---
title: Backward Pullback Recursion
url: https://www.emergentmind.com/topics/backward-pullback-recursion
type: topic
---

# Backward Pullback Recursion

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 [1009.1235].

## 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 $\theta\colon\Omega\to\Omega$ is a measurable bijection, $\mathbf P\circ\theta^{-1}=\mathbf P$, and $\theta$ is ergodic. The recursion is driven by an $M(E)$-valued random mapping $\varphi$, meaning that for each $\omega\in\Omega$, the map $\varphi_\omega\colon E\to E$ is measurable in $x$, and $\omega\mapsto\varphi_\omega$ is measurable from $\Omega$ into $M(E)$ [1009.1235].

The discrete-time stochastic recursion is
\[
X_{n+1}
=
\varphi_{\theta^n\omega}(X_n),
\qquad X_0\in E.
\]
A stationary solution is an $E$-valued random variable $X$ such that
\[
X\circ\theta
=
\varphi(X)
\quad\text{$\mathbf P$-a.s.}
\]
The backward pullback viewpoint does not begin from a forward trajectory. Instead, it considers iterates transported from time $-n$ to time $0$, and asks whether a limit exists as $n\to\infty$ for suitable seeds $x_n\in E$, often chosen in an envelope $G(\theta^{-n}\omega)$.

This formulation isolates the difficulty specific to non-monotonic recursions. When $\varphi$ 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 $G(\omega)\subset E$ satisfying the stability condition
\[
\varphi_\omega(G(\omega))
\subset
G(\theta\omega)
\quad\text{a.s.}
\]
Typical examples stated in the source are $G\equiv E$ or a finite deterministic set stable by each $\varphi$. From this envelope one defines the decreasing pullback sets
\[
H_n(\omega)
:=
\Bigl\{
\varphi_{\theta^{-1}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}(x)
\;;\;
x\in G(\theta^{-n}\omega)
\Bigr\},
\qquad n\ge 1.
\]
Because of the envelope stability, $H_{n+1}(\omega)\subset H_n(\omega)$ almost surely. The pullback limit set is then
\[
H(\omega)=\bigcap_{n\ge 1}H_n(\omega).
\]

The standing finiteness condition is
\[
\mathbf P\bigl\{\,\omega\;;\;0<|H(\omega)|<\infty\,\bigr\}>0.
\]
Under the hypotheses of the generalized scheme, $H(\omega)$ is a $\mathbf P$-a.s. non-empty finite subset of $E$, and for each $\omega$ the restriction
\[
\varphi_\omega\colon H(\omega)\longrightarrow H(\theta\omega)
\]
is a bijection. Consequently, $|H(\omega)|$ is $\mathbf P$-a.s. constant; the source denotes this constant by $c$ [1009.1235].

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

| Object | Definition | Role |
|---|---|---|
| $\Omega^*$ | $\{(\omega,x):\omega\in\Omega,\ x\in H(\omega)\}$ | Enriched state space |
| $\theta^*$ | $\theta^*(\omega,x)=(\theta\omega,\varphi_\omega(x))$ | Shift on the extension |
| $\mathbf P^*$ | Equal weighting on each fiber $H(\omega)$ | Stationary measure |
| $X^*$ | $X^*(\omega,x)=x$ | Canonical stationary solution |

The measure $\mathbf P^*$ is defined by weighting each fiber $H(\omega)$ equally:
\[
\mathbf P^*(A)
=
\int_\Omega
\frac{1}{c}\;
\#\{x\in H(\omega):(\omega,x)\in A\}\;
\mathbf P(d\omega).
\]
The resulting system $(\Omega^*,\mathscr F^*,\mathbf P^*,\theta^*)$ is stationary and ergodic, and the canonical projection satisfies
\[
X^*\circ\theta^*=\varphi(X^*)
\quad\text{a.s.}
\]
Thus $X^*$ is a stationary solution on the extension [1009.1235].

## 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 $M(E)$-valued random mapping $V_\omega$ such that $0_E\preceq\varphi_\omega(x)\preceq V_\omega(x)$ for all $x\in E$ a.s.; $x\mapsto V_\omega(x)$ is $\preceq$-nondecreasing and continuous a.s.; and the recursion
\[
Y\circ\theta=V(Y)
\]
admits at least one $E$-valued stationary solution $Y$.

