Backward Pullback Recursion

• Backward pullback recursion is a method to obtain stationary solutions for stochastic recursions on partially ordered Polish spaces without assuming monotonicity.
• It uses pullback iterates from a stable envelope to explicitly construct an enriched probability space that captures solution multiplicity.
• Applications, such as in non-monotonic queueing systems, demonstrate its utility in stability analysis and in unifying diverse recursive methods.

Searching arXiv for the core paper and closely related pullback/backward-recursion work to ground the article in current metadata. Backward pullback recursion is the backward, or pullback, construction used to obtain stationary solutions of stochastic recursions of the form $X\circ\theta=\varphi(X)$ on a partially-ordered Polish space when monotonicity of $\varphi$ is not assumed. In the generalized backwards scheme of Moyal, the central object is the pullback iterate

$$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$

and the main result is an explicit construction of an extension of the original probability space on which a stationary solution is well-defined, together with conditions under which the solution is already defined on the original space. The framework is directed toward non-monotonic stochastic recursions and is applied to the stability study of two non-monotonic queueing systems (Moyal, 2010).

1. Formal setting of the recursion

The state space is a partially ordered Polish space $(E,\preceq)$ with Borel $\sigma$-field $\mathscr B(E)$, a bottom element $0_E$, and the property that every $\preceq$-increasing sequence converges in $E$. The driving environment is an ergodic, stationary dynamical system $(\Omega,\mathscr F,\mathbf P,\theta)$, where $\varphi$0 is a measurable bijection, $\varphi$1, and $\varphi$2 is ergodic. The recursion is driven by an $\varphi$3-valued random mapping $\varphi$4, meaning that for each $\varphi$5, the map $\varphi$6 is measurable in $\varphi$7, and $\varphi$8 is measurable from $\varphi$9 into $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$0 (Moyal, 2010).

The discrete-time stochastic recursion is

$$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$1

A stationary solution is an $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$2-valued random variable $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$3 such that

$$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$4

The backward pullback viewpoint does not begin from a forward trajectory. Instead, it considers iterates transported from time $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$5 to time $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$6, and asks whether a limit exists as $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$7 for suitable seeds $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$8, often chosen in an envelope $$X^{(-n)}(\omega;x) = \varphi_{\theta^{-1}\omega}\circ\varphi_{\theta^{-2}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x),$$9.

This formulation isolates the difficulty specific to non-monotonic recursions. When $(E,\preceq)$0 is not assumed monotone, the existence of a stationary solution cannot be reduced to the usual monotone comparison mechanism alone. The generalized backwards scheme therefore replaces a single backward trajectory by a family of pullback images indexed by admissible seeds.

2. Pullback sets and the explicit enriched space

The construction begins with a random set $(E,\preceq)$1 satisfying the stability condition

$(E,\preceq)$2

Typical examples stated in the source are $(E,\preceq)$3 or a finite deterministic set stable by each $(E,\preceq)$4. From this envelope one defines the decreasing pullback sets

$(E,\preceq)$5

Because of the envelope stability, $(E,\preceq)$6 almost surely. The pullback limit set is then

$(E,\preceq)$7

The standing finiteness condition is

$(E,\preceq)$8

Under the hypotheses of the generalized scheme, $(E,\preceq)$9 is a $\sigma$0-a.s. non-empty finite subset of $\sigma$1, and for each $\sigma$2 the restriction

$\sigma$3

is a bijection. Consequently, $\sigma$4 is $\sigma$5-a.s. constant; the source denotes this constant by $\sigma$6 (Moyal, 2010).

The enriched probability space is built explicitly from these pullback limit sets.

Object	Definition	Role
$\sigma$7	$\sigma$8	Enriched state space
$\sigma$9	$\mathscr B(E)$0	Shift on the extension
$\mathscr B(E)$1	Equal weighting on each fiber $\mathscr B(E)$2	Stationary measure
$\mathscr B(E)$3	$\mathscr B(E)$4	Canonical stationary solution

The measure $\mathscr B(E)$5 is defined by weighting each fiber $\mathscr B(E)$6 equally: $\mathscr B(E)$7 The resulting system $\mathscr B(E)$8 is stationary and ergodic, and the canonical projection satisfies

$\mathscr B(E)$9

Thus $0_E$0 is a stationary solution on the extension (Moyal, 2010).

3. Standing assumptions and the existence mechanism

The generalized backwards scheme is organized around four groups of hypotheses. First are the state-space and measurability assumptions already described. Second is the dominating recursion hypothesis (H1): there exists a second $0_E$1-valued random mapping $0_E$2 such that $0_E$3 for all $0_E$4 a.s.; $0_E$5 is $0_E$6-nondecreasing and continuous a.s.; and the recursion

$0_E$7

admits at least one $0_E$8-valued stationary solution $0_E$9.

Third is a regularity alternative. Either (H2) $\preceq$0 is continuous on $\preceq$1 a.s., or (H3) $\preceq$2 has only finitely many discontinuities and there is a $\preceq$3-locally-finite $\preceq$4 a.s. such that $\preceq$5. Fourth are the envelope stability condition for $\preceq$6 and the finiteness of the pullback limit set $\preceq$7 (Moyal, 2010).

