[Paper Review] Ergodicity results for the open KPZ equation

Shalin Parekh|arXiv (Cornell University)|Dec 16, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the existence and uniqueness of an invariant measure for the open KPZ equation on [0,1] with arbitrary inhomogeneous Neumann boundary conditions, using two novel proofs based on a support theorem and a polymer coupling argument. It further proves exponential ergodicity in total variation, leveraging the strong Feller property and a compact state space construction that extends to Hölder spaces modulo constants.

ABSTRACT

We give a new proof of existence as well as two proofs of uniqueness of the invariant measure of the open-boundary KPZ equation on [0,1], for all possible choices of inhomogeneous Neumann boundary data. Both proofs yield an exponential convergence result in total variation when combined with the strong Feller property which was recently established in [Knizel-Matetski, arXiv:2211.04466]. An important ingredient in both proofs is the construction of a compact state space for the Markov operator to act on, which measurably contains all of the usual Hölder spaces modulo constants. Along the way, the strong Feller property is extended to this larger class of initial conditions. The arguments do not rely on exact descriptions of the invariant measures, and some of the results generalize to the case of a spatially colored noise and other boundary conditions.

Motivation & Objective

• To provide a new proof of existence of an invariant measure for the open KPZ equation with arbitrary inhomogeneous Neumann boundary data.
• To establish uniqueness of the invariant measure without relying on explicit descriptions of the invariant law, resolving a long-standing conjecture.
• To prove exponential convergence in total variation to the invariant measure, establishing geometric ergodicity for all boundary parameters.
• To extend the strong Feller property to a larger class of initial conditions, including Hölder spaces modulo constants.
• To generalize results to spatially colored noise and other boundary conditions, broadening applicability beyond white noise.

Proposed method

• Construct a compact state space $\mathcal{X}$, isometric to the space of Borel probability measures on [0,1] under Prohorov distance, which contains all Hölder spaces modulo constants.
• Use the Hopf-Cole transform to map the KPZ equation to a stochastic heat equation with Robin boundary conditions, enabling well-posedness via mild formulation.
• Apply the Feynman-Kac formula and Cameron-Martin theorem in the support-theoretic proof to analyze the transition kernel and establish weak convergence of differences in initial conditions.
• Employ a polymer coupling argument based on the convolution property of propagators for the stochastic heat equation, enabling a one-force one-solution principle.
• Leverage the strong Feller property (recently established in [KM22]) on $\mathcal{X}$ to ensure regularity of transition probabilities and support the convergence arguments.
• Use discrete-time Markov semigroups to simplify analysis and prove exponential decay of the operator norm $\|\mathfrak{P}_*^N\|_{\text{op}} \to 0$ as $N \to \infty$, implying geometric ergodicity.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of the invariant measure for the open KPZ equation be proven without relying on exact descriptions of the invariant law?
• RQ2 Does the Markov process associated with the open KPZ equation exhibit exponential mixing in total variation for all boundary parameters $A, B \in \mathbb{R}$?
• RQ3 Can the strong Feller property be extended to a larger class of initial conditions, including Hölder continuous functions modulo constants?
• RQ4 Is the convergence to the invariant measure uniform across all initial data, and can this be quantified via a spectral gap?
• RQ5 Can the results be generalized to spatially colored noise and other boundary conditions beyond Neumann?

Key findings

• The paper establishes existence of an invariant measure for the open KPZ equation on [0,1] with arbitrary inhomogeneous Neumann boundary data, independent of prior constructions via ASEP limits.
• Uniqueness of the invariant measure is proven for all $A, B \in \mathbb{R}$ without requiring explicit knowledge of the invariant law, resolving a conjecture from [Cor22, CK20].
• Exponential convergence in total variation to the invariant measure is established, with $\|\mathfrak{P}_*^N \delta_\phi - \mathfrak{P}_*^N \delta_\psi\|_{TV} \leq Ce^{-cN}$ for all $\phi, \psi \in \mathcal{X}$, implying a spectral gap.
• The strong Feller property is extended to the compact state space $\mathcal{X}$, which includes all Hölder spaces modulo constants, enabling regularity of transition kernels.
• The results generalize to the case of spatially colored noise, as shown in Corollary 2.5, indicating robustness beyond white noise.
• The compact state space $\mathcal{X}$, constructed as the completion of $\mathcal{X}_0$ under a metric derived from the Prohorov distance, is isometric to the space of Borel probability measures on [0,1], ensuring compactness and facilitating the ergodicity proof.