p-th Moment Lyapunov Exponent in Random Systems

• The p-th moment Lyapunov exponent is a growth rate that quantifies asymptotic behavior of moments in multiplicative random evolutions, reflecting intermittency and stability in stochastic models.
• Exact formulas are obtained using techniques like contour-integral representations, steepest descent, and the Bethe ansatz, particularly in integrable stochastic PDEs and random matrix products.
• Generalizations of the exponent to fractional, colored-noise, and manifold settings extend its application to large deviation theory, spectral analysis, and multiplicative ergodic problems.

The $p$-th moment Lyapunov exponent is an asymptotic growth rate for moments of multiplicative random evolutions. Across stochastic PDEs, random media, random matrix products, and derived tangent flows, it appears in limits of the form

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$

with the normalization $a$ determined by the model. In the literature it plays several roles simultaneously: an annealed growth exponent, a scaled cumulant-generating function, a principal eigenvalue of a twisted operator, and, via Legendre transform, an input to large-deviation theory for finite-time Lyapunov exponents (Borodin et al., 2012, Das et al., 2019, Charbonneau et al., 2017, Baxendale, 21 Jul 2025).

1. Definitions and normalization conventions

The notion is not tied to a single normalization. In discrete and continuum parabolic Anderson models with white-in-time forcing, the natural scaling is linear in time. In long-range temporal-covariance models and fractional parabolic equations, the logarithm of the $p$-th moment grows superlinearly, so the correct normalization is $t^{-\rho}$ or $t^{-\kappa}$. In random matrix products, the corresponding object is usually the scaled cumulant-generating function of the finite-sample Lyapunov exponent. Some authors use the unnormalized quantity $L(p)$, while others divide by $p$ (Borodin et al., 2012, Lê, 2015, Charbonneau et al., 2017, Chen et al., 2016).

Setting	Definition	Time/length scale
Discrete-space PAM	$$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$	$t$
SHE with delta data	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$0	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$1
Fractional-time Gaussian PAM	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$2	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$3
Fractional PAM	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$4	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$5
Random matrices	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$6	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$7
Hybrid stochastic heat equation	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$8	$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$9

These conventions are not interchangeable without renormalization. The hybrid stochastic heat equation paper explicitly states that the $a$0-th moment exponent is defined with no $a$1 normalization, while the dynamic random environment PAM defines $a$2 with a prefactor $a$3 (Bao et al., 2011, Erhard et al., 2012). In random matrix theory, the paper on cycle expansions notes that some authors define $a$4, whereas others use $a$5 as a normalized exponent (Charbonneau et al., 2017).

2. Exact formulas in integrable stochastic PDEs

In the $a$6-dimensional parabolic Anderson model with nearest-neighbor jumps and space-time white noise, Borodin and Corwin obtain exact contour-integral formulas for moments and then extract moment Lyapunov exponents by steepest descent and residue analysis. For the nearest-neighbor model with $a$7, they prove an explicit two-contour formula for $a$8. In the symmetric case $a$9 and velocity $p$0, the first moment exponent is $p$1, while the second moment exponent is

$p$2

with small-$p$3 asymptotics $p$4 and large-$p$5 asymptotics $p$6. In the totally asymmetric right-jump-only model, they derive all $p$7-point moments by nested contour integrals and compute

$p$8

where $p$9 solves $t^{-\rho}$0 (Borodin et al., 2012).

The same paper compares the discrete one-sided model with the continuum stochastic heat equation and records the continuum value

$t^{-\rho}$1

for integer $t^{-\rho}$2 at $t^{-\rho}$3 and $t^{-\rho}$4 (Borodin et al., 2012). A later paper establishes the continuum result for all real $t^{-\rho}$5: for the one-dimensional SHE with delta initial data,

$t^{-\rho}$6

Equivalently,

$t^{-\rho}$7

The same asymptotics imply the one-point upper-tail large deviation principle for the centered Hopf–Cole height $t^{-\rho}$8, with speed $t^{-\rho}$9 and rate function $t^{-\kappa}$0 (Das et al., 2019).

A central point in these integrable models is that the exponent is not merely asymptotic but explicitly computable. In the discrete PAM this occurs through contour deformation, residues, and saddle-point analysis; in the SHE it occurs through refined asymptotics for fractional moments, so that the exact cubic law in $t^{-\kappa}$1 persists beyond integer replica indices (Borodin et al., 2012, Das et al., 2019).

