--- title: Hyperbolic Anderson Model in SPDE Analysis url: https://www.emergentmind.com/topics/hyperbolic-anderson-model type: topic --- # Hyperbolic Anderson Model in SPDE Analysis Searching arXiv for recent 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 \(u\,\dot W\) [1602.07004][2112.04954]. 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 [2604.24917][2312.11857]. 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 \(u(0,x)=1\), \(\partial_tu(0,x)=0\) used in much of the fluctuation literature [1602.07004][2112.04954]. The mild form is \[ u(t,x)=w(t,x)+\int_0^t\int_{\mathbb R^d}G_{t-s}(x-y)u(s,y)\,W(ds,dy), \] with \[ w(t,x)=\frac{\partial}{\partial t}(G_t*u_0)(x)+(G_t*u_1)(x), \] and wave kernel Fourier transform \[ \widehat{G_t}(\xi)=\frac{\sin(t|\xi|)}{|\xi|}. \] For \(d=1,2,3\), the fundamental solutions are explicitly available; in particular, \[ G_t(x)=\frac12\,\mathbf 1_{\{|x|1\) closes the fixed point [2205.15773]. A later Stratonovich analysis for time-dependent Gaussian noise in \(d=1,2,3\) established the optimal existence condition \[ \int_{\mathbb R^d}\left(\frac{1}{1+|\xi|^2}\right)^{\frac{2-\alpha_0}{2}}\mu(d\xi)<\infty, \] and identifies it as necessary and sufficient for the square-integrable Stratonovich solution considered there [2510.01412]. That paper explicitly contrasts this with the Skorohod threshold \((3-\alpha_0)/2\), so in that regime the Stratonovich solvability condition is weaker than the Skorohod one [2510.01412]. In the Lévy setting, the baseline model is the one-dimensional stochastic wave equation \[ (\partial_t^2-\partial_x^2)u=u\,L \] with \(L\) a space-time pure-jump Lévy white noise of finite variance, realized through \[ L(A)=\int_{A\times\mathbb R_0} z\,\widehat N(ds,dy,dz), \qquad m_2=\int_{\mathbb R_0}|z|^2\,\nu(dz)<\infty \] [2302.14178][2310.10784]. Lévy colored noise is obtained by spatial convolution, \[ X_t(\varphi)=L_t(\varphi*k), \] with covariance kernel \(f=k*\widetilde k\) and either \(k\in L^1(\mathbb R)\) or \(k=R_{1,\alpha/2}\) in the Riesz case [2602.23137][2602.24189]. ## 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 \(H\in(1/4,1/2)\), existence of a Skorohod solution holds precisely for \(H>1/4\), and this threshold is necessary [1605.00024]. In that regime the \(p\)-th moments satisfy an exponential upper bound, and the solution is weakly intermittent in the sense that \[ \underline\gamma(2)>0,\qquad \overline\gamma(p)<\infty\ \text{for all }p\ge 2 \] [1605.00024]. 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 [1704.02411]. In the Riesz-kernel case \(f(x)=|x|^{-\alpha}\), the hyperbolic exponent is \[ \lim_{t\to\infty}\frac1t\log \mathbb E|u^w(t,x)|^2 = \big(2^{1-\alpha}\rho\big)^{1/(3-\alpha)}, \] with \(\rho\) defined variationally through the Riesz interaction [1704.02411]. A different asymptotic regime appears in the Stratonovich theory with time-dependent Gaussian noise. Under homogeneous spatial covariance \(\gamma(cx)=c^{-\alpha}\gamma(x)\) and \(\alpha_0+\alpha<2\), the expectation grows according to \[ \lim_{t\to\infty} t^{-\frac{4-\alpha-\alpha_0}{3-\alpha}} \log E\,u(t,x) = (3-\alpha)\Big(\cdots\Big)^{1/(3-\alpha)}, \] where the constant is expressed through a variational functional \(\mathcal M\) or \(\mathcal E_0\) built from a time-randomized Brownian intersection local time [2510.01412]. That analysis also shows that the expected even Stratonovich chaos levels can be represented through \[ \mathcal I_t^{(\theta)} = \int_0^t\!