Stochastic PDEs arise naturally in out-of-equilibrium statistical physics in diverse contexts.

First (from a critical phenomena perspective, inspired from the equilibrium Ginzburg-Landau theory) one can consider dynamic universality classes of Langevin equations as in Hohenberg & Halperin's paper: model A (system with no conservation laws), model B (conserved order parameter),... Model A (inspired from spin Glauber dynamics)  is a gradient flow with additive noise. Model B (whose prototypical example is Cahn-Hilliard model) is a conservation equation with gradient noise.

Second, one can try to derive them directly from stochastic models of interacting particles: interacting particle systems (lattice gases), Dyson's Brownian motion (log gas dynamics), etc. Starting e.g.  from lattice gases, hydrodynamic fluctuation theory yields linear, Ornstein-Uhlenbeck like, or non-linear flows (e.g. KPZ equation, as proved originally by Bertini & Giacomin).

In either case there is strong interest in understanding the ultra-violet (short scale) and/or infra-red (large scale) behavior of these equations.

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○ Large-scale diffusive limit for Kardar-Parisi-Zhang (KPZ) equation in space dimension d≥3.

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 Motivated in particular by the study of the large-scale behavior of the KPZ equation in d space dimensions, I started developing functional spaces of unbounded solutions, locally modelled on W^{1,∞}, in which KPZ solutions live, and

functional bounds derived from maximum/comparison principle for Hamilton-Jacobi equations & Hamilton-Jacobi-Bellman principle.

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In the case of a quadratic nonlinearity (linearizable through exponential Cole-Hopf transformation, which relates the solution to the partition function of directed polymers), we showed with J. Magnen a large-scale diffusive limit. In other words, at large scale the field behaves like a solution of the linearized, Ornstein-Uhlenbeck like equation, called Edwards-Wilkinson model in the physics literature.  For that we used the Cole-Hopf transform and used rigorous renormalization techniques inspired from previous work by Iagolnitzer-Magnen on weakly self-avoiding walks in 4d (Commun. Math. Phys., 1994). Assuming a more general macroscopically varying condition instead,  Y. Gu, L. Ryzhik & O. Zeitouni proved convergence of leading order of large scale fluctuations of the exponential field to the same renormalized Ornstein-Uhlenbeck model; the proof, based on explicit moment expressions obtained through Itô's formula, relies on homogenization-type results and a martingale central limit theorem.

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In the case of a general (convex, smooth, quadratically bounded at infinity) nonlinearity, we are completing a paper to the same conclusion, based on a mixture between rigorous, multi-scale renormalization techniques and the above functional bounds.  Thus the Edwards-Wilkinson scaling limit holds "universally",  a much stronger result which cannot be proved without a careful use of perturbation theory & expansions. An animation explaining the dynamical, step-by-step evolution of the cluster expansion is available here.

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○ Study of the d=2 KPZ equation.

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Perturbative Feynman diagram expansions to first loop order show that the KPZ equation in its critical dimension is asymptotically free at short scale. The coefficient of the nonlinearity is scale-independent because of a Ward identity due to Galilei invariance I, while the coupling constant g decreases logarithmically when the space-time scale ℓ goes to 0,

g(ℓ) ~ (log(1/ℓ))^{-1/2}. See F. Caravenna,  R. Sun and N. Zygouras for the linearizable case and directed polymers, and upcoming article with J. Magnen for the general case.

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Large scale asymptotic analysis is difficult in general because the theory is strongly coupled, see detailed numerical simulations yielding as of today a dynamical exponent z≈1.62 (see Halpin-Healy, PRL 109, '12), compared to exact result z=3/2 in d=1, with seemingly roughly Airy-like queue of height distribution as in d=1. However, departure from Edwards-Wilkinson model is partly understood via an effective potential approach as symmetry breaking for the gradient ∇h whose module fluctuates in the neighborhood of a circle of small radius ~ exp(-1/g(ℓ=1)). Coupling this with an O(1/N) expansion, one can work out the large-scale limit for the associated N-component vector equation for N large enough (work in progress).

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