3. Fractional and colored-noise generalizations

For the parabolic Anderson model driven by Gaussian noise fractional in time and with spatial covariance $t^{-\kappa}$2, the correct time normalization is

$t^{-\kappa}$3

equivalently $t^{-\kappa}$4 in terms of the Hurst index $t^{-\kappa}$5. Under the stated scaling and technical assumptions, Xia Chen’s integer-moment formula extends to all real $t^{-\kappa}$6: $t^{-\kappa}$7 with

$t^{-\kappa}$8

The lower bound in fact holds for all $t^{-\kappa}$9, but the exact upper bound in the note is proved for $L(p)$0 by hypercontractivity of the Ornstein–Uhlenbeck semigroup and the comparison

$L(p)$1

(Lê, 2015).

A different fractional generalization replaces the Laplacian by $L(p)$2 and allows Gaussian noise colored in both time and space. In that setting the unified Stratonovich/Skorohod family $L(p)$3 satisfies

$L(p)$4

and the exact asymptotic constant is

$L(p)$5

Hence

$L(p)$6

The difference reflects the renormalization of diagonal terms in the Skorohod interpretation. The basic integrability threshold for the $L(p)$7-stable path functional is $L(p)$8 (Chen et al., 2016).

These two lines of work show that the label “$L(p)$9-th moment Lyapunov exponent” covers both linear-time and superlinear-time growth laws. The shared structure is the extraction of a deterministic rate from $p$0-moments, while the scaling and the $p$1-dependence encode the underlying covariance singularities, spatial homogeneity, and interpretation of the noise (Lê, 2015, Chen et al., 2016).

4. Analytical mechanisms for computing the exponent

In integrable stochastic PDEs, the starting point is often a Feynman–Kac or directed-polymer representation. For the discrete PAM, moments reduce to expectations over multiple random walks with pairwise local-time interactions, yielding a “quantum delta Bose gas” structure. For $p$2 in the nearest-neighbor case, and for all $p$3 in the one-sided case, the moment equations close on ordered domains and are solved explicitly by coordinate Bethe ansatz and nested contour integrals. The large-$p$4 exponent then emerges from contour deformation, simple-pole residues, and steepest descent; in the one-sided model, the dominant term comes from the “ground state” partition $p$5 (Borodin et al., 2012).

For the SHE with delta initial data, the Laplace transform of $p$6 admits a Fredholm determinant representation. Fractional moments are accessed through the identity

$p$7

and the dominant contribution is the $p$8 trace term in the Fredholm expansion. Higher wedge terms are shown to be exponentially smaller, which isolates the leading growth $p$9 and hence $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$0 (Das et al., 2019).

In non-integrable Gaussian-noise PAMs, explicit formulas are replaced by variational principles and semigroup methods. The extension from integer to real $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$1 in the fractional-time model relies on Mehler’s formula and hypercontractivity for the Ornstein–Uhlenbeck semigroup (Lê, 2015). In the fractional Laplacian model, the exponent is identified through a variational inequality and a Feynman–Kac type large deviation principle for additive functionals of symmetric $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$2-stable processes, producing the variational constant $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$3 (Chen et al., 2016).

The same broad theme persists in dynamical-systems formulations. In cycle expansions, the exponent is encoded by the zero of a Ruelle dynamical zeta function. In transfer-operator approaches for random matrices, it is the logarithm of the largest eigenvalue of an averaged operator. In random dynamical systems on manifolds, it is the principal eigenvalue of a twisted generator acting on a weighted function space (Charbonneau et al., 2017, Texier, 2019, Baxendale, 21 Jul 2025).

5. Random matrices, maps, and generalized moment exponents

For products of i.i.d. random matrices, the generalized Lyapunov exponent is the scaled cumulant-generating function

$$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$4

and its derivatives at $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$5 generate the usual quenched Lyapunov exponent, the Lyapunov susceptibility, and higher cumulant densities. The cycle-expansion formalism expresses the relevant zeta function as a sum over pseudocycles,

$$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$6

so that $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$7 and $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$8 are computable as formally exact cycle sums. The method is particularly effective for susceptibilities and higher moments in disordered transfer-matrix models (Charbonneau et al., 2017).

