Probabilistic Carleman Linearization (PCL)

• Probabilistic Carleman Linearization (PCL) is a method that lifts nonlinear stochastic dynamics into an infinite-dimensional deterministic linear system using Kronecker powers, enabling uncertainty quantification.
• It applies to discrete-time stochastic polynomial systems and continuous-time SDEs, offering techniques like exact finite-horizon recovery and high-probability truncation-error bounds.
• PCL balances the computational challenges of high-dimensional lifts with practical safety certification through controlled moment propagation and explicit error analysis.

Searching arXiv for papers on probabilistic Carleman linearization and closely related Carleman methods. Probabilistic Carleman Linearization (PCL) denotes a class of methods that extend Carleman linearization from deterministic polynomial dynamics to uncertainty-aware settings by lifting nonlinear dynamics into linear systems over Kronecker powers and then propagating either moments or high-probability truncation bounds. In the discrete-time stochastic polynomial literature, PCL is the transformation of a stochastic polynomial system into an infinite-dimensional deterministic linear moment system, followed by truncation and probabilistic safety analysis (Pruekprasert et al., 2022). In work on nonlinear stochastic differential equations driven by Ornstein–Uhlenbeck noise, the same term refers to Carleman truncation equipped with probabilistic error control rather than purely worst-case bounds (Li et al., 12 Mar 2026). Across these variants, the central objective is unchanged: replace nonlinear evolution by lifted linear dynamics while retaining quantitatively controlled approximation error.

1. Scope, problem classes, and relation to classical Carleman lifting

Classical Carleman linearization starts from polynomial dynamics and introduces lifted coordinates built from tensor or Kronecker powers of the state. For quadratic ordinary differential equations of the form

$\frac{d x(t)}{dt} = F_1 x + F_2 x^{\otimes 2},$

the lift

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$

produces an infinite-dimensional linear system, which is then truncated at order $N$ and solved as a finite linear ODE in the lifted variables (Forets et al., 2021). Explicit truncation-error analysis for such polynomial systems was developed in deterministic form by Forets and Pouly, who studied polynomial ODEs with equilibrium at the origin,

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$

and derived explicit time-dependent error bounds for finite Carleman truncations (Forets et al., 2017).

Within the probabilistic literature considered here, PCL appears in two closely related senses. The first is moment propagation for discrete-time stochastic polynomial systems with random coefficients, where expectation converts the lifted stochastic recursion into a deterministic linear recursion for raw Kronecker moments (Pruekprasert et al., 2019). The second is probabilistic truncation control for nonlinear stochastic differential equations, where the lifted linear system remains random and the truncation error is treated as a random variable controlled by concentration arguments (Li et al., 12 Mar 2026).

This dual usage matters conceptually. In the discrete-time moment-propagation setting, probabilistic structure enters through expectations and safety probabilities. In the stochastic-differential setting, probabilistic structure enters through tail bounds on the truncation residual. A plausible implication is that PCL is better viewed as a methodology family than as a single canonical algorithm.

2. Lifted moment dynamics for discrete-time stochastic polynomial systems

The discrete-time PCL framework studies stochastic polynomial dynamics of the form

$x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$

where $x(t)\in \mathbb{R}^n$, the $F_i(t)$ are random matrix-valued processes, $F_0(t)$ represents additive-noise structure, and $F_1(t),\dots,F_d(t)$ encode multiplicative random polynomial coefficients (Pruekprasert et al., 2022). The method assumes that the initial condition and all coefficient processes are mutually independent, and that the coefficient processes are identically distributed over time (Pruekprasert et al., 2022).

The lift is constructed from Kronecker powers. For truncation level $k$, the lifted vector is

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$0

The combinatorial expansion of

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$1

yields block matrices

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$2

with the sparsity property

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$3

Consequently, the lifted system is an infinite-dimensional linear stochastic system in the monomial coordinates (Pruekprasert et al., 2022).

Taking expectation and using independence between the current lifted coefficient block and the past lifted state produces an exact deterministic moment recursion,

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$4

where

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$5

This is the central structural step of discrete-time PCL: nonlinear stochastic dynamics are recast as an infinite deterministic linear system for moments (Pruekprasert et al., 2022).

The practical approximation truncates at order $\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$6, defining

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$7

which evolves according to the finite-dimensional linear time-invariant system

$\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$8

This is the form used for fast offline/online propagation in the safety-analysis papers (Pruekprasert et al., 2022).

