Lyapunov Moment Function

• Lyapunov moment function is defined as the limit of the normalized exponential growth rate of the p-th moment of observables in random systems.
• It is convex and analytic under regular conditions, enabling applications in large deviations, central limit theorems, and moderate deviation analysis.
• The function applies to Markov processes, stochastic differential equations, and Hamiltonian systems, with numerical bounds obtained via optimization methods like SDP and SOS.

The Lyapunov moment function generalizes Lyapunov exponents to encode the asymptotic exponential growth rate of the $p$-th moment (or more generally, the cumulant-generating function) of observables along stochastic or random dynamical trajectories. This concept unifies and extends ideas from stability theory, large deviations, random matrix products, stochastic processes, and statistical mechanics. In contemporary mathematics and physics, the Lyapunov moment function appears in random matrix theory, Markov processes, stochastic differential equations, and the paper of high-dimensional dynamical and Hamiltonian systems.

1. Fundamental Definition and Characterizations

Let $(X_t)_{t\ge0}$ be a Markov or stochastic process or, in discrete time, a sequence of matrix products, observables, or solutions to a dynamical system. The Lyapunov moment function, commonly denoted $\Lambda(p)$ or $L(p)$, captures the normalized exponential growth rate of the $p$-th moment of a multiplicative functional. For a sequence of random matrices $A_n$ (i.i.d. or with stationary increments) and a vector $v$, it is defined as

$$\Lambda(p) = \lim_{n\to\infty} \frac{1}{n}\log \mathbb{E}\left[\|A_nA_{n-1}\cdots A_1 v\|^{p}\right],$$

provided the limit exists. More generally, for stochastic processes with an additive observable $A_T$, the definition extends as

$$\Lambda(p) = \lim_{T\to\infty} \frac{1}{T}\log \mathbb{E}\left[e^{pA_T}\right].$$

This function is convex and, under suitable regularity and integrability assumptions, differentiable and analytic in a neighborhood of $p=0$.

In the context of continuous-time Markov chains with affine transition rates, Lyapunov moment functions are constructed to bound moments of the invariant distribution via optimization-based Lyapunov function construction (Milias-Argeitis et al., 2014).

For random matrix products, the joint analyticity of the Lyapunov moment function in both the exponent $p$ and the driving measure (law) $\mu$ is established under strong irreducibility and proximality (Chalhoub et al., 4 Dec 2025). For random dynamical systems, $\Lambda(p)$ also governs exponential tail probabilities via large deviations principles (Baxendale, 21 Jul 2025).

2. Analytical Properties and Cumulant Expansion

Analyticity of the Lyapunov moment function plays a foundational role in statistical and ergodic properties of the underlying stochastic system. Under irreducibility and spectral gap hypotheses, the function $\Lambda(p)$ extends to a Fréchet-holomorphic function of both $p$ and the driving measure $\mu$ in a neighborhood of $p=0$ (Chalhoub et al., 4 Dec 2025). The derivatives of $\Lambda$ at $p=0$ have concrete probabilistic interpretations: $$\Lambda'(0) = \text{(mean Lyapunov exponent)} \qquad \Lambda''(0) = \text{(asymptotic variance)} > 0,$$ and higher derivatives correspond to joint cumulants: $$\Lambda^{(k)}(0) = \lim_{n\to\infty} \frac{1}{n}\kappa_k\left(\log \|A_n\cdots A_1 v\|\right),$$ where $\kappa_k$ denotes the $k$-th cumulant.

The Taylor expansion near $p=0$ gives

$$\Lambda(p) = \lambda p + \frac{\sigma^2}{2}p^2 + \frac{\kappa_3}{6}p^3 + \cdots,$$

enabling applications of central limit theorems and moderate deviation principles for the finite-time Lyapunov exponent (Baxendale, 21 Jul 2025).

3. Lyapunov Moment Functions in Markov and Stochastic Processes

In the paper of continuous-time Markov chains (CTMC) with affine transition rates, Lyapunov moment functions underlie algorithms for explicit upper bounds on stationary moments. The optimization-based methodology uses parameterized Lyapunov functions $V(x)$ to enforce the Foster–Lyapunov drift condition: $QV(x) \leq -\|x\|^p + b,$ where $Q$ is the generator. If such a $V$ is found, $\mathbb{E}_\pi[\|X\|^p] \leq b$ for the stationary law $\pi$ (Milias-Argeitis et al., 2014). Quadratic test functions and polynomial inequalities are reduced to semidefinite programming (SDP), with sum-of-squares (SOS) or S-procedure relaxations.

