---
title: Hyperbolic Anderson Equation
url: https://www.emergentmind.com/topics/hyperbolic-anderson-equation
type: topic
---

# Hyperbolic Anderson Equation

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 [2112.04954, 2312.11857, 2604.24917].

## 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 [2112.04954, 2205.15773, 1605.00024].

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 \(\mathbb H^d\) rather than to a wave operator [2604.24917, 2312.11857, 2507.05530, 2506.20147].

## 2. Canonical stochastic-wave formulations

The deterministic backbone is the wave propagator. For \(d=1,2\), the fundamental solution is
\[
G_t(x)=
\begin{cases}
\frac12\,\mathbf 1_{\{|x|<t\}, & d=1,\\[1mm]
\frac{1}{2\pi\sqrt{t^2-|x|^2}}\,\mathbf 1_{\{|x|<t\}, & d=2,
\end{cases}
\]
while in all dimensions its Fourier transform is
\[
\mathcal F G_t(\xi)=\frac{\sin(t|\xi|)}{|\xi|}.
\]
Accordingly, the mild form is the natural starting point. In the Skorohod setting one writes
\[
u(t,x)=1+\int_0^t\int_{\mathbb R^d}G_{t-s}(x-y)\,u(s,y)\,W(ds,dy),
\]
or, with general initial data,
\[
u(t,x)=\partial_t G_t*u_0(x)+G_t*u_1(x)+\int_0^t\int_{\mathbb R^d}G_{t-s}(x-y)\,u(s,y)\,W(ds,dy).
\]
In the pathwise Stratonovich framework the same equation is written as
\[
u_t=(\partial_t G)_t u_0+G_tu_1+\int_0^t G_{t-r}(u_r\,dW_r).
\]
For time-independent spatial noise, the mild form becomes
\[
u(t,x)=1+\int_0^t\int_{\mathbb R^d}G_{t-s}(x-y)u(s,y)\,W(\delta y)\,ds,
\]
with \(W(\delta y)\) understood as a spatial Skorohod integral [2205.15773, 2112.04954, 2201.02319].

The solution theory is commonly organized through Wiener chaos. A typical expansion is
\[
u(t,x)=1+\sum_{n\ge 1}I_n(\widetilde f_{t,x,n}),
\]
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 [2101.10957, 2112.04954].

## 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,
\[
\int_{\mathbb R^d}\frac{1}{1+|\xi|^2}\,\mu(d\xi)<\infty,
\]
and Hölder continuity under the stronger condition
\[
\int_{\mathbb R^d}\frac{1}{(1+|\xi|^2)^\beta}\,\mu(d\xi)<\infty,\qquad \beta\in(0,1),
\]
regardless of the temporal covariance function \(\gamma\) [1602.07004].

A more recent Skorohod analysis treats the covariance
\[
\mathbb E[\dot W(s,x)\dot W(t,y)]=|s-t|^{-\alpha_0}\gamma(x-y),\qquad \alpha_0\in[0,1),
\]
in dimensions \(d\in\{1,2,3\}\). For \(\alpha_0\in(0,1)\), the sharp criterion for a mild Skorohod solution is
\[
\int_{\mathbb R^d}\left(\frac{1}{1+|\xi|^2}\right)^{\frac{3-\alpha_0}{2}}\mu(d\xi)<\infty,
\]
and in the homogeneous spatial case \(\gamma(cx)=c^{-\alpha}\gamma(x)\) this becomes
\[
\alpha_0+\alpha<3.
\]
For \(\alpha_0=0\), the same condition is sufficient [2112.04954].

In the Stratonovich regime with time-independent spatial Gaussian noise,
\[
\partial_t^2 u=\Delta u+W(x)u,
\]
the noise is regularized by
\[
W_\varepsilon(x)=\int_{\mathbb R^d}W(y)p_\varepsilon(y-x)\,dy,
\]
and the Stratonovich integral is defined as the \(L^2\)-limit
\[
\int_{\mathbb R^d}V(x)\,W(dx):=\lim_{\varepsilon\to0+}\int_{\mathbb R^d}V(x)W_\varepsilon(x)\,dx.
\]
For \(d=1,2,3\), Dalang’s condition
\[
\int_{\mathbb R^d}\frac{1}{1+|\xi|^2}\,\mu(d\xi)<\infty
\]
is both necessary and sufficient for solvability [2403.08603].

