Stationary directed polymers and energy solutions of the Burgers equation☆
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 where is a space–time white noise, i.e. a distribution-valued centered Gaussian process with covariance . More precisely, acts on in such a way that the random variables are jointly
System of SDEs and the martingale decomposition
An application of Itô’s formula shows that, under , the collection satisfies the system of SDEs: where and . As it will be noticed later, . Writing and setting , the system above can be summarized as The initial condition is taken as where is an i.i.d. family of random variables. Hence, the
Static estimates
We briefly recall some facts about the Gamma and log-Gamma distributions. If , then where is the Gamma function. By explicit computations, Now, if we take , and let , we obtain as, under , with . Here, denotes the variance with respect to . On the other hand, for ,
Dynamical estimates
Recall that denotes the collection of cylindrical functions of the form for some and some with polynomial growth of its derivatives up to order . We recall the Kipnis–Varadhan estimate: where the -norm is defined through the variational formula The proof is a straightforward adaptation of [14], Corollary 3.5. Note that so that
Tightness
We will use Mitoma’s criterion [37]: a sequence is tight in if and only if is tight in for all .
Identification of the limit
By tightness, we obtain processes and such that along a subsequence that we still denote by .
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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2023, Communications in Mathematical PhysicsKPZ-TYPE FLUCTUATION EXPONENTS FOR INTERACTING DIFFUSIONS IN EQUILIBRIUM
2023, Annals of ProbabilityCENTRAL MOMENTS OF THE FREE ENERGY OF THE STATIONARY O'CONNELL-YOR POLYMER
2022, Annals of Applied ProbabilityMean Curvature Interface Limit from Glauber+Zero-Range Interacting Particles
2022, Communications in Mathematical Physics
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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’.
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Partially supported by CNPq and FAPERJ.