Hypothesis	Content
(H1)	Dominating recursion via $\preceq$8 and a stationary solution $\preceq$9
(H2)	$E$0 continuous on $E$1 a.s.
(H3)	Finitely many discontinuities plus invariant locally finite $E$2
(H4)	$E$3

Under (H1) plus (H2) or (H3), the source states that an extension exists by a Prohorov-tightness argument associated with Anantharam–Konstantopoulos. Under (H1), (H2/H3), the envelope stability condition, and (H4), the explicit enrichment described above is valid. This divides the theory into two layers: an abstract existence result for an extension, and an explicit fiberwise construction once the pullback limit set is controlled.

A plausible implication is that the generalized scheme shifts the analytical burden away from global monotonicity and toward the structure of finite pullback images inside a stable envelope. That inference is consistent with the organization of the hypotheses but is not itself stated as a theorem.

4. Projection to the original space, multiplicity, and backward coupling

The enriched construction projects to the original base through $E$4. The invariant events of the enriched system are of the form

$E$5

The cardinality $E$6 determines whether the stationary solution already lives on the original probability space and whether it is unique (Moyal, 2010).

If $E$7 a.s., then $E$8 and $E$9 is the unique $(\Omega,\mathscr F,\mathbf P,\theta)$0-valued solution of

$(\Omega,\mathscr F,\mathbf P,\theta)$1

on $(\Omega,\mathscr F,\mathbf P,\theta)$2. In this case the pullback iterates couple strongly backward: $(\Omega,\mathscr F,\mathbf P,\theta)$3 This is the strongest conclusion in the scheme: the extension collapses to a single measurable selector on the original space, and every admissible pullback sequence converges to it.

If $(\Omega,\mathscr F,\mathbf P,\theta)$4, one does not obtain uniqueness on the original space. Instead, the source states that there are exactly $(\Omega,\mathscr F,\mathbf P,\theta)$5 distinct solutions on the original space, corresponding to the $(\Omega,\mathscr F,\mathbf P,\theta)$6 orbits in $(\Omega,\mathscr F,\mathbf P,\theta)$7 under successive $(\Omega,\mathscr F,\mathbf P,\theta)$8-applications. In this regime the enrichment is not merely a technical device; it records the multiplicity that remains invisible from the base alone.

5. Operational form and queueing applications

The construction can be read algorithmically. One computes a stable envelope $(\Omega,\mathscr F,\mathbf P,\theta)$9, initializes $\varphi$00, and then iterates

$\varphi$01

until stabilization or until a finite cardinality condition is reached with positive probability. One then sets

$\varphi$02

forms the extended space $\varphi$03, defines $\varphi$04, and sets $\varphi$05. If $\varphi$06 a.s., then $\varphi$07 is obtained directly on the original space and the backward coupling conclusion follows (Moyal, 2010).

The illustrative example in the source is the Loss-Queue $\varphi$08. Here $\varphi$09 with the usual order, $\varphi$10 denotes the interarrival time of $\varphi$11, and $\varphi$12 the service time of $\varphi$13, with the marks jointly ergodic. The source states that if $\varphi$14 then Loynes’ classical stability implies a unique solution on the original space. Otherwise one takes $\varphi$15 and constructs

$\varphi$16

Under mild ergodicity and moment conditions, $\varphi$17 is a finite nonempty set a.s., leading to an enriched space of cardinal $\varphi$18 and exactly $\varphi$19 stationary workload distributions.

The same section states that this framework recovers the constructions of Flipo–Neveu and of Anantharam–Konstantopoulos as special or dominated cases. This situates backward pullback recursion as a unifying method for non-monotonic queueing recursions rather than a queue-specific ad hoc construction.

6. Terminological scope and neighboring uses of “pullback” and “backward”

A common source of confusion is the breadth of the words “pullback” and “backward” across different areas. In the stochastic-recursion setting, the pullback object is the iterated composition

$\varphi$20

applied to admissible seeds inside an envelope, with the goal of constructing stationary solutions (Moyal, 2010).

By contrast, “Pullback parking functions” uses “pullback” for a combinatorial parking rule in which cars may move backwards up to $\varphi$21 spots and forwards up to $\varphi$22 spots on a one-way street; the main result counts such parking functions by final parking outcome and via a recursive formula (Elder et al., 21 Mar 2025). “A Differential-form Pullback Programming Language for Higher-order Reverse-mode Automatic Differentiation” uses pullbacks of differential $\varphi$23-forms, with

$\varphi$24

to formulate reverse-mode AD in a higher-order setting (Mak et al., 2020). “Getting There and Back Again” uses “back again” for a recursion pattern in which recursive calls traverse one data structure and returns traverse another, formalized in Coq and accompanied by a tail-recursive variant called There and Forth Again (Danvy, 2022).

These uses share vocabulary but not the same construction. This suggests terminological overlap rather than methodological identity. The stochastic notion of backward pullback recursion is specifically a stationary-solution method for random recursions over an ergodic dynamical system, with explicit dependence on pullback iterates, invariant envelopes, and extension of the probability space.