Hyperbolic Anderson Model in SPDE Analysis

• Hyperbolic Anderson Model is a class of stochastic wave equations with the hyperbolic operator (∂²t - Δ) and linear multiplicative noise, linking randomness with wave dynamics.
• It uses Wiener and Poisson chaos expansions along with Malliavin techniques to analyze well-posedness, moment growth, and intermittency in various noise regimes.
• Recent work extends the framework to negatively curved spaces, demonstrating how hyperbolic geometry alters Anderson localization on lattices and manifolds.

Searching arXiv for papers on the hyperbolic Anderson model and related hyperbolic Anderson localization. First, I’ll retrieve the core papers on the SPDE “hyperbolic Anderson model” and the separate hyperbolic-plane Anderson localization work. The hyperbolic Anderson model is a family of Anderson-type random evolution equations whose defining deterministic operator is hyperbolic rather than parabolic. In the stochastic-partial-differential-equation literature represented here, it is the stochastic wave equation with linear multiplicative noise,

$$\partial_t^2 u(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$

or its Lévy-noise analogue, typically with initial data $u(0,x)=1$ and $\partial_tu(0,x)=0$; the term “hyperbolic” refers to the wave operator $\partial_t^2-\Delta$, and “Anderson” to the linear multiplicative potential $u\,\dot W$ (Balan et al., 2016, Chen et al., 2021). A distinct recent usage places Anderson localization directly on negatively curved spaces such as $\mathbb H^2$ or on hyperbolic lattices, where curvature changes the localization mechanism itself (Altland et al., 27 Apr 2026, Li et al., 2023). Taken together, these literatures make the hyperbolic Anderson model a junction point between stochastic wave equations, Malliavin analysis, intermittency theory, limit theorems for spatial averages, and disorder-driven localization in negatively curved geometry.

1. SPDE formulation and basic structure

In the SPDE sense, the model is the stochastic wave equation

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$

with either general initial data $u(0,x)=u_0(x)$, $\partial_tu(0,x)=u_1(x)$, or the constant data $u(0,x)=1$, $u(0,x)=1$0 used in much of the fluctuation literature (Balan et al., 2016, Chen et al., 2021). The mild form is

$u(0,x)=1$1

with

$u(0,x)=1$2

and wave kernel Fourier transform

$u(0,x)=1$3

For $u(0,x)=1$4, the fundamental solutions are explicitly available; in particular,

$u(0,x)=1$5

in $u(0,x)=1$6, and

$u(0,x)=1$7

in $u(0,x)=1$8 (Chen et al., 2021, Balan et al., 2016).

The solution theory depends on the stochastic interpretation. For Gaussian noises colored in time, the mild integral is typically taken in the Skorohod sense, or equivalently with Wick product notation $u(0,x)=1$9 (Balan et al., 2016, Xia et al., 2024). For pure-jump Lévy white noise and Lévy colored noise, the product is interpreted in the Itô sense via a compensated Poisson random measure (Balan et al., 2023, Balan et al., 26 Feb 2026). A separate Stratonovich regime constructs the stochastic convolution as a Young-type pathwise limit of Riemann sums (Chen et al., 2022, Chen, 1 Oct 2025).

Wiener or Poisson chaos expansions are central throughout this theory. In the Gaussian setting,

$\partial_tu(0,x)=0$0

and in the Lévy setting,

$\partial_tu(0,x)=0$1

with explicit kernels built from iterated wave propagators (Balan et al., 2016, Balan et al., 2023). This converts existence, moment estimates, and fluctuation problems into summability and derivative estimates on deterministic kernels.

2. Noise classes and well-posedness regimes

For Gaussian noise homogeneous in space and time, a basic existence threshold is the Dalang-type condition

$\partial_tu(0,x)=0$2

where $\partial_tu(0,x)=0$3 is the spatial spectral measure (Balan et al., 2016). For the more singular case of temporal covariance

$\partial_tu(0,x)=0$4

the sharp Skorohod criterion in $\partial_tu(0,x)=0$5 becomes

$\partial_tu(0,x)=0$6

which is necessary and sufficient for $\partial_tu(0,x)=0$7, and sufficient when $\partial_tu(0,x)=0$8 (Chen et al., 2021). In homogeneous spatial classes this becomes the simple inequality $\partial_tu(0,x)=0$9 (Chen et al., 2021).

