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.

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

 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.

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.

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.

○ Study of the d=2 KPZ equation.

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.

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).

Renormalization, Stochastic PDEs &

 Statistical Physics

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