Periodic Pitman transforms and jointly invariant measures

TL;DR

The paper constructs explicit jointly invariant initial data for the periodic KPZ equation (and related stochastic PDEs) by transforming independent Brownian bridges through a periodic Pitman-like transform, establishing a one force–one solution principle for uniqueness. It develops a semi-discrete (periodic) Burgers’ framework, proves a Burke-type invariance and a generator intertwining via a bijection D^{N,k}, and uses these tools to pass from discrete polymer models to the continuum KPZ limit. The main contributions include the explicit descriptions of joint invariant measures, scaling limits to the periodic KPZ horizon, a Gaussian process limit for long-time height fluctuations with a covariance expressed in terms of the joint measures, and a detailed treatment of the periodic O’Connell–Yor polymer with convergence to the stochastic heat equation. Together, these results link discrete periodic polymers and invariant measures to the continuum KPZ dynamics, providing both structural understanding and quantitative limits. The methods open avenues for exact computations of fluctuations and cumulants in periodic directed polymers and related stochastic growth models.

Abstract

We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their uniqueness via a one force--one solution principle. The measures are given by polymer-like transforms of independent Brownian bridges. We describe several properties and limits of these measures, including an extension to a continuous process in the slope parameter that we term the periodic KPZ horizon. As an application of our construction, we prove a Gaussian process limit theorem with an explicit covariance function for the long-time height function fluctuations of the periodic KPZ equation when started from varying slopes. In connection with this, we conjecture a formula for the fluctuations of cumulants of the endpoint distribution for the periodic continuum directed random polymer. To prove joint invariance, we address the analogous problem for a semi-discrete system of SDEs related to the periodic O'Connell-Yor polymer model and then perform a scaling limit of the model and jointly invariant measures. For the semi-discrete system, we demonstrate a bijection that maps our systems of SDEs to another system with product invariant measure. Inverting the map on this product measure yields our invariant measures. This map relates to a periodic version of the discrete geometric Pitman transform that we introduce and probe. As a by-product of this, we show that the jointly invariant measures for a periodic version of the inverse-gamma polymer are the same as those for the O'Connell-Yor polymer.

Figures (12)

• Figure 1: The evolution of the periodic KPZ equation (time and space in the vertical and horizontal directions, respectively). The periodic space-time white noise is depicted through random disks of varying radii and repeat in each horizontal width one strip. The initial data $h_{\beta}(0,x)$ is drawn out of the paper (sheared to illustrate this dimension) and repeats with the same periodicity up to a vertical shift by the slope $\theta$ (i.e., $h_{\beta}(0,x+1)-h_{\beta}(0,x)=\theta$ for all $x\in \mathbb{R}$). This evolves using the noise up to time $t$ to yield $h_{\beta}(t,x)$ likewise drawn. The slope $\theta$ is preserved by the evolution, as well as the periodicity.
• Figure 2: On the left are $k=5$ independent Brownian bridges with slopes $(\theta_1,\ldots,\theta_5) = (-5,-2.5,0,2.5,5)$. These are mapped via $\Psi^5$ to the output on the right, giving a sample of the invariant measure $\mathcal{P}_\beta^{(\theta_1,\ldots,\theta_5)}$ for $\beta = 1$. Each trajectory/color represents a different value of $\theta$.
• Figure 3: Simulation of $\widetilde{g}_{\beta,\theta}$ from Proposition \ref{['prop:betalim']} for $\beta = 0.01$ (left) and $\beta = 30$ (right), and $\theta \in \{-5,-2.5,0,2.5,5\}$. The pictures are produced by applying $\Psi^5$ to the same set of $5$ independent Brownian bridges (as in Theorem \ref{['thm:KPZ_invar_main']}), with the scaling factor $\beta$ changed. Consequently the bottom (blue) curve is the same in each picture.
• Figure 4: On the left is an illustration of the polymer defined in \ref{['eq:Zdis']}. The bulk weights are given by the vectors $\mathbf W_1,\ldots, \mathbf W_5\in \mathbb{R}^{\mathbb{Z}_N}$ with $N=4$. The horizontal gray lines signify where this environment periodically repeats. The first column (shaded in gray) contains the initial condition $\mathbf U^{(0)}$, also periodically repeated. The allowable paths (thick black) considered in defining the partition function for this model involve up or up/right steps and the weight assigned to a path is the exponential of the sum of all bulk weights along the path and the cumulative sum (measured from height zero) of the initial condition. The figure on the right is the result of shearing this model so that the allowable paths are those that go up or right. This is the periodic inverse-gamma polymer.
• Figure 5: The first figure on the left graphically defines via a circle the transformation from the pair $(\mathbf X,\mathbf X')$ to $(D^{N,2}(\mathbf X,\mathbf X'),\mathbf X)$ (we drop the $N$ superscript here an in all of the graphical map figures). The direction of the arrows go from input to output, and at times below we will use rotated or reflected versions of this and the $J$ building blocks to mean the same map. We will, however, always maintain the convention that the blocks are oriented so that the horizontal input passes through unchanged to become the horizontal output. The map $\mathcal{D}^{N,k}$ with $k=4$ is depicted graphically in the second figure on the left as a composition of these building block maps. In particular $(\mathbf U_1,\mathbf U_2,\mathbf U_3,\mathbf U_4):=\mathcal{D}^{N,k}(\mathbf X_1,\mathbf X_2,\mathbf X_3,\mathbf X_4)$ is the output, given input $(\mathbf X_1,\mathbf X_2,\mathbf X_3,\mathbf X_4)$. The output of each circle maps into the inputs of the next map. The first figure on the right depicts via a diamond the transformation of a pair $(\mathbf U,\mathbf U')$ to $J^{N,2}(\mathbf U,\mathbf U')$ with the input down on the left and top and outputs on the right and bottom. The map $\mathcal{J}^{N,k}$ for $k=4$ is depicted in the rightmost figure as a composition of these building block maps. In particular $(\mathbf X_1,\mathbf X_2,\mathbf X_3,\mathbf X_4):=\mathcal{J}^{N,k}(\mathbf U_1,\mathbf U_2,\mathbf U_3,\mathbf U_4)$ is the output, given input $(\mathbf U_1,\mathbf U_2,\mathbf U_3,\mathbf U_4)$.

Theorems & Definitions (187)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Jointly invariant measure for periodic KPZ
• Theorem 1.4: 1F1S principle
• Proposition 2.1
• Corollary 2.2: Periodic KPZ horizon
• proof
• Corollary 2.3
• proof
• Proposition 2.4