Hyperbolic Anderson Equation

• The hyperbolic Anderson equation is a stochastic wave model with linear multiplicative noise that generalizes the parabolic Anderson model and introduces concepts like intermittency and Gaussian fluctuations.
• It employs advanced analytical methods such as Wiener chaos expansions, Malliavin calculus, and fixed-point arguments under Skorohod, Itô, and Stratonovich frameworks.
• The model reveals rich phenomena including moment asymptotics, intermittency, and localization effects in both Euclidean and hyperbolic geometries.

The hyperbolic Anderson equation most commonly denotes the stochastic wave equation with linear multiplicative noise,

$\partial_t^2 u=\Delta u+u\,\dot W,$

together with suitable initial conditions and a stochastic interpretation of the product $u\,\dot W$. In this usage, “hyperbolic” refers to the wave operator $\partial_t^2-\Delta$, while “Anderson” refers to the linear multiplicative random-potential structure familiar from the parabolic Anderson model. The term is not uniform across the literature, however. It is also used, more loosely, for Anderson localization problems on negatively curved lattices or on the hyperbolic plane, and for parabolic Anderson equations posed on hyperbolic spaces (Chen et al., 2021, Li et al., 2023, Altland et al., 27 Apr 2026).

1. Terminology and scope

In the SPDE literature, the hyperbolic Anderson model is the wave-equation analogue of the parabolic Anderson model. A representative form is

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

or, in one space dimension with constant initial data,

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$

This is the sense used in the Skorohod/Wick, Itô, and Stratonovich studies surveyed below. The multiplicative coefficient is exactly $\sigma(u)=u$, and the central questions are solvability, regularity, moment growth, intermittency, and fluctuation theory (Chen et al., 2021, Chen et al., 2022, Balan et al., 2016).

A distinct line of work uses “hyperbolic” to describe the underlying geometry rather than the PDE type. There the relevant equations are the disordered Schrödinger operator on $\mathbb H^2$,

$$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$

or the lattice Anderson eigenvalue problem on a hyperbolic graph,

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$

Related but separate are parabolic Anderson models on hyperbolic spaces,

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$

where “hyperbolic” refers to $u\,\dot W$0 rather than to a wave operator (Altland et al., 27 Apr 2026, Li et al., 2023, Geng et al., 7 Jul 2025, Geng et al., 25 Jun 2025).

2. Canonical stochastic-wave formulations

The deterministic backbone is the wave propagator. For $u\,\dot W$1, the fundamental solution is

$u\,\dot W$2

while in all dimensions its Fourier transform is

$u\,\dot W$3

Accordingly, the mild form is the natural starting point. In the Skorohod setting one writes

$u\,\dot W$4

or, with general initial data,

$u\,\dot W$5

In the pathwise Stratonovich framework the same equation is written as

$u\,\dot W$6

For time-independent spatial noise, the mild form becomes

$u\,\dot W$7

with $u\,\dot W$8 understood as a spatial Skorohod integral (Chen et al., 2022, Chen et al., 2021, Balan et al., 2022).

The solution theory is commonly organized through Wiener chaos. A typical expansion is

$u\,\dot W$9

with kernels built by iterating the wave propagator along ordered times. This is the main analytic mechanism in the Skorohod literature, and it is also the basis for Malliavin derivative estimates and fluctuation results (Balan et al., 2021, Chen et al., 2021).

3. Solvability and interpretation frameworks

Several distinct solvability theories coexist. In the Skorohod theory for space-time homogeneous Gaussian noise, one formulation proves existence and uniqueness under the same condition on the spatial spectral measure as in the white-noise-in-time case,

$\partial_t^2-\Delta$0

and Hölder continuity under the stronger condition

$\partial_t^2-\Delta$1

regardless of the temporal covariance function $\partial_t^2-\Delta$2 (Balan et al., 2016).

A more recent Skorohod analysis treats the covariance

$\partial_t^2-\Delta$3

in dimensions $\partial_t^2-\Delta$4. For $\partial_t^2-\Delta$5, the sharp criterion for a mild Skorohod solution is

$\partial_t^2-\Delta$6

and in the homogeneous spatial case $\partial_t^2-\Delta$7 this becomes

$\partial_t^2-\Delta$8

For $\partial_t^2-\Delta$9, the same condition is sufficient (Chen et al., 2021).

In the Stratonovich regime with time-independent spatial Gaussian noise,

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

the noise is regularized by

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

and the Stratonovich integral is defined as the $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$2-limit

$$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$3

For $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$4, Dalang’s condition

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

is both necessary and sufficient for solvability (Chen et al., 2024).

