Random Time Evolution of Infinite Particle Systems · 1974-prob-200

Remaining questions recorded by the source

These are retained within the parent record and are not counted as additional index entries.

Question 1

Determine whether there exist an irreducible probability kernel $p:\mathbb Z\to[0,1]$ satisfying $\sum_{z\in\mathbb Z}p(z)=1$, $\sum_{z\in\mathbb Z}|z|p(z)<\infty$, and $\sum_{z\in\mathbb Z}zp(z)>0$, together with an extremal stationary probability measure $\mu$ for the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$, such that $\mu$ is a profile measure but not a blocking measure. Here “profile measure” means that $\mu\{\eta:\eta(x)=1\}\to0$ as $x\to-\infty$ and $\mu\{\eta:\eta(x)=1\}\to1$ as $x\to+\infty$, while “blocking measure” means that $\mu$ is concentrated on configurations satisfying $\sum_{x<0}\eta(x)<\infty$ and $\sum_{x>0}(1-\eta(x))<\infty$.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Introduction and Section 10 (open question on nonblocking stationary profile measures). DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.

Question 2

Let $d>1$, and let $p:\mathbb Z^d\to[0,1]$ be an irreducible probability kernel satisfying $\sum_{z\in\mathbb Z^d}p(z)=1$ and $\sum_{z\in\mathbb Z^d}|z|p(z)<\infty$. Consider the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z^d}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$, and write $m=\sum_z zp(z)$. Prove or disprove that every extremal stationary measure $\mu$ that is not invariant under all lattice translations is $v$-homogeneous for some $v\in\mathbb R^d$ satisfying $\langle m,v\rangle>0$, where $v$-homogeneous means that, for every finite $A\subset\mathbb Z^d$, there is a continuous function $f_A:\mathbb R\to[0,1]$ such that $\mu\{\eta:\eta=1\text{ on }x+A\}=f_A(\langle x,v\rangle)$ for every $x\in\mathbb Z^d$.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Section 10, Open Problem 1. DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.

Question 3

Let $d>1$, and let $p:\mathbb Z^d\to[0,1]$ be an irreducible probability kernel satisfying $\sum_{z\in\mathbb Z^d}p(z)=1$ and $\sum_{z\in\mathbb Z^d}|z|p(z)<\infty$. Consider the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z^d}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$, and write $m=\sum_z zp(z)$. Prove or disprove that every extremal stationary measure $\mu$ that is not invariant under all lattice translations is a $v$-profile measure for some $v\in\mathbb R^d$ satisfying $\langle m,v\rangle>0$. Here $\mu$ is a $v$-profile measure if, for every finite $A\subset\mathbb Z^d$, there is a continuous function $f_A:\mathbb R\to[0,1]$ such that $\mu\{\eta:\eta=1\text{ on }x+A\}=f_A(\langle x,v\rangle)$ for every $x\in\mathbb Z^d$, and, for every nonempty $A$, $f_A(s)\to0$ as $s\to-\infty$ and $f_A(s)\to1$ as $s\to+\infty$.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Section 10, Open Problem 2. DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.

Question 4

Let $d>1$, and let $p:\mathbb Z^d\to[0,1]$ be an irreducible probability kernel satisfying $\sum_{z\in\mathbb Z^d}p(z)=1$ and $\sum_{z\in\mathbb Z^d}|z|p(z)<\infty$. Consider the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z^d}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$, and write $m=\sum_z zp(z)$. Fix $v\in\mathbb R^d$ with $\langle m,v\rangle>0$. Prove or disprove that there is a weakly continuous one-parameter family $(\mu_\alpha)_{\alpha\in(0,1)}$ of inhomogeneous extremal stationary $v$-profile measures satisfying $\mu_\alpha\{\eta:\eta(0)=1\}=\alpha$. Here $\mu$ is a $v$-profile measure if, for every finite $A\subset\mathbb Z^d$, there is a continuous function $f_A:\mathbb R\to[0,1]$ such that $\mu\{\eta:\eta=1\text{ on }x+A\}=f_A(\langle x,v\rangle)$ for every $x\in\mathbb Z^d$, and, for every nonempty $A$, $f_A(s)\to0$ as $s\to-\infty$ and $f_A(s)\to1$ as $s\to+\infty$.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Section 10, Open Problem 3. DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.

Question 5

Let $d>1$, and let $p:\mathbb Z^d\to[0,1]$ be an irreducible, nonsymmetric probability kernel satisfying $\sum_{z\in\mathbb Z^d}p(z)=1$, $\sum_{z\in\mathbb Z^d}|z|p(z)<\infty$, and $\sum_{z\in\mathbb Z^d}zp(z)=0$. Consider the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z^d}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$. Prove or disprove that its extremal stationary probability measures are exactly the homogeneous Bernoulli product measures $\nu_\rho$, $0\le\rho\le1$, under which the occupation variables are independent with common mean $\rho$.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Section 10, Open Problem 4. DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.

Question 6

Let $d>1$, and let $p:\mathbb Z^d\to[0,1]$ be an irreducible probability kernel satisfying $\sum_{z\in\mathbb Z^d}p(z)=1$, $\sum_{z\in\mathbb Z^d}|z|p(z)<\infty$, and $m:=\sum_{z\in\mathbb Z^d}zp(z)\ne0$. Consider the continuous-time simple exclusion process on $\{0,1\}^{\mathbb Z^d}$ in which a particle at $x$ attempts to jump to $x+z$ at rate $p(z)$. Prove or disprove that there exists an extremal stationary probability measure for this process that is not a product measure.

Source reference: Maury Bramson and Thomas M. Liggett, “Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence,” The Annals of Probability 33(6) (2005), 2255–2313, Section 10, Open Problem 5. DOI: https://doi.org/10.1214/009117905000000341; arXiv:math/0602098, https://arxiv.org/abs/math/0602098.