KPZ Reloaded

Abstract

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer.Apart from a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Gonçalves and Jara and paracontrolled solutions.Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation.Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation which does not rely on the Cole-Hopf transform.

The Dynamic $${\Phi^4_3}$$ Φ 3 4 Model Comes Down from Infinity

“…We showed in [31] that the 4 2 model can be obtained as the scaling limit of Ising-Kac models near criticality, as anticipated in [12]. Related results were obtained for the KPZ equation, first in [2] via a Cole-Hopf transformation, and then, following [22], in a series of works including [11,17,19,20,27,29]. See also the survey articles [6,24] for a summary of the work on the 4 model with regularity structures.…”

“…20). Now that we have controlled the second sum in (A.19), we will argue that the first sum is close to f (g = h).…”

The Work of the 2014 Fields Medalists

“…On the other hand, the set of parameter values corresponding to stochastic maps has positive Lebesgue measure, as proved in [20], and the complement has measure zero [22] but positive Hausdorff dimension. The main result in [5], [9] W. de Melo is a professor at the Instituto de Matemática Pura e Aplicada. His email address is demelo.impa@gmail.com.…”

“…If the regularization of the noise were just in space, this could be done using the Hopf-Cole + Itô approach, but the key point is that the new method is general and robust. To give just one example, if the regularization of the noise is on a scale ε in space and ε 2 in time, the weak KPZ universality suggests that the classical solution to (9) ∂ tĥε = ε(∂ xĥε ) 4 − C ε + ∂ 2 xĥε + ξ ε should converge to the Hopf-Cole solution of (5) with a highly nontrivial λ, the exponent 4 miraculously becoming a 2 in the limit. Hairer's method can be adapted to prove such things (see [13] for a review) as well as providing a good approximation theory for equations like (7) [14].…”

“…But (11) just opened a new can of worms; for twenty years nobody could even make sense of it in d = 3 until Hairer constructed the appropriate regularity structure, in which the solution exists as an abstract model to which lifts of appropriate renormalized solutions of regularized versions of (11) converge. Since Hairer's first well-posedness results, several proofs have appeared for KPZ [9] and dynamic Φ 4 3 [16], [5]. However these appear to be far more closely tied to the particular regularization as well as the specifics of the equation.…”