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Abstract
We consider a large class of$1+1$-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf–Cole solutions to the KPZ equation.
Highlights
If one tries to brutally cast this into the framework of [Hai14], one might try to deal with arbitrary polynomial nonlinearities and view the multiplicative ε as a parameter of the equation. This is bound to fail since the KPZ equation with a higher than quadratic nonlinearity fails to satisfy the assumption of local subcriticality which is key to the analysis of [Hai14]
Given a kernel K as above, we introduce a set M of admissible models which are analytical objects built upon our regularity structure (T, G) that will play a role for our solutions that is similar to that of the usual Taylor polynomials for smooth functions
It is very important to keep in mind that not every admissible model is obtained in this way, or even as a limit of such models! This will be apparent in Section B below where we describe the renormalization group
Summary
Where ξ denotes space–time white noise and λ ∈ R is a parameter describing the strength of its ‘asymmetry’. Ε1/4λ(∂x h)2 + ε1/4a(∂x h)4 + ε1/4ξε, ε1/4h(ε−1x, ε−2t ) − Cεt will converge to h for an appropriate These regimes are really just versions of intermediate disorder in that in the formal rescaling the higher order error terms are small as in (1.9) as opposed to relevant as in (1.20), and they only contribute to the global shift Cε. [HL15] introduced weighted spaces allowing the extension of the results on convergence of smoothed noise approximations of the quadratic KPZ equation to the whole line These use in an essential way the Hopf–Cole transformation, which is not available for the nonquadratic versions considered in this article.
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