Stationary directed polymers and energy solutions of the Burgers equation

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Abstract

We consider the stationary O’Connell–Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation.

The proof does not rely on the Cole–Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann–Gibbs principle.

Section snippets

KPZ equation and stochastic Burgers equation

The Kardar–Parisi–Zhang equation [33], or KPZ equation, was introduced in the physics literature as a model for interface motions in generic situations. The typical physical set-up is the following: suppose we have a thin physical system where a stable and a meta-stable phase can coexist and suppose both phases are separated by an interface. We are concerned with the behavior of such an interface as the stable phase invades the meta-stable region. The first thing one observes is a net motion of

Energy solutions of the Burgers equation

We will present the basics of the theory of energy solutions of the stochastic Burgers equation as it was introduced in [17] and further developed in [20], [23] (see also [19], [24], [25]). Recall we are concerned with the equation tu=12x2u+cxu12xu2+xW,where W is a space–time white noise, i.e. a distribution-valued centered Gaussian process with covariance E[W(t,x)W(s,y)]=δ(ts)δ(yx). More precisely, W acts on L2(R+×R) in such a way that the random variables {W(f):fL2(R+×R)} are jointly

System of SDEs and the martingale decomposition

An application of Itô’s formula shows that, under Pn, the collection {uj:j1} satisfies the system of SDEs: duj=Wj1Wjdt+βdBt(j)dBt(j1),j2,du1=(β22W1)dt+βdBt(1)dBt(0), where Wj=1euj and β=n14. As it will be noticed later, En[Wj]=β22. Writing W¯j=WjEn[Wj] and setting W¯0=0, the system above can be summarized as duj=W¯j1W¯jdt+βdBt(j)dBt(j1),j1.The initial condition is taken as uj(0)=logXj+12logn,j1,where (Xj)j is an i.i.d. family of Gamma(n+12) random variables. Hence, the

Static estimates

We briefly recall some facts about the Gamma and log-Gamma distributions. If XGamma(ν), then PXx=1Γ(ν)xyν1eydy,where Γ(ν)=0yν1eydy is the Gamma function. By explicit computations, E[X]=νandVar[X]=ν.Now, if we take β=n14, θ=1+1(2n) and let ν=β2θ=n+12, we obtain En[W]=En[1eu]=12n=β22,Varn[W]=Varn[eu]=1n+12n=β2θ, as, under Pn, euβ2X with XGamma(n+12). Here, Varn denotes the variance with respect to Pn. On the other hand, for XGamma(n+12), PlogXlogβ2x=1β2νΓ(ν)xeνyβ2

Dynamical estimates

Recall that C denotes the collection of cylindrical functions F of the form F(u)=f(un,,un) for some n0 and some fC2(R2n+1) with polynomial growth of its derivatives up to order 2. We recall the Kipnis–Varadhan estimate: EnsuptT|0tF(s,u(sn))ds|2C0TF(s,)1,nds,where the 1,n-norm is defined through the variational formula F1,n=supfC2F(u)fdμn+nfLfdμnThe proof is a straightforward adaptation of [14], Corollary 3.5. Note that fLfdμn=β22j(j+1j)f2dμnso that F1,n2=supfC2F

Tightness

We will use Mitoma’s criterion [37]: a sequence (Yn)n is tight in C([0,T],S(R)) if and only if (Yn(φ))n is tight in C([0,T],R) for all φS(R).

Identification of the limit

By tightness, we obtain processes X,S,B˜ and M such that limnXn=X,blallimnSn=S,limnB˜n=B˜,blalimnMn=M, along a subsequence that we still denote by n.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement No 715734), and from MATH Amsud ‘Random Structures and Processes in Statistical Mechanics’.

    1

    Partially supported by CNPq and FAPERJ.

    2

    Partially supported by Fondecyt grant 1171257 and Núcleo Milenio ‘Modelos Estocásticos de Sistemas Complejos y Desordenados’ .

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