A complementary operator-theoretic construction for $$\gamma_p(\beta;\nu)=\lim_{t\to\infty}\frac{1}{t}\log \mathbb{E}[u(t,\lfloor \nu t\rfloor)^p]$$9 products defines

$t$0

and reduces its computation to the largest eigenvalue of an averaged transfer operator. In the solvable continuum-limit $t$1 subgroup case, the generalized moment exponent is obtained in closed form: $t$2 The same framework yields exact single-integral formulas for the variance rates of $t$3 and $t$4 in one-dimensional Schrödinger transfer-matrix problems with Lévy noise (Texier, 2019).

For deterministic one-dimensional maps, the paper on generalized Lyapunov exponents shifts attention from the linear clock $t$5 to a dynamical instability sequence $t$6. It defines

$t$7

and states that the deterministic framework suggests the natural extension

$t$8

This accommodates super-exponential and sub-exponential instability classes. The cited examples include the infinite Bernoulli scheme with Lyapunov pair $t$9, the ant-lion map with $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$00, Thaler maps with $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$01 or $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$02, and the log-Weibull map with $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$03 (Akimoto et al., 2014).

6. Spectral characterization on manifolds

For random dynamical systems generated by SDEs on a manifold $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$04, the derived process on the tangent bundle leads to a canonical finite-time Lyapunov observable. If $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$05 is the linearized flow and $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$06 is the induced diffusion on the unit sphere bundle $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$07, then

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$08

The $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$09-th moment Lyapunov exponent is

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$10

Its twisted generator is

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$11

or, on $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$12,

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$13

(Baxendale, 21 Jul 2025).

On compact manifolds, the principal eigenvalue of $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$14 equals $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$15. The non-compact case requires a weighted Banach space

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$16

where $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$17 satisfies the growth condition (G). Under the paper’s hypotheses (H), (P), and (G), the twisted semigroup

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$18

is bounded and compact on $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$19, possesses a positive eigenfunction $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$20, and satisfies

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$21

The map $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$22 is convex and real analytic on intervals where the hypotheses hold (Baxendale, 21 Jul 2025).

This spectral identification feeds directly into finite-time fluctuation theory. When $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$23 is finite and essentially smooth, the Gärtner–Ellis theorem gives the large deviation rate function

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$24

At $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$25, one has $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$26, the almost-sure top Lyapunov exponent, and

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$27

If $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$28, moderate deviations occur with speed $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$29 and quadratic rate $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$30. The Ornstein–Uhlenbeck quadratic-functional example in the paper shows why the weighted-space formulation is essential on non-compact state spaces: without it, the eigenvalue problem may admit multiple positive solutions that do not characterize $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$31 (Baxendale, 21 Jul 2025).

7. Intermittency, stability, and unresolved directions

A recurrent qualitative signature is intermittency. In the discrete PAM with delta initial data, the paper proves

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$32

and equivalently that $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$33 is strictly increasing. The one-sided model and the symmetric nearest-neighbor model both exhibit this strict ordering, meaning that large moments are dominated by rare high peaks. The fractional PAM with colored noise yields the same qualitative picture through the superlinear factor $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$34, so that $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$35 is strictly increasing when $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$36 (Borodin et al., 2012, Chen et al., 2016).

In hybrid stochastic heat equations with Markovian switching, the $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$37-th moment exponent is an exact variational quantity. For deterministic $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$38,

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$39

and in the $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$40-dimensional noise case one replaces $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$41 by $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$42. The paper also gives the almost-sure exponent

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$43

It follows that $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$44 can exceed $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$45, and the paper provides an explicit two-regime example in which the solution is almost surely exponentially stable while the second moment grows exponentially (Bao et al., 2011).

The dynamic random environment PAM adds a different perspective. There the annealed exponent is defined, when it exists, by

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$46

while the main proved results concern the quenched exponent $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$47. The paper formulates the conjecture

$$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$48

motivated by the heuristic that large diffusion lets the random walk spend most of its time in the largest favorable space-time clumps, thereby suppressing the difference between quenched and annealed exponential growth rates (Erhard et al., 2012).

Taken together, these results show that the $$\lim_{t\to\infty} t^{-a}\log \mathbb{E}[X_t^p] \quad\text{or}\quad \lim_{N\to\infty}\frac{1}{N}\log \mathbb{E}[e^{pS_N}],$$49-th moment Lyapunov exponent is not a single formula but a structural object. Its normalization reflects the temporal scaling of the model, its explicit form ranges from closed contour-integral saddle values to variational constants and principal eigenvalues, and its probabilistic content spans intermittency, moment stability, susceptibility, and large deviations (Borodin et al., 2012, Das et al., 2019, Charbonneau et al., 2017, Baxendale, 21 Jul 2025).