In stochastic differential equations (SDEs) on manifolds, the Lyapunov moment function is characterized as the principal eigenvalue of a twisted elliptic operator on the unit sphere bundle. For an SDE with generator $L$ and additive functional $A_T$, the operator

$L_p = L + pY + pQ + \frac{p^2}{2}R$

acts on smooth functions, and the principal eigenvalue is $\psi(p) = \Lambda(p)$ (Baxendale, 21 Jul 2025). The existence and uniqueness of solutions are ensured by Hörmander hypoellipticity, positivity, and Lyapunov growth conditions.

4. Applications in Statistical Mechanics and Random Fields

In statistical mechanics, moment Lyapunov exponents appear in models such as the monomer-dimer problem. The thermodynamic free energy per site can be explicitly written in terms of the moment Lyapunov exponent (MLE) of an associated Gaussian random field: $f(\beta) = -T \Lambda(\beta),$ where $$\Lambda(\beta)=\lim_{| \Lambda | \to \infty} (1/|\Lambda|) \log \mathbb{E}\prod_{x\in\Lambda}\xi_x$$ for a Gaussian field $\xi$ (Vladimirov, 2012). In the one-dimensional case, the MLE is computed as the logarithm of the dominant eigenvalue of a transfer/pantograph operator, with explicit connections to creation and annihilation operators in Fock space.

This framework generalizes to interacting particle systems and models with infinite-range correlations, where the spectral representation of the covariance function allows a variational or large-deviation formulation for $\Lambda$.

5. Large Deviations, Central Limit Theorem, and Moderate Deviations

The Lyapunov moment function serves as the cumulant-generating function in the context of large deviations: if $\Lambda(p)$ is differentiable, the Gartner–Ellis theorem yields the large deviation rate function for the finite-time Lyapunov exponent $\Lambda_T$: $I(s) = \sup_{p\in\mathbb{R}} (ps - \Lambda(p)),$ and

$\Pr(\Lambda_T \approx s) \asymp e^{-T I(s)}.$

The second derivative $\Lambda''(0)$ governs both the CLT and moderate deviation regimes: $$\sqrt{T}\left(\frac{A_T}{T} - \lambda\right) \xrightarrow{d} N(0, \Lambda''(0))$$ and for scaling sequences $\sqrt{T}\ll b_T\ll T$,

$\frac{A_T - T\lambda}{b_T}$

satisfies an LDP with speed $b_T^2/T$ and rate function $s^2/(2\Lambda''(0))$ (Baxendale, 21 Jul 2025).

6. Lyapunov-Moment Methods in Hamiltonian Systems

A distinct approach, the energy-second-moment map for Hamiltonian systems, defines Lyapunov functions in terms of the evolution of moments of energy and quadratic phase-space coordinates. For generic $H=p^2/2+V(q,t)$, the map between the initial and evolved second moments is linear, governed by a third-order linear ODE for the associated fundamental matrix. The corresponding Lyapunov functions,

$$\lambda_k(t) = \frac{1}{t}\int_0^t b_{kk}(\tau)d\tau,$$

where $b_{kk}(t)$ are entries from the $QR$ decomposition of the fundamental matrix, yield a simple characterization of regular (integrable) versus irregular (chaotic) behavior depending on whether $\lambda_k(t)$ converge or fluctuate indefinitely (Struckmeier et al., 2023).

This technique has been demonstrated in practical computations for the Hénon–Heiles oscillator, coupled quartic oscillators, and restricted three-body problems, where persistent nonconvergence of finite-time Lyapunov moment functions serves as a robust diagnostic of phase-space chaos.

7. Perturbative and Numerical Methods, Optimization, and Further Generalizations

Lyapunov moment functions admit local power series and perturbative expansions around $p=0$, with coefficients determined by cumulants or solutions to associated spectral problems (Chalhoub et al., 4 Dec 2025, Baxendale, 21 Jul 2025). In Markovian, chemical, or queuing models with polynomial or affine transition structure, moment function bounds are realized via convex optimization (SDP or SOS), which can be solved efficiently and yield sharp moment inequalities for the stationary distribution (Milias-Argeitis et al., 2014).

An immediate application is to stochastic optimization and learning algorithms where quadratic Lyapunov functions have been constructed to yield rates and regularity criteria for accelerated methods such as Polyak’s heavy-ball method (Orvieto, 2023).

A plausible implication is that further extensions of the Lyapunov moment function will underpin future developments in high-dimensional stochastic stability, statistical mechanics of disordered systems, and data-driven methods for large-scale random dynamical systems.