A different Stratonovich/pathwise program considers $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$6 with fractional-in-time Gaussian noise and proves existence and uniqueness by combining Strichartz-type estimates for the wave kernel in weighted Besov spaces with a Young-integration fixed-point argument. Under the homogeneous assumption $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$7, the explicit criterion is

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

The solution lives in a weighted Besov-Hölder space $$\frac{\partial^2 u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\,\dot W(t,x),$$9 (Chen et al., 2022).

For one-dimensional rough spatial noise corresponding to a fractional Brownian motion with Hurst index

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$0

the Skorohod solution exists uniquely, and the threshold $\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$1 is both sufficient and necessary. The key integrability condition is driven by

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$2

with

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$3

This setting lies outside the standard Dalang framework with a regular spatial covariance kernel (Balan et al., 2016).

4. Intermittency and moment asymptotics

Moment growth is one of the defining features of the hyperbolic Anderson model. In the one-dimensional rough-noise setting with constant initial data and $\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$4, the Skorohod solution satisfies, for every $\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$5,

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$6

with positive constants $\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$7 depending only on $\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$8. The same work proves weak intermittency, in the sense that

$\begin{cases} \dfrac{\partial^2 u}{\partial t^2}(t,x)=\dfrac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\,\dot X(t,x),\[1mm] u(0,x)=1,\qquad \dfrac{\partial u}{\partial t}(0,x)=0. \end{cases}$9

where $\sigma(u)=u$0 and $\sigma(u)=u$1 are lower and upper Lyapunov exponents (Balan et al., 2016).

In the Stratonovich regime with time-independent spatial Gaussian noise, the intermittency theory is much sharper. Under the homogeneity assumption

$\sigma(u)=u$2

or the borderline case $\sigma(u)=u$3, the long-time and high-moment asymptotics are governed by a variational quantity $\sigma(u)=u$4 over $\sigma(u)=u$5. For fixed integer $\sigma(u)=u$6,

$\sigma(u)=u$7

has order

$\sigma(u)=u$8

while for fixed $\sigma(u)=u$9,

$\mathbb H^2$0

has order

$\mathbb H^2$1

In the one-dimensional white-noise case $\mathbb H^2$2, these become the $\mathbb H^2$3 and $\mathbb H^2$4 scales. The same paper emphasizes that the Stratonovich and Skorohod/Wick models have genuinely different intermittency behavior, especially in the fixed-$\mathbb H^2$5 long-time regime (Chen et al., 2024).

A noteworthy structural device in that analysis is the Laplace-transform representation of Stratonovich chaos terms by Brownian motion in Gaussian potential. One formulation is

$\mathbb H^2$6

which is central to the exact asymptotic analysis (Chen et al., 2024).

5. Malliavin derivatives and Gaussian fluctuations

A large part of the modern theory is built on sharp Malliavin derivative estimates. For the hyperbolic Anderson model driven by colored Gaussian homogeneous noise in dimensions $\mathbb H^2$7, one fundamental result states that

$\mathbb H^2$8

and for every integer $\mathbb H^2$9, every $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$0, and almost all $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$1,

$$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$2

These bounds feed directly into quantitative CLTs via a second-order Gaussian Poincaré inequality (Balan et al., 2021).

For spatial averages

$$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$3

the same work derives total-variation CLTs and functional CLTs. When the spatial covariance is integrable, $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$4 and

$$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$5

For Riesz covariance $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$6, $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$7 and

$$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$8

The corresponding process-level limits hold in $$\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),$$9 (Balan et al., 2021).

The time-independent Gaussian setting admits an analogous fluctuation theory. For

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$0

with $$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$1, integrable spatial covariance yields

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$2

while the Riesz case $$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$3 gives

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$4

In the Riesz regime the first Wiener chaos determines the leading covariance asymptotics, and the paper also proves the corresponding functional CLT (Balan et al., 2022).

For time-independent rough spatial noise associated with fractional Brownian motion of Hurst index $$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$5, the centered spatial integral

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$6

satisfies

$$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$7

and $$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$8 converges in $$E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i.$$9 to a centered Gaussian process. A distinctive point here is that the limiting covariance is not determined solely by the first chaos; higher chaoses survive in the limit (Balan et al., 2023).

At the almost-sure level, the colored-in-time Gaussian theory proves that the normalized spatial averages over balls satisfy an ASCLT: $$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$0 The proof combines quantitative Gaussian approximation via second-order Gaussian Poincaré inequalities with the Ibragimov–Lifshits criterion, precisely because Itô/Clark–Ocone methods are unavailable in the colored-in-time setting (Xia et al., 2024).

6. Lévy-driven hyperbolic Anderson models

The hyperbolic Anderson equation has also been developed for Lévy noise. In the finite-variance Lévy white-noise setting,

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$1

the spatial average

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$2

satisfies an ASCLT under

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$3

The note gives two proofs: one based on the Clark–Ocone formula and martingale structure, and one based on a second-order Gaussian Poincaré inequality together with the Ibragimov–Lifshits method (Balan et al., 2023).

For Lévy colored noise in dimension $$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$4, the mild equation is

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$5

with $$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$6. The large-scale spatial integral

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$7

has variance exponent

$$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$8

so that $$\partial_t u=\Delta u+\xi u \qquad\text{or}\qquad \partial_t u=\Delta u+\beta u\,\dot W,$$9. Under

$u\,\dot W$00

the paper proves

$u\,\dot W$01

with rates in the Fortet–Mourier, $u\,\dot W$02-Wasserstein, or Kolmogorov distances, and also proves a functional CLT: $u\,\dot W$03 (Balan et al., 26 Feb 2026).

The same Lévy-colored setting also satisfies an ASCLT. For fixed $u\,\dot W$04, if $u\,\dot W$05 or $u\,\dot W$06, then

$u\,\dot W$07

obeys

$u\,\dot W$08

The proof again combines covariance estimates for Clark–Ocone functionals with variance asymptotics imported from the corresponding CLT theory (Balan et al., 27 Feb 2026).

7. Hyperbolic geometry interpretations beyond the wave equation

A different body of work studies Anderson-type equations on negatively curved spaces. On regular hyperbolic lattices $u\,\dot W$09, the relevant single-particle problem is

$u\,\dot W$10

with onsite disorder

$u\,\dot W$11

Using randomized boundary reconnection and finite-size scaling of the adjacent-gap ratio and the inverse participation ratio, one finds finite-disorder Anderson transitions and mobility edges on $u\,\dot W$12 and $u\,\dot W$13. At the band center, the reported critical disorders are

$u\,\dot W$14

and

$u\,\dot W$15

depending on the observable, with $u\,\dot W$16. In the $u\,\dot W$17 limit, the cavity method gives

$u\,\dot W$18

close to the random-regular-graph value $u\,\dot W$19 (Li et al., 2023).

For the continuum hyperbolic plane $u\,\dot W$20, the disorder problem is formulated as

$u\,\dot W$21

or, at the field-theory level, by the nonlinear sigma model

$u\,\dot W$22

The central result is a two-parameter flow in conductivity and curvature scale,

$u\,\dot W$23

with an extended critical line separating metallic and insulating phases. Here “hyperbolic” refers to $u\,\dot W$24, not to a wave operator (Altland et al., 27 Apr 2026).

Related parabolic Anderson models on hyperbolic spaces show that negative curvature changes both phase structure and asymptotic scales. For

$u\,\dot W$25

with spatial covariance $u\,\dot W$26, the critical decay exponent is

$u\,\dot W$27

If $u\,\dot W$28, sufficiently small $u\,\dot W$29 yields a bounded $u\,\dot W$30-region; if $u\,\dot W$31, the second moment diverges for every $u\,\dot W$32, and for small $u\,\dot W$33

$u\,\dot W$34

while for large $u\,\dot W$35 the growth is exponential (Geng et al., 7 Jul 2025).

For the parabolic Anderson model

$u\,\dot W$36

on $u\,\dot W$37 with a stationary Gaussian potential of finite correlation length, the quenched asymptotic is

$u\,\dot W$38

with $u\,\dot W$39 obtained from an explicit variational problem. The optimal strategy reaches a peak at distance of order $u\,\dot W$40, then remains there for the rest of the time (Geng et al., 25 Jun 2025).

Taken together, these works show that “hyperbolic Anderson equation” has two stable meanings in current research. In the narrow SPDE sense, it is the stochastic wave equation with multiplicative noise and a rich theory of Skorohod, Itô, and Stratonovich solutions, intermittency, and Gaussian fluctuations. In the geometric sense, it refers to Anderson-type random Schrödinger operators or parabolic Anderson equations on negatively curved spaces, where exponential volume growth, spectral gaps, and non-Euclidean transport radically alter localization and growth phenomena (Chen et al., 2021, Altland et al., 27 Apr 2026).