3. Truncation, exactness, and computational structure

The defining approximation question in PCL is how much accuracy is lost when the infinite lifted hierarchy is cut at finite order. For the $\hat y_j := x^{\otimes j}, \qquad j\in \mathbb{N},$9-th moment, the discrete-time moment error is

$N$0

The 2022 moment-propagation paper derives an exact expansion in which the error consists only of contributions from missing higher-order moments beyond the truncation limit, and then proves the global bound

$N$1

with

$N$2

It also gives tighter partially exact bounds based on selected index subsets $N$3 or coordinate subsets $N$4, which are used for efficient online error estimation (Pruekprasert et al., 2022).

A distinctive feature of the discrete-time theory is exact finite-horizon recovery for sufficiently large truncation. The paper states that

$N$5

This sharply distinguishes the discrete-time moment framework from continuous-time Carleman truncation, where exact recovery generally does not occur (Pruekprasert et al., 2022). The earlier 2019 paper established the same general phenomenon: sufficiently large truncation limits can precisely compute moments for sufficiently small degrees and numbers of time steps (Pruekprasert et al., 2019).

Because the ambient dimension of unreduced Kronecker powers grows rapidly, the 2022 formulation introduces reduced Kronecker powers. For $N$6, the reduced $N$7-th Kronecker power keeps only distinct monomials indexed by multi-indices of total degree $N$8. The paper illustrates the reduction with

$N$9

in place of the unreduced

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$0

This reduces offline matrix construction time, memory use, and online propagation cost (Pruekprasert et al., 2022).

The same computational logic already appeared in the 2019 predecessor, which organized the method as an iterative propagation of approximate raw Kronecker moments through a truncated deterministic linear system and emphasized that the main cost driver is the rapid growth of lifted dimension with truncation order and state dimension (Pruekprasert et al., 2019).

4. Probabilistic safety analysis and certified regions

In the discrete-time PCL literature, propagated moments are used not only as descriptive statistics but as ingredients for safety certification. The 2022 paper defines an ellipsoid by a positive semidefinite matrix $x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$1 and seminorm

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$2

with region

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$3

The central quantity is the deviation of the state from the propagated mean proxy $x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$4, together with a global mean-approximation error bound of the form

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$5

Markov’s inequality then yields a probabilistic bound of the form

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$6

and, when first and second moments are sufficiently accurate, a covariance-based surrogate using $x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$7 and $x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$8 (Pruekprasert et al., 2022).

The same paper formulates online safe-ellipsoid construction as an approximate three-step optimization procedure. First, it solves a convex surrogate

$x'(t)=f(x(t))=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$9

subject to a chance-bound constraint expressed through propagated second moments. Second, it scales $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$0 by a scalar $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$1 using a moment-error over-approximation. Third, it enlarges the resulting ellipsoid by the mean-error bound to obtain a certified region that contains the state with probability at least $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$2 (Pruekprasert et al., 2022). This yields a fully moment-based safety pipeline with an explicit offline/online split.

The earlier 2019 paper used the same moment machinery for safety analysis, though with Euclidean-ball regions centered at the approximate mean $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$3 and tail bounds derived from Chebyshev’s inequality and second moments (Pruekprasert et al., 2019). In both formulations, the safety certificate is only as strong as the moment approximation and its truncation-error bound. This suggests that error calibration, rather than lifting alone, is the decisive element in probabilistic use of Carleman methods.

The example systems in this line of work are a stochastic logistic map and a vehicle dynamics model under stochastic disturbance. The logistic example uses

$x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$4

with $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$5 in the 2022 paper, while the vehicle example uses a polynomialized and discretized kinematic bicycle model with stochastic acceleration and demonstrates that larger truncation limits produce moment trajectories closer to Monte Carlo means (Pruekprasert et al., 2022).

5. Continuous-time stochastic PCL and high-probability truncation error

A different PCL construction appears in the quantum-simulation paper on nonlinear stochastic differential equations driven by Ornstein–Uhlenbeck noise. The system is

$x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$6

where $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$7 is an $x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$8-dimensional OU process satisfying

$x(t+1)=\sum_{i=0}^{d} F_i(t)\,x^{[i]}(t),$9

Here the Carleman lift again uses tensor powers,

$x(t)\in \mathbb{R}^n$0

and truncation yields a finite linear stochastic system in the lifted variables (Li et al., 12 Mar 2026).

The paper characterizes the truncation residual by

$x(t)\in \mathbb{R}^n$1

Unlike deterministic Carleman analysis, the state norm is now random because the forcing is random. The paper therefore introduces a Lyapunov-norm framework with $x(t)\in \mathbb{R}^n$2, logarithmic stability parameter $x(t)\in \mathbb{R}^n$3, one-sided quadratic growth parameter $x(t)\in \mathbb{R}^n$4, and

$x(t)\in \mathbb{R}^n$5

Under these conditions it proves the pathwise estimate

$x(t)\in \mathbb{R}^n$6

which reduces truncation control to concentration of the OU supremum (Li et al., 12 Mar 2026).