A different Stratonovich/pathwise program considers \(d\in\{1,2\}\) 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 \(\gamma(x)=c|x|^{-a}\), the explicit criterion is
\[
a_0+a<p_d,\qquad p_1=1,\quad p_2=\frac12.
\]
The solution lives in a weighted Besov-Hölder space \(\mathcal E_T^{\gamma,\kappa}\) [2205.15773].

For one-dimensional rough spatial noise corresponding to a fractional Brownian motion with Hurst index
\[
H\in\left(\frac14,\frac12\right),
\]
the Skorohod solution exists uniquely, and the threshold \(H>\frac14\) is both sufficient and necessary. The key integrability condition is driven by
\[
\int_{\mathbb R} |\mathcal F G(t,\cdot)(\xi)|^2\,|\xi|^{2(1-2H)}\,d\xi<\infty,
\]
with
\[
\mathcal F G(t,\cdot)(\xi)=\frac{\sin(t|\xi|)}{|\xi|}.
\]
This setting lies outside the standard Dalang framework with a regular spatial covariance kernel [1605.00024].

## 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 \(H\in(\frac14,\frac12)\), the Skorohod solution satisfies, for every \(p\ge2\),
\[
E|u(t,x)|^p \le |\eta|^p\, C_1 \exp\!\Big(C_2\,|\lambda|^{\frac{2}{2H+1}}\, p^{\frac{2H+2}{2H+1}}\, t\Big),
\]
with positive constants \(C_1,C_2\) depending only on \(H\). The same work proves weak intermittency, in the sense that
\[
\underline\gamma(2)>0
\qquad\text{and}\qquad
\overline\gamma(p)<\infty\quad\text{for all }p>2,
\]
where \(\underline\gamma\) and \(\overline\gamma\) are lower and upper Lyapunov exponents [1605.00024].

In the Stratonovich regime with time-independent spatial Gaussian noise, the intermittency theory is much sharper. Under the homogeneity assumption
\[
\gamma(cx)=c^{-\alpha}\gamma(x),\qquad 0<\alpha<2\wedge d,
\]
or the borderline case \(\alpha=d=1\), the long-time and high-moment asymptotics are governed by a variational quantity \(\mathcal M\) over \(W^{1,2}(\mathbb R^d)\). For fixed integer \(p\),
\[
\log \mathbb E\,u^p(t,x)
\]
has order
\[
t^{\frac{4-\alpha}{3-\alpha}},
\]
while for fixed \(t>0\),
\[
\log \mathbb E|u(t,x)|^p
\]
has order
\[
p^{\frac{4-\alpha}{3-\alpha}}.
\]
In the one-dimensional white-noise case \(\alpha=1\), these become the \(t^{3/2}\) and \(p^{3/2}\) scales. The same paper emphasizes that the Stratonovich and Skorohod/Wick models have genuinely different intermittency behavior, especially in the fixed-\(p\) long-time regime [2403.08603].

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
\[
\int_0^\infty e^{-\lambda t}S_n(g_n(\cdot,t,x))\,dt
=
\frac{1}{n!}\Big(\frac{2}{\lambda}\Big)^{n+1}
\int_0^\infty e^{-\lambda^2 t/2}\,
\mathbb E_x\!\left[\Big(\int_0^t W(B(s))\,ds\Big)^n\right]dt,
\]
which is central to the exact asymptotic analysis [2403.08603].

## 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 \(d=1,2\), one fundamental result states that
\[
u(t,x)\in\mathbb D^\infty,
\]
and for every integer \(m\ge1\), every \(p\in[2,\infty)\), and almost all \((\mathbf s,\mathbf y)\in[0,t]^m\times\mathbb R^{md}\),
\[
m!\,f_{t,x,m}(\mathbf s,\mathbf y)\le \|D^m_{\mathbf s,\mathbf y}u(t,x)\|_p \le C\,f_{t,x,m}(\mathbf s,\mathbf y).
\]
These bounds feed directly into quantitative CLTs via a second-order Gaussian Poincaré inequality [2101.10957].

For spatial averages
\[
F_R(t)=\int_{B_R}[u(t,x)-1]\,dx,
\qquad
\sigma_R(t)=\sqrt{\operatorname{Var}(F_R(t))},
\]
the same work derives total-variation CLTs and functional CLTs. When the spatial covariance is integrable, \(\sigma_R(t)\asymp R^{d/2}\) and
\[
d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\le C R^{-d/2}.
\]
For Riesz covariance \(\gamma(x)=|x|^{-\beta}\), \(\sigma_R(t)\asymp R^{d-\beta/2}\) and
\[
d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\le C R^{-\beta/2}.
\]
The corresponding process-level limits hold in \(C(\mathbb R_+)\) [2101.10957].

The time-independent Gaussian setting admits an analogous fluctuation theory. For
\[
F_R(t)=\int_{B_R}(u(t,x)-1)\,dx
\]
with \(d\le2\), integrable spatial covariance yields
\[
\mathbb E[F_R(t)F_R(s)]\sim K(t,s)R^d,
\qquad
d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\le C_t R^{-d/2},
\]
while the Riesz case \(\gamma(x)=|x|^{-\beta}\) gives
\[
\mathbb E[F_R(t)F_R(s)]\sim K'(t,s)R^{2d-\beta},
\qquad
d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\le C_t R^{-\beta/2}.
\]
In the Riesz regime the first Wiener chaos determines the leading covariance asymptotics, and the paper also proves the corresponding functional CLT [2201.02319].

For time-independent rough spatial noise associated with fractional Brownian motion of Hurst index \(H\in(\frac14,\frac12)\), the centered spatial integral
\[
F_{R,\theta}(t)=\int_{-R}^R (u_\theta(t,x)-1)\,dx
\]
satisfies
\[
\frac{1}{R}\,\mathbb E[F_{R,\theta}(t)F_{R,\theta}(s)]\to K_\theta(t,s),
\qquad
d_{\mathrm{TV}}\!\left(\frac{F_{R,\theta}(t)}{\sigma_{R,\theta}(t)},Z\right)\le C_{t,H,\theta}R^{-1/2},
\]
and \(R^{-1/2}F_{R,\theta}(\cdot)\) converges in \(C([0,\infty))\) 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 [2305.05043].

At the almost-sure level, the colored-in-time Gaussian theory proves that the normalized spatial averages over balls satisfy an ASCLT:
\[
\frac1{\log T}\int_1^T \frac1R\,\delta_{F_R(\omega)}\,dR \Rightarrow N(0,1)
\quad\text{a.s.}
\]
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 [2409.07358].

## 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,
\[
(\partial_t^2-\partial_x^2)u=u\,L,
\qquad
(u(0,\cdot),\partial_tu(0,\cdot))=(1,0),
\]
the spatial average
\[
F_\theta=\int_{-\theta}^{\theta}\big(u(t_0,x)-1\big)\,dx,
\qquad
\widetilde F_\theta=\frac{F_\theta}{\sigma_\theta},
\]
satisfies an ASCLT under
\[
m_{1+a}+m_{2+2a}<\infty
\qquad\text{for some }a\in(0,1].
\]
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 [2310.10784].

For Lévy colored noise in dimension \(d=1\), the mild equation is
\[
u(t,x)=1+\int_0^t \int_{\mathbb R}G_{t-s}(x-y)u(s,y)X(ds,dy),
\]
with \(X_t(\varphi)=L_t(\varphi*k)\). The large-scale spatial integral
\[
F_R(t)=\int_{-R}^{R}\big(u(t,x)-1\big)\,dx
\]
has variance exponent
\[
\beta=
\begin{cases}
1, & k\in L^1(\mathbb R),\\[1mm]
\alpha+1, & k=R_{1,\alpha/2},\ \alpha\in(0,1),
\end{cases}
\]
so that \(\sigma_R^2(t)\asymp R^\beta\). Under
\[
m_p<\infty,\qquad m_{2p}<\infty
\qquad\text{for some }p\in(1,2],
\]
the paper proves
\[
\frac{F_R(t)}{\sigma_R(t)}\xrightarrow{d}N(0,1),
\]
with rates in the Fortet–Mourier, \(1\)-Wasserstein, or Kolmogorov distances, and also proves a functional CLT:
\[
R^{-\beta/2}F_R(\cdot)\Rightarrow G(\cdot)\quad\text{in }C[0,\infty)
\]
[2602.23137].

The same Lévy-colored setting also satisfies an ASCLT. For fixed \(t>0\), if \(k\in L^1(\mathbb R)\) or \(k=R_{1,\alpha/2}\), then
\[
\widetilde F_R(t)=\frac{F_R(t)}{\sigma_R(t)}
\]
obeys
\[
\nu_T^\omega=\frac{1}{\log T}\int_1^T \frac1\theta\,\delta_{\widetilde F_\theta(t,\omega)}\,d\theta
\Longrightarrow N(0,1)
\quad\text{a.s.}
\]
The proof again combines covariance estimates for Clark–Ocone functionals with variance asymptotics imported from the corresponding CLT theory [2602.24189].

## 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 \(\{p,q\}\), the relevant single-particle problem is
\[
H\psi=E\psi,\qquad
E\psi_i=\sum_{j\in\partial i}\psi_j+\epsilon_i\psi_i,
\]
with onsite disorder
\[
\epsilon_j\in[-W/2,W/2].
\]
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 \(\{3,8\}\) and \(\{4,8\}\). At the band center, the reported critical disorders are
\[
W_c=77.2\pm3\ \text{or}\ 83.0\pm4 \quad \text{for }\{3,8\},
\]
and
\[
W_c=88.2\pm3\ \text{or}\ 94\pm4 \quad \text{for }\{4,8\},
\]
depending on the observable, with \(\nu\approx1\). In the \(\{\infty,8\}\) limit, the cavity method gives
\[
W_c=102.5,
\]
close to the random-regular-graph value \(W_c=105\) [2312.11857].

For the continuum hyperbolic plane \(\mathbb H^2\), the disorder problem is formulated as
\[
\left[-\frac{1}{2m}\Delta_{\mathbb H^2}+V(x)\right]\psi(x)=E\psi(x),
\]
or, at the field-theory level, by the nonlinear sigma model
\[
S[Q]=-\frac{\sigma}{16}\int d^2x\sqrt g\,\mathrm{str}(Q\Delta Q).
\]
The central result is a two-parameter flow in conductivity and curvature scale,
\[
\frac{ d \sigma}{d \ell}= -\frac{1}{2\pi} \frac{u^2\tanh(\pi u)}{u^2+\frac{1}{4} },
\qquad
\frac{d u}{d \ell}= - u,
\]
with an extended critical line separating metallic and insulating phases. Here “hyperbolic” refers to \(\mathbb H^2\), not to a wave operator [2604.24917].

Related parabolic Anderson models on hyperbolic spaces show that negative curvature changes both phase structure and asymptotic scales. For
\[
\partial_t u=\Delta u+\beta\,u\,\dot W
\quad\text{on }\mathbb H^d,
\]
with spatial covariance \(f(x,y)\asymp \rho(x,y)^{-\alpha}\), the critical decay exponent is
\[
\alpha_c=1.
\]
If \(\alpha>1\), sufficiently small \(\beta\) yields a bounded \(L^2\)-region; if \(0<\alpha<1\), the second moment diverges for every \(\beta>0\), and for small \(\beta\)
\[
\log \mathbf E[u(t,x)^2]\asymp t^{1-\alpha},
\]
while for large \(\beta\) the growth is exponential [2507.05530].

For the parabolic Anderson model
\[
\partial_t u(t,x)=\Delta u(t,x)+\xi(x)\,u(t,x),
\qquad u(0,x)\equiv1,
\]
on \(\mathbb H^d\) with a stationary Gaussian potential of finite correlation length, the quenched asymptotic is
\[
u(t,x)=\exp\!\big(L^* t^{5/3}+o(t^{5/3})\big)\qquad\text{a.s.},
\]
with \(L^*\) obtained from an explicit variational problem. The optimal strategy reaches a peak at distance of order \(t^{4/3}\), then remains there for the rest of the time [2506.20147].

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 [2112.04954, 2604.24917].

Source: https://www.emergentmind.com/topics/hyperbolic-anderson-equation