The Stratonovich theory is different. In dimensions $\partial_t^2-\Delta$0, a pathwise Young formulation was developed using weighted Besov spaces and Strichartz-type estimates for the wave kernel (Chen et al., 2022). There the key smoothing index is

$\partial_t^2-\Delta$1

and the Gaussian noise is required to have enough temporal Hölder regularity and spatial Besov regularity so that the Young condition $\partial_t^2-\Delta$2 closes the fixed point (Chen et al., 2022). A later Stratonovich analysis for time-dependent Gaussian noise in $\partial_t^2-\Delta$3 established the optimal existence condition

$\partial_t^2-\Delta$4

and identifies it as necessary and sufficient for the square-integrable Stratonovich solution considered there (Chen, 1 Oct 2025). That paper explicitly contrasts this with the Skorohod threshold $\partial_t^2-\Delta$5, so in that regime the Stratonovich solvability condition is weaker than the Skorohod one (Chen, 1 Oct 2025).

In the Lévy setting, the baseline model is the one-dimensional stochastic wave equation

$\partial_t^2-\Delta$6

with $\partial_t^2-\Delta$7 a space-time pure-jump Lévy white noise of finite variance, realized through

$\partial_t^2-\Delta$8

(Balan et al., 2023, Balan et al., 2023). Lévy colored noise is obtained by spatial convolution,

$\partial_t^2-\Delta$9

with covariance kernel $u\,\dot W$0 and either $u\,\dot W$1 or $u\,\dot W$2 in the Riesz case (Balan et al., 26 Feb 2026, Balan et al., 27 Feb 2026).

3. Moments, intermittency, and Lyapunov growth

Moment growth is one of the oldest structural questions for the model. For the one-dimensional white-in-time Gaussian equation with rough spatial covariance corresponding to fractional Brownian motion with Hurst index $u\,\dot W$3, existence of a Skorohod solution holds precisely for $u\,\dot W$4, and this threshold is necessary (Balan et al., 2016). In that regime the $u\,\dot W$5-th moments satisfy an exponential upper bound, and the solution is weakly intermittent in the sense that

$u\,\dot W$6

(Balan et al., 2016).

For white-in-time Gaussian noise in arbitrary spatial dimension, the second moment admits a close Laplace-transform relation to the parabolic Anderson model. This permits exact computation of the upper second-order Lyapunov exponent for the hyperbolic model under homogeneous spatial covariance and in the one-dimensional rough fractional case (Balan et al., 2017). In the Riesz-kernel case $u\,\dot W$7, the hyperbolic exponent is

$u\,\dot W$8

with $u\,\dot W$9 defined variationally through the Riesz interaction (Balan et al., 2017).

A different asymptotic regime appears in the Stratonovich theory with time-dependent Gaussian noise. Under homogeneous spatial covariance $\mathbb H^2$0 and $\mathbb H^2$1, the expectation grows according to

$\mathbb H^2$2

where the constant is expressed through a variational functional $\mathbb H^2$3 or $\mathbb H^2$4 built from a time-randomized Brownian intersection local time (Chen, 1 Oct 2025). That analysis also shows that the expected even Stratonovich chaos levels can be represented through

$\mathbb H^2$5

which is the mechanism by which time dependence modifies both solvability and large-$\mathbb H^2$6 growth (Chen, 1 Oct 2025).

4. Spatial averages, ergodicity, and quantitative Gaussian fluctuations

A major recent direction studies the spatial integral

$\mathbb H^2$7

and its normalized fluctuations as $\mathbb H^2$8 (Balan et al., 2021, Balan et al., 2023). For Gaussian colored noise in $\mathbb H^2$9, quantitative central limit theorems were obtained by coupling first- and second-order Malliavin derivative bounds with a second-order Gaussian Poincaré inequality (Balan et al., 2021). In the integrable spatial-covariance regime,

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$0

while for the Riesz kernel $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$1,

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$2

(Balan et al., 2021). The corresponding functional CLTs hold in $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$3 (Balan et al., 2021).

For time-independent rough Gaussian noise in one dimension, the normalized spatial integral also converges to $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$4, with

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$5

and a functional limit

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$6

(Balan et al., 2023). A notable feature there is that the first Wiener chaos does not contribute to the limiting covariance (Balan et al., 2023).

In the Lévy white-noise setting, the one-dimensional solution field is strictly stationary and ergodic in space for each fixed time, and

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$7

(Balan et al., 2023). The variance satisfies

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$8

with explicit covariance kernel

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$9

and, assuming $u(0,x)=u_0(x)$0 and $u(0,x)=u_0(x)$1,

$u(0,x)=u_0(x)$2

in Fortet–Mourier, $u(0,x)=u_0(x)$3-Wasserstein, and Kolmogorov distance (Balan et al., 2023).