This is the point at which the method becomes explicitly probabilistic. By combining the pathwise state bound with Gaussian concentration for the OU process, the paper derives a high-probability truncation bound with a simplified asymptotic form

$x(t)\in \mathbb{R}^n$7

for constants $x(t)\in \mathbb{R}^n$8, and, in the steady-state perturbation setting, a sharper bound

$x(t)\in \mathbb{R}^n$9

The paper interprets this as probabilistic exponential convergence for Carleman linearization, provided the nonlinear stochastic system is stable and reaches a steady state (Li et al., 12 Mar 2026).

In that work, PCL is coupled to stochastic linear combination of Hamiltonian simulations (SLCHS) for quantum simulation. The quantum component is separate from the mathematical definition of PCL itself. The PCL contribution is the nonlinear-to-linear reduction together with probabilistic truncation-error control; the SLCHS contribution is the simulation of the resulting linear stochastic system (Li et al., 12 Mar 2026).

6. Deterministic foundations, misconceptions, and limitations

PCL is not identical to the broader Carleman literature. Several prominent papers are deterministic counterparts or precursors rather than probabilistic methods. The reachability paper on weakly nonlinear polynomial ODEs embeds the nonlinear system into a high-dimensional linear system via Carleman linearization, propagates sets using support functions, and obtains sound dense-time reachable sets by bloating the projected linear flowpipe with a global truncation-error bound (Forets et al., 2021). That framework already separates uncertainty in the initial condition set, deterministic lifted linear dynamics, and explicit truncation error, and the paper notes that this structure naturally suggests propagation of random initial conditions or probabilistic reachable sets. However, it also states explicitly that there are no stochastic disturbances and that the method as written is not probabilistic (Forets et al., 2021).

The same distinction applies to the 2024 quantum-analysis paper, which studies deterministic Carleman linearization beyond the dissipative regime. It proves that under a no-resonance condition, truncation error satisfies

$F_i(t)$0

with a rate controlled by the resonance gap $F_i(t)$1, and emphasizes that there is no randomized lifting, no Monte Carlo sampling of coefficients, no stochastic truncation rule, and no expectation-based estimator (Wu et al., 2024). This directly addresses a common misconception: not every modern extension of Carleman truncation to broader dynamical regimes is a form of PCL.

Deterministic error theory remains foundational for PCL. Forets and Pouly derived two explicit truncation-error bounds, one requiring an a priori trajectory bound and one based only on $F_i(t)$2, $F_i(t)$3, and $F_i(t)$4, thereby supplying computable worst-case envelopes for finite Carleman truncations (Forets et al., 2017). More recent work has refined this theory by producing degree-aware observable-specific estimates via Dyson–Duhamel expansions (Boreale et al., 23 Jun 2026), semigroup-based convergence results for unbounded generators and semi-discretized evolution equations (Gakkhar et al., 5 May 2026), and well-posedness and convergence theorems for parabolic PDEs with a clean separation between linearization error and discretization error (Heinzelreiter et al., 1 Oct 2025). These works are deterministic, but they identify structural mechanisms—degree propagation, dissipativity, semigroup generation, and discretization independence—that any probabilistic extension must still respect.

The main limitations of PCL as currently represented in the literature are consistent across deterministic and probabilistic variants. The lifted dimension grows rapidly, typically as

$F_i(t)$5

in unreduced tensor coordinates (Forets et al., 2021). The discrete-time stochastic moment framework assumes polynomial dynamics, temporal independence, and identical distribution of the coefficient processes (Pruekprasert et al., 2022). The continuous-time stochastic framework requires stability assumptions and is strongest when the system reaches a steady state (Li et al., 12 Mar 2026). Safety bounds may become conservative when moment-error estimates are loose (Pruekprasert et al., 2019). Exact finite-horizon recovery is a discrete-time phenomenon and does not transfer directly to continuous-time Carleman truncation (Pruekprasert et al., 2022).

A final misconception concerns the source of probabilistic content. PCL is not, in the papers considered here, a randomized embedding or stochastic truncation heuristic. In the discrete-time literature, the probabilistic aspect lies in moment propagation and safety certification for stochastic systems (Pruekprasert et al., 2022). In the OU-driven NSDE literature, it lies in high-probability truncation-error bounds derived from stochastic forcing concentration (Li et al., 12 Mar 2026). This suggests that the defining feature of PCL is not randomness in the lifting map itself, but the systematic integration of probability into the lifted linear approximation and its error calculus.