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I'm reading Hairer's notes on SPDEs: http://www.hairer.org/notes/SPDEs.pdf

He says on page 6 that "the stationary solution to the stochastic heat equation is Gaussian free field". He never defines what "stationary solution" means and it's not obvious. Does it mean that the solution doesn't depend on time? Since the noise depends on time this can never happen.

What is the stationary solution to a SPDE?

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    $\begingroup$ Loosely, it means that if you have some initial distribution and let it evolve according to the SDE, at any point in time the distribution will be the same. $\endgroup$ – tch Dec 30 '18 at 2:40
  • $\begingroup$ @TylerChen So the distribution is stationary. Do you have a source for this? $\endgroup$ – Rptoughs Dec 30 '18 at 2:41
  • $\begingroup$ Yes. Except in trivial cases, it only makes sense in the context of distributions (as you noted a single trajectory will generally depend on time because of the randomness). This book has a decent intro in 6.9. $\endgroup$ – tch Dec 30 '18 at 5:36

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