carry through?

An area with many results of that flavour is of course stochastic quantization, which was designed for that purpose to begin with. However, in particular I would like to know the state of the art for long-term behaviour of parabolic spdes with distributional solutions.

Answer 1

The situation can be a bit trickier because we are dealing with distribution valued objects and thus the need to first smoothen the equations.

For example, as explained in "Fluctuation and Rate of Convergence of the Stochastic Heat Equation in Weak Disorder", for the stochastic heat equation on $R^{d}$ with smoothened noise

$$du_{\epsilon,t} = \frac{1}{2}\Delta u_{\epsilon,t} dt + \beta \epsilon^{(d-2)/2} u_{\epsilon,t} dB_{\epsilon,t}$$

depending on the tuning of the parameter $\beta$ being small or large, they get different asymptotic limits

But in terms of which theory to use (Walsh or Hilbert framework from "Stochastic integrals for spde’s: A comparison"), as explained in ,say, "Invariant measures for the nonlinear stochastic heat equation with no drift term" 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.