Solutions to the SPDE ∂_t u = 1/2 Δu + σ(u)η with u(0)≡1 are spatially stationary and ergodic for t>0 under mild decay on the spatial correlation f of the noise.
Spatial asymptotics for the paraboli c Anderson models with generalized time-space Gaussian noise
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Spatial ergodicity for SPDEs via Poincar\'e-type inequalities
Solutions to the SPDE ∂_t u = 1/2 Δu + σ(u)η with u(0)≡1 are spatially stationary and ergodic for t>0 under mild decay on the spatial correlation f of the noise.