Stochastic zeta functions

Definitions and mathematical statement

Let $P=(p_{ij})$ be a finite stochastic matrix: $p_{ij}\ge0$ and every row sum is one. Require $P$ to be primitive, meaning some power has all entries positive. Its stationary Markov measure on bi-infinite admissible state sequences, together with the shift, is a mixing Markov shift. For $t\in\mathbb R$, form the matrix $P^{[t]}$ by replacing each positive entry $p_{ij}$ by $p_{ij}^t$ and leaving zero entries zero.

Using the usual generating-variable normalization, its stochastic zeta function is \[ Z_P(z,t)=\det(I-zP^{[t]})^{-1} =\exp\!\left(\sum_{r\ge1}\frac{z^r}{r} \operatorname{tr}((P^{[t]})^r)\right). \] The equality holds as a formal power series in $z$, and analytically for sufficiently small $z$ at each fixed $t$. The trace sums $t$th powers of weights of closed state paths, where a path's weight is the product of its transition probabilities.

The problem is to characterize intrinsically all two-variable functions that occur as $Z_P$ for some finite primitive stochastic $P$ of arbitrary size. The equivalent source convention uses $\det(wI-P^{[t]})^{-1}$; the two are related by $w=1/z$ and the evident power of $z$. A characterization should give checkable structural conditions and a realization theorem, beyond restating existence of a matrix. The coefficient functions retain all path weights, so the ordinary unweighted zeta function or the entropy alone does not supply the requested data.

Short English statement

Characterize exactly which weighted periodic-path zeta functions arise from finite mixing Markov chains.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600010 (AMR-045-0010), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 10.1. Original source indexed by AMR-045-0010. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.