Stationary distributions in one dimension

Definitions and mathematical statement

Let $p:\mathbb Z\to[0,1]$ have $\sum_zp(z)=1$, and assume the random walk with jump law $p$ is irreducible on $\mathbb Z$. Consider the continuous-time exclusion process on configurations $\eta\in\{0,1\}^{\mathbb Z}$. For each ordered pair $(x,x+z)$, an independent clock of rate $p(z)$ attempts to move a particle from $x$ to $x+z$. The move occurs only if $x$ is occupied and $x+z$ is vacant.

Suppose \[ \sum_z|z|p(z)<\infty,\qquad \sum_z z p(z)>0,\qquad \sum_{z<0}z^2p(z)=\infty. \] A probability measure $\mu$ on the configuration space is stationary if starting with $\eta_0\sim\mu$ gives $\eta_t\sim\mu$ for every $t\ge0$. Does every such kernel admit a stationary $\mu$ satisfying \[ \lim_{x\to-\infty}\mu(\eta(x)=1)=0, \qquad \lim_{x\to+\infty}\mu(\eta(x)=1)=1? \] The requested measure describes a stationary transition from empty sites on the left to occupied sites on the right, despite the positive mean motion of individual particles. It need not be invariant under spatial translations or be a product measure. The conclusion only prescribes one-site marginal limits: it does not require almost surely finitely many particles on the left or finitely many holes on the right. The infinite second moment is imposed only on negative jumps; the first absolute moment remains finite on both sides. A negative answer requires an admissible irreducible kernel for which no measure with these marginal limits exists.

Short English statement

Does one-dimensional exclusion with rightward drift and an infinite negative-jump second moment admit a stationary empty-to-full interface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9600001 (AMR-095-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Liggett - Some Open Problems (2012), Problem 1, PDF page 1. Original source indexed by AMR-095-0001. https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf.