Multi-Time Markov Renewal Equations

• Multi-Time Markov Renewal Equations are renewal identities indexed by multiple time coordinates, integrating Markov or semi-Markov dynamics.
• They employ multidimensional convolutional inverses and algebraic techniques—such as FFT and Gauss–Jordan methods—for efficient computation.
• Their applications span communication theory, reliability, and biological systems, offering insights into joint-time distributions and infinite-memory processes.

Searching arXiv for the cited papers to ground the article in current sources. Multi-Time Markov Renewal Equations are renewal identities in which the renewal mechanism is indexed by several time coordinates, several observation times, or an infinite ordered family of elapsed-time variables, while the underlying dynamics retain a Markov-renewal or semi-Markov structure. A central recent formulation is the discrete $d$-dimensional equation

$L = G + q * L,$

with solution

$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$

for matrix-valued sequences on $\mathbb{N}^d$ (Kordalis et al., 3 Aug 2025). Related arXiv strands treat multi-time joint laws at several calendar times for continuous-time Markov renewal processes (Gehri et al., 2024), infinite-times renewal equations driven by the sequence of elapsed times since past events (Dou et al., 2023), and matrix Markov renewal equations with one time variable but multidimensional state space, which are explicitly distinguished from genuinely multi-parameter multi-time models (Kolesko et al., 7 Mar 2025).

1. Terminological scope and basic meanings

Current arXiv usage assigns the expression “multi-time” to several closely related constructions. In the discrete multidimensional setting, time is indexed by $k_{1:d}\in \mathbb{N}^d$ with the componentwise partial order $k \le l$ iff $k_u \le l_u$ for all $u=1,\dots,d$. The process is a $d$-dimensional multi-time Markov renewal chain $(J,S) := (J_n,S_n)_{n\ge 0}$, where $L = G + q * L,$0 is a finite-state embedded chain and $L = G + q * L,$1 is a multi-time jump epoch (Kordalis et al., 3 Aug 2025).

In continuous time, the phrase appears in the analysis of joint laws at several observation times $L = G + q * L,$2. There the state is described by the current state $L = G + q * L,$3 and backward recurrence time $L = G + q * L,$4, and multi-time distributions are written through iterated convolutions of the semi-Markov kernel and survival factors (Gehri et al., 2024).

A third meaning arises in infinite-memory models. The state is then the ordered sequence $L = G + q * L,$5, where $L = G + q * L,$6 is the elapsed time since the $L = G + q * L,$7-th most recent event and $L = G + q * L,$8. The corresponding renewal equation becomes an infinite-dimensional transport-plus-renewal PDE or an equivalent hierarchy of marginal equations (Dou et al., 2023).

Setting	Time/state structure	Representative relation
Discrete multi-time Markov renewal chain	$L = G + q * L,$9, finite $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$0	$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$1
Continuous-time multi-time joint laws	$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$2, $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$3	Iterated convolutions of $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$4 with survival factors
Infinite-times renewal equation	$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$5 with $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$6	$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$7 with renewal via shift

A recurrent misconception is to identify “multi-time” with “multidimensional” in the sense of vector-valued state alone. The paper on asymptotic expansions of Markov renewal equations states explicitly that it studies a single time variable $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$8 but a multidimensional state space $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$9 types, and that extending to genuinely multi-parameter renewal equations would require analytic tools for several complex variables (Kolesko et al., 7 Mar 2025).

2. Discrete multi-time Markov renewal chains and the fundamental equation

The discrete algebraic formulation begins with a finite state space $\mathbb{N}^d$0, the matrix space $\mathbb{N}^d$1, and the set $\mathbb{N}^d$2 of $\mathbb{N}^d$3-valued sequences $\mathbb{N}^d$4. For $\mathbb{N}^d$5, the $\mathbb{N}^d$6-dimensional convolution product is

$\mathbb{N}^d$7

Entrywise,

$\mathbb{N}^d$8

so convolution acts as matrix multiplication over the commutative ring of real $\mathbb{N}^d$9-dimensional sequences (Kordalis et al., 3 Aug 2025).

The identity sequence $k_{1:d}\in \mathbb{N}^d$0 is defined by $k_{1:d}\in \mathbb{N}^d$1 and $k_{1:d}\in \mathbb{N}^d$2 otherwise. Convolutional powers are

$k_{1:d}\in \mathbb{N}^d$3

equivalently

$k_{1:d}\in \mathbb{N}^d$4

The algebra $k_{1:d}\in \mathbb{N}^d$5 is a unital associative $k_{1:d}\in \mathbb{N}^d$6-algebra with identity $k_{1:d}\in \mathbb{N}^d$7, and $k_{1:d}\in \mathbb{N}^d$8 has a convolutional inverse iff $k_{1:d}\in \mathbb{N}^d$9 is nonsingular (Kordalis et al., 3 Aug 2025).

A $k \le l$0-dimensional multi-time Markov renewal chain is a pair $k \le l$1 with inter-jump times $k \le l$2, $k \le l$3, and counting process

$k \le l$4

Its time-homogeneous semi-Markov kernel is

$k \le l$5

with $k \le l$6, $k \le l$7, and optionally $k \le l$8 to exclude instantaneous transitions across all dimensions (Kordalis et al., 3 Aug 2025).

The fundamental multi-time Markov renewal equation is

$k \le l$9

Its componentwise form is

$k_u \le l_u$0

If $k_u \le l_u$1, then $k_u \le l_u$2, so $k_u \le l_u$3 is invertible in the convolution algebra. Hence the equation has the unique solution

$k_u \le l_u$4

and $k_u \le l_u$5 satisfies

$k_u \le l_u$6

This resolvent form is the direct multi-time analogue of the classical one-dimensional renewal matrix (Kordalis et al., 3 Aug 2025).

3. Canonical renewal objects: transition, renewal, and first passage

Once the resolvent $k_u \le l_u$7 is available, the principal quantities of the theory are written as explicit multi-time renewal solutions. The associated multi-time semi-Markov chain is

$k_u \le l_u$8

Its transition function

$k_u \le l_u$9

satisfies

$u=1,\dots,d$0

where $u=1,\dots,d$1 is the diagonal matrix sequence of complementary cdfs of sojourn times by current state. This is precisely of the form $u=1,\dots,d$2 with $u=1,\dots,d$3 (Kordalis et al., 3 Aug 2025).

The multi-time Markov renewal function $u=1,\dots,d$4 is defined by

$u=1,\dots,d$5

where $u=1,\dots,d$6 counts visits to state $u=1,\dots,d$7 up to multi-time $u=1,\dots,d$8. It satisfies

$u=1,\dots,d$9

so, up to the diagonal structure made explicit in the paper,

$d$0

This is the multi-time analogue of the classical Markov renewal function (Kordalis et al., 3 Aug 2025).

First-passage laws admit both an algebraic closed form and a recursive multi-time renewal equation. If $d$1 is the first-passage pmf to state $d$2, then

$d$3

Conditioning on the first step yields the alternative recursion

$d$4

The second formula makes the structural analogy with the classical one-dimensional first-passage renewal equation explicit, the difference being that convolution is now multidimensional and the order on indices is componentwise (Kordalis et al., 3 Aug 2025).

The same framework also produces ergodic passage-time identities. For an irreducible ergodic multi-time Markov renewal chain with stationary distribution $d$5 of the embedded chain $d$6, the recurrence-time moments satisfy

$d$7

together with the stated formula for $d$8. These identities generalize one-dimensional ergodic relations to the multi-time setting (Kordalis et al., 3 Aug 2025).

4. Continuous-time multi-time laws and renewal-preserving filtering

In continuous time, Markov renewal theory is formulated through an embedded chain $d$9, jump times $(J,S) := (J_n,S_n)_{n\ge 0}$0, holding times $(J,S) := (J_n,S_n)_{n\ge 0}$1, and the semi-Markov kernel

$(J,S) := (J_n,S_n)_{n\ge 0}$2

assumed absolutely continuous with density $(J,S) := (J_n,S_n)_{n\ge 0}$3. The renewal measure is

$(J,S) := (J_n,S_n)_{n\ge 0}$4

and it satisfies the Volterra equation

$(J,S) := (J_n,S_n)_{n\ge 0}$5

When densities exist, the renewal density matrix satisfies

$(J,S) := (J_n,S_n)_{n\ge 0}$6

(Gehri et al., 2024).

The continuous-time multi-time content enters through the backward recurrence time

$(J,S) := (J_n,S_n)_{n\ge 0}$7

and the current state $(J,S) := (J_n,S_n)_{n\ge 0}$8. A classical Markov renewal equation for the joint law of $(J,S) := (J_n,S_n)_{n\ge 0}$9 is

$L = G + q * L,$00

with density

$L = G + q * L,$01

For $L = G + q * L,$02, the joint law over $L = G + q * L,$03 is expressed via iterated convolutions of $L = G + q * L,$04 and survival factors. Compactly, matrix-Volterra formulations use $L = G + q * L,$05 and survival multipliers to express multi-time distributions; practically, the expression is organized as “renewal blocks” $L = G + q * L,$06 separated by survival factors (Gehri et al., 2024).

A major structural result is Anderson’s filtering theorem. If the original state space is partitioned as $L = G + q * L,$07 and the semi-Markov kernel has block form

$L = G + q * L,$08

then the filtered process on $L = G + q * L,$09 is again a Markov renewal process with kernel

$L = G + q * L,$10

and, in Laplace domain,

$L = G + q * L,$11

Two subclasses are singled out: one in which $L = G + q * L,$12 is itself a renewal process, and one in which $L = G + q * L,$13 remains a Markov renewal process after coarse-graining (Gehri et al., 2024).

This filtered continuous-time theory supports exact information-theoretic calculations for Poisson-type channels. For a counting output $L = G + q * L,$14 with predictable intensity $L = G + q * L,$15,

$L = G + q * L,$16

and the filtered hazard into state $L = G + q * L,$17 is

$L = G + q * L,$18

or, equivalently in the paper’s notation,

$L = G + q * L,$19

The framework is applied to bacterial gene expression, where filtering is analytically tractable (Gehri et al., 2024).

5. Algebraic inversion, FFT computation, and asymptotics on large multi-index domains

The multidimensional convolution algebra is designed for computation as well as theory. If $L = G + q * L,$20 is nonsingular, the convolutional inverse admits the representation

$L = G + q * L,$21

with coefficientwise truncation finite at each fixed $L = G + q * L,$22. The same paper gives the classical recurrence for $L = G + q * L,$23, Newton’s inversion method for formal power series, and a convolutional Gauss–Jordan algorithm based on elementary row operations implemented by convolution with elementary matrix sequences (Kordalis et al., 3 Aug 2025).

The computational complexity reported for FFT-based inversion is

$L = G + q * L,$24

while direct convolution scales as $L = G + q * L,$25. The reported experiments are as follows (Kordalis et al., 3 Aug 2025):

Experiment	FFT-based method	Direct method
1D, $L = G + q * L,$26, up to $L = G + q * L,$27	convolution $L = G + q * L,$28s	$L = G + q * L,$29s
2D, $L = G + q * L,$30, $L = G + q * L,$31	convolution $L = G + q * L,$32s	$L = G + q * L,$33s
2D inverse, $L = G + q * L,$34, $L = G + q * L,$35	Gauss–Jordan $L = G + q * L,$36s	$L = G + q * L,$37s

The paper states the corresponding speedups as $L = G + q * L,$38 in the one-dimensional convolution example, $L = G + q * L,$39 in the two-dimensional convolution example, and Gauss–Jordan $L = G + q * L,$40 faster than direct in the two-dimensional inversion example. It also reports a larger two-dimensional run, $L = G + q * L,$41, with FFT $L = G + q * L,$42s versus direct $L = G + q * L,$43s while operating on much larger domains (Kordalis et al., 3 Aug 2025).

A parallel development based on multi-index convolution and multivariate formal power series writes scalar renewal and matrix Markov renewal equations as

$L = G + q * L,$44

with generating functions

$L = G + q * L,$45

respectively. The practical inversion schemes combine FFT-based multidimensional convolution with Newton-type reciprocal iteration, including the matrix update

$L = G + q * L,$46

and convolution on a padded grid of size $L = G + q * L,$47 is stated to have complexity

$L = G + q * L,$48

(Kordalis et al., 30 Jun 2026).

The same framework provides asymptotics under proportional growth. If $L = G + q * L,$49 with $L = G + q * L,$50 and $L = G + q * L,$51, and $L = G + q * L,$52, then the directional rate is

$L = G + q * L,$53

The strong laws are

$L = G + q * L,$54

For additive functionals,

$L = G + q * L,$55

If there is a unique rate-determining coordinate $L = G + q * L,$56, then

$L = G + q * L,$57

The paper also states that fixed-horizon observations induce a genuinely multivariate right-censoring mechanism, leading to an exact nonparametric maximum likelihood estimator and its asymptotic normality (Kordalis et al., 30 Jun 2026).

6. Infinite-times renewal equations and memory of the whole past

The infinite-times formulation replaces finitely many time coordinates by the whole ordered sequence of elapsed times since previous events. For finite $L = G + q * L,$58, with state $L = G + q * L,$59, the renewal PDE is

$L = G + q * L,$60

together with the renewal boundary condition

$L = G + q * L,$61

The shift operator is

$L = G + q * L,$62

and in the infinite-dimensional case $L = G + q * L,$63. The formal generator on test functions is

$L = G + q * L,$64

(Dou et al., 2023).

The hazards are assumed to have the finite-window decomposition

$L = G + q * L,$65

with $L = G + q * L,$66, tail norms

$L = G + q * L,$67

and uniform lower and upper bounds

$L = G + q * L,$68

Under these conditions, for each finite $L = G + q * L,$69 there exists a unique weak solution $L = G + q * L,$70 that preserves positivity, mass, and support (Dou et al., 2023).

Two rigorous notions are given for the infinite-times equation. One is a hierarchy formulation for all finite marginals $L = G + q * L,$71, with coupling terms

$L = G + q * L,$72

satisfying

$L = G + q * L,$73

The other is an infinite-dimensional measure formulation on $L = G + q * L,$74 using test functions depending on finitely many coordinates. Their equivalence is proved via the Kolmogorov extension theorem (Dou et al., 2023).

Long-time behavior is established in both strong and weak metrics. For finite $L = G + q * L,$75, Doeblin bounds yield exponential convergence in $L = G + q * L,$76 to a unique steady state. For the infinite-times model, uniform-in-time strong approximation of marginals and convergence of steady states hold when the hazard tail decays fast enough so that $L = G + q * L,$77. A second route uses a Monge–Kantorovich distance built from the weighted truncated cost

$L = G + q * L,$78

together with the Lipschitz condition

$L = G + q * L,$79

If $L = G + q * L,$80, then

$L = G + q * L,$81

and the paper gives the corresponding infinite-times analogue (Dou et al., 2023).

7. Related matrix theories, precursors, and application domains

A related but distinct line of work studies one-time vector-valued Markov renewal equations of the form

$L = G + q * L,$82

where $L = G + q * L,$83 is a $L = G + q * L,$84 matrix of locally finite measures on $L = G + q * L,$85. The renewal matrix is

$L = G + q * L,$86

and the unique solution is

$L = G + q * L,$87

The asymptotic behavior is governed by the characteristic equation

$L = G + q * L,$88

the pole orders $L = G + q * L,$89 of $L = G + q * L,$90, and the Malthusian parameter $L = G + q * L,$91 determined by $L = G + q * L,$92. In the primitive case, $L = G + q * L,$93 is a simple pole; in reducible cases $L = G + q * L,$94 can exceed $L = G + q * L,$95, which yields polynomial corrections $L = G + q * L,$96 (Kolesko et al., 7 Mar 2025). The same paper states explicitly that extending this residue-calculus approach to genuinely multi-parameter renewal equations would require several complex variables, multidimensional Laplace transforms, and spectral analysis on product half-spaces (Kolesko et al., 7 Mar 2025).

An earlier probabilistic precursor is the quasi-stochastic matrix framework. Given a quasi-stochastic matrix $L = G + q * L,$97, a matrix of distribution functions $L = G + q * L,$98, and the matrix kernel

$L = G + q * L,$99

the matrix renewal measure is

$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$00

with renewal equation

$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$01

The harmonic transform

$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$02

converts $$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$03 into a stochastic matrix, and in the nonarithmetic positive-drift case the renewal theorem gives

$$L = u * G,\qquad u := (\mathbb{I}_s - q)^{(-1)} = \sum_{n\ge 0} q^{(n)},$$04

The exposition connecting this paper to multi-time Markov renewal equations states that the framework naturally yields multi-time integral equations via iterated matrix convolution (Alsmeyer, 2013).

Applications are spread across several domains. In communication theory and stochastic filtering, the continuous-time framework supports simulation-free or reduced-simulation evaluation of mutual information and mutual information rate for Poisson-type channels, including an application to bacterial gene expression (Gehri et al., 2024). In discrete multi-time renewal theory, the stated applications include a binomial--multiset identity, two-attribute warranty evaluation, alternating-renewal availability computation, and discretization-based approximations of continuous-time bivariate renewal and availability models (Kordalis et al., 30 Jun 2026). The infinite-times PDE was motivated by neuroscience, where prediction of the next discharge can depend on the last two discharge times and possibly more; the same paper also mentions epidemics/contact tracing and connections to Hawkes and Wold processes (Dou et al., 2023). The algebraic multidimensional Markov renewal chain framework identifies reliability, maintenance, and biological growth as natural application areas for heterogeneous clocks and jointly distributed waiting times (Kordalis et al., 3 Aug 2025).

Taken together, these works show that Multi-Time Markov Renewal Equations are not a single formalism but a family of renewal structures built around the same core ingredients: a Markovian state update, renewal kernels or hazards, convolutional resolvents, and explicit control of multi-time dependence. This suggests that the subject is best understood as a common operator-theoretic and probabilistic language spanning discrete multi-index time, continuous multi-time observation, and infinite-memory renewal dynamics.