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I basically asked this question over on Physics Stack Exchange, but that went nowhere and I've tried to refine my question to bring it here.

When numerically modeling a natural system with turbulence, some modelers use linear stochastic PDEs. For example, a noisy diffusion equation has been studied to describe the statistical evolution of the ocean surface temperature field $T(x,t)$ $$\frac{dT(x,t)}{dt}=\kappa\nabla^2 T(x,t)+\eta(x,t)$$ $$<\eta>=0$$ $$<\eta(x,y)\eta(x',t')>\propto \delta(x-x')\delta(t-t')$$ where $\eta$ is white noise representing turbulent weather forcing from the atmosphere, or likewise you could have a conservative noise term appearing in the temperature flux.

A semi-recent paper titled A Fundamental Limitation of Markov Models compares the use of Markov models and high dimensional deterministic models of turbulence. The authors claim that because Markov models are not infinitely differentiable, but deterministic systems are, there always exists a time lag below which the covariance of the Markov model differs from the deterministic model. If this is a fundamental limitation of Markov models, does that invalidate stochastic approaches to modeling turbulence?

My question then is this, can the misrepresentation of the smallest scales in a linear stochastic PDE affect the solution at larger/longer scales? Is it possible to say that such models might not reproduce emergent behavior? Said another way, is emergent behavior in a model at all sensitive to inaccuracies in the way the smaller scales are modeled?

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  • $\begingroup$ but that went no where – Well it seems to have helped you to make your question more focussed. Still: A stochastic PDE is not a Markov model (at least in the sense of that paper as far as I understood it). The former is continuous in time; the latter is discrete. $\endgroup$ – Wrzlprmft Jul 26 '17 at 19:15

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