Third is a regularity alternative. Either (H2) $\varphi_\omega(\cdot)$ is continuous on $E$ a.s., or (H3) $\varphi_\omega$ has only finitely many discontinuities and there is a $\preceq$-locally-finite $L(\omega)\subset E$ a.s. such that $\varphi_\omega(L(\omega))\subset L(\theta\omega)$. Fourth are the envelope stability condition for $G$ and the finiteness of the pullback limit set $H$ [1009.1235].

| Hypothesis | Content |
|---|---|
| (H1) | Dominating recursion via $V_\omega$ and a stationary solution $Y$ |
| (H2) | $\varphi_\omega(\cdot)$ continuous on $E$ a.s. |
| (H3) | Finitely many discontinuities plus invariant locally finite $L(\omega)$ |
| (H4) | $\mathbf P\{0<|H(\omega)|<\infty\}>0$ |

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 $(\omega,x)\mapsto\omega$. The invariant events of the enriched system are of the form
\[
\{(\omega,x):x\in I(\omega)\}
\quad\text{with }I(\omega)\subset H(\omega)\text{ constant on }\theta^*\text{-orbits.}
\]
The cardinality $c=|H(\omega)|$ determines whether the stationary solution already lives on the original probability space and whether it is unique [1009.1235].

If $c=1$ a.s., then $H(\omega)=\{X(\omega)\}$ and $X$ is the unique $E$-valued solution of
\[
X\circ\theta=\varphi(X)
\]
on $(\Omega,\mathscr F,\mathbf P,\theta)$. In this case the pullback iterates couple strongly backward:
\[
X^{(-n)}(\omega;x)\longrightarrow X(\omega)
\quad\text{a.s. for any seed }x\in G(\theta^{-n}\omega).
\]
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 $c>1$, one does not obtain uniqueness on the original space. Instead, the source states that there are exactly $c$ distinct solutions on the original space, corresponding to the $c$ orbits in $H(\omega)$ under successive $\varphi$-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 $G(\omega)$, initializes $H_0(\omega)\leftarrow G(\omega)$, and then iterates
\[
H_n(\omega)\leftarrow
\varphi_{\theta^{-1}\omega}(H_{n-1}(\theta^{-1}\omega))
\]
until stabilization or until a finite cardinality condition is reached with positive probability. One then sets
\[
H(\omega)\leftarrow \bigcap_{k=1}^n H_k(\omega),
\]
forms the extended space $\Omega^*=\{(\omega,x):x\in H(\omega)\}$, defines $\theta^*(\omega,x)=(\theta\omega,\varphi_\omega(x))$, and sets $X^*(\omega,x)=x$. If $|H(\omega)|\equiv 1$ a.s., then $X(\omega)$ is obtained directly on the original space and the backward coupling conclusion follows [1009.1235].

The illustrative example in the source is the Loss-Queue $G/G/1/1$. Here $E=\mathbb R_+$ with the usual order, $\xi_n$ denotes the interarrival time of $C_n$, and $\sigma_n$ the service time of $C_n$, with the marks jointly ergodic. The source states that if $\mathbf P\{\sigma<\xi\}>0$ then Loynes’ classical stability implies a unique solution on the original space. Otherwise one takes $G(\omega)=E=\mathbb R_+$ and constructs
\[
H_n(\omega)
=
\{\text{possible residual workloads at time 0 if at $-n$ the system was in }G(\theta^{-n}\omega)\}.
\]
Under mild ergodicity and moment conditions, $H(\omega)$ is a finite nonempty set a.s., leading to an enriched space of cardinal $c$ and exactly $c$ 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
\[
\varphi_{\theta^{-1}\omega}\circ\cdots\circ\varphi_{\theta^{-n}\omega}
\]
applied to admissible seeds inside an envelope, with the goal of constructing stationary solutions [1009.1235].

By contrast, “Pullback parking functions” uses “pullback” for a combinatorial parking rule in which cars may move backwards up to $k$ spots and forwards up to $\ell$ spots on a one-way street; the main result counts such parking functions by final parking outcome and via a recursive formula [2503.17256]. “A Differential-form Pullback Programming Language for Higher-order Reverse-mode Automatic Differentiation” uses pullbacks of differential $1$-forms, with
\[
(f^*\omega)(x)=(Jf(x))^\top\circ\omega(f(x)),
\]
to formulate reverse-mode AD in a higher-order setting [2002.08241]. “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 [2203.00145].

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.

Source: https://www.emergentmind.com/topics/backward-pullback-recursion