For Lévy colored noise in $u(0,x)=u_0(x)$4, the variance exponent is

$u(0,x)=u_0(x)$5

so $u(0,x)=u_0(x)$6 converges to a centered Gaussian process, and the one-time normalized variable converges quantitatively to $u(0,x)=u_0(x)$7 in $u(0,x)=u_0(x)$8, $u(0,x)=u_0(x)$9, or $\partial_tu(0,x)=u_1(x)$0 (Balan et al., 26 Feb 2026).

5. Almost sure central limit theorems and Malliavin mechanisms

The almost sure central limit theorem strengthens ordinary convergence in law by proving that, for almost every sample,

$\partial_tu(0,x)=u_1(x)$1

For Lévy white noise, this was proved using two different methods: a Clark–Ocone argument exploiting the white-in-time martingale structure, and an Ibragimov–Lifshits characteristic-function criterion combined with a second-order Poincaré inequality on Poisson space (Balan et al., 2023). The same paper emphasizes that the Clark–Ocone route is not applicable for colored-in-time noises, whereas the second route is expected to extend (Balan et al., 2023).

That extension was carried out in both the Gaussian-colored and Lévy-colored settings. For Gaussian colored noises, the ASCLT for parabolic and hyperbolic Anderson models is proved by combining the Ibragimov–Lifshits criterion with a second-order Gaussian Poincaré inequality, precisely because “the lack of Itô tools in this colored-in-time setting” blocks the earlier white-noise methods (Xia et al., 2024). For the hyperbolic model, this applies in $\partial_tu(0,x)=u_1(x)$2, with either $\partial_tu(0,x)=u_1(x)$3 or $\partial_tu(0,x)=u_1(x)$4, and uses the wave-specific bound

$\partial_tu(0,x)=u_1(x)$5

to control mixed covariances across scales (Xia et al., 2024).

For Lévy colored noise, the ASCLT holds for

$\partial_tu(0,x)=u_1(x)$6

under the same two kernel classes, $\partial_tu(0,x)=u_1(x)$7 or $\partial_tu(0,x)=u_1(x)$8, and is proved by adapting the Clark–Ocone covariance-decay method from the white-noise paper (Balan et al., 27 Feb 2026). Across these results, the decisive estimates are bounds on first and second Malliavin derivatives of the solution, often expressed through auxiliary wave equations with Dirac delta initial velocity (Balan et al., 2023, Balan et al., 27 Feb 2026).

6. Distinct usage: Anderson localization on negatively curved spaces

A separate literature uses closely related terminology for quantum disorder on hyperbolic geometry itself. In the continuum setting, the model is a random Schrödinger operator on the hyperbolic plane,

$\partial_tu(0,x)=u_1(x)$9

with Gaussian white-noise potential

$u(0,x)=1$0

posed on $u(0,x)=1$1 of curvature radius $u(0,x)=1$2 and scalar curvature $u(0,x)=1$3 (Altland et al., 27 Apr 2026). The key geometric fact is that $u(0,x)=1$4 is locally Euclidean but has exponential volume growth at large distance, so returns are suppressed and the Anderson problem is no longer the flat-$u(0,x)=1$5 one. The resulting nonlinear sigma-model analysis yields a two-parameter renormalization flow

$u(0,x)=1$6

with $u(0,x)=1$7, and, after matching to lattice strong-disorder physics at the curvature scale, an extended critical line separating metallic and insulating phases (Altland et al., 27 Apr 2026).

On hyperbolic lattices, the tight-binding Anderson Hamiltonian

$u(0,x)=1$8

shows a finite-disorder localization transition and mobility edges, in sharp contrast with ordinary $u(0,x)=1$9 Euclidean lattices (Li et al., 2023, Chen et al., 2023). On randomly boundary-connected $u(0,x)=1$00 and $u(0,x)=1$01 lattices, finite-size scaling of the adjacent-gap ratio and inverse participation ratio gives band-center estimates

$u(0,x)=1$02

and

$u(0,x)=1$03

with $u(0,x)=1$04 and multifractal critical states of dimension $u(0,x)=1$05 (Li et al., 2023). A complementary periodic-boundary construction on $u(0,x)=1$06 and $u(0,x)=1$07 likewise finds finite critical disorder, approximately

$u(0,x)=1$08

together with strong finite-size drift in level statistics (Chen et al., 2023).

This dual usage of the phrase “hyperbolic Anderson model” marks a genuine split in current literature. In one branch, hyperbolicity refers to the operator $u(0,x)=1$09; in the other, it refers to the underlying negatively curved space. A plausible implication is that the common label reflects two different ways randomness and hyperbolicity can interact: through hyperbolic dynamics of stochastic waves, or through hyperbolic geometry of the configuration space itself.