\!\int_0^t \big(\theta|s-r|+i(\beta(s)-\beta(r))\big)^{-\alpha_0} \gamma(B(s)-B(r))\,ds\,dr, \] which is the mechanism by which time dependence modifies both solvability and large-\(t\) growth [2510.01412]. ## 4. Spatial averages, ergodicity, and quantitative Gaussian fluctuations A major recent direction studies the spatial integral \[ F_R(t)=\int_{B_R}\big(u(t,x)-1\big)\,dx \quad\text{or in }d=1,\quad F_R(t)=\int_{-R}^{R}\big(u(t,x)-1\big)\,dx, \] and its normalized fluctuations as \(R\to\infty\) [2101.10957][2302.14178]. For Gaussian colored noise in \(d=1,2\), quantitative central limit theorems were obtained by coupling first- and second-order Malliavin derivative bounds with a second-order Gaussian Poincaré inequality [2101.10957]. In the integrable spatial-covariance regime, \[ \sigma_R(t)\asymp R^{d/2}, \qquad d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\lesssim R^{-d/2}, \] while for the Riesz kernel \(\gamma(x)=|x|^{-\beta}\), \[ \sigma_R(t)\asymp R^{d-\beta/2}, \qquad d_{\mathrm{TV}}\!\left(\frac{F_R(t)}{\sigma_R(t)},Z\right)\lesssim R^{-\beta/2} \] [2101.10957]. The corresponding functional CLTs hold in \(C(\mathbb R_+)\) [2101.10957]. For time-independent rough Gaussian noise in one dimension, the normalized spatial integral also converges to \(N(0,1)\), with \[ \frac{\sigma_{R,\theta}^2(t)}{R}\to K_\theta(t,t)>0, \qquad d_{\mathrm{TV}}\!\left(\frac{F_{R,\theta}(t)}{\sigma_{R,\theta}(t)},Z\right)\le C_{t,H,\theta}R^{-1/2}, \] and a functional limit \[ \{R^{-1/2}F_{R,\theta}(t):t\ge 0\}\Rightarrow \{G_\theta(t):t\ge 0\} \] [2305.05043]. A notable feature there is that the first Wiener chaos does not contribute to the limiting covariance [2305.05043]. In the Lévy white-noise setting, the one-dimensional solution field is strictly stationary and ergodic in space for each fixed time, and \[ \frac{F_R(t)}{R}\to 0 \quad\text{in }L^2\text{ and a.s.} \] [2302.14178]. The variance satisfies \[ \operatorname{Var}(F_R(t))\sim \mathcal E_{t,t}R, \] with explicit covariance kernel \[ \mathcal E_{t,s} = 2m_2\int_0^{t\wedge s}(t-r)(s-r)\cosh\!\left(r\sqrt{\frac{m_2}{2}}\right)\,dr, \] and, assuming \(m_{2+2a}<\infty\) and \(m_{1+a}<\infty\), \[ \mathrm{dist}\!\left(\frac{F_R(t)}{\sigma_R(t)},N(0,1)\right)\lesssim R^{-a} \] in Fortet–Mourier, \(1\)-Wasserstein, and Kolmogorov distance [2302.14178]. For Lévy colored noise in \(d=1\), the variance exponent is \[ \beta= \begin{cases} 1,& k\in L^1(\mathbb R),\\ \alpha+1,& k=R_{1,\alpha/2}, \end{cases} \] so \(R^{-\beta/2}F_R(\cdot)\) converges to a centered Gaussian process, and the one-time normalized variable converges quantitatively to \(N(0,1)\) in \(d_{FM}\), \(d_W\), or \(d_K\) [2602.23137]. ## 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, \[ \frac{1}{\log T}\int_1^T \frac{1}{R}\,\delta_{\widetilde F_R(\omega)}\,dR \Rightarrow N(0,1) \qquad (T\to\infty). \] 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 [2310.10784]. 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 [2310.10784]. 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 [2409.07358]. For the hyperbolic model, this applies in \(d=1,2\), with either \(\gamma_1\in L^1\) or \(\gamma_1(x)=|x|^{-\alpha}\), and uses the wave-specific bound \[ (1_{B_R}*G_r)(y)\le r\,1_{B_{R+r}}(y) \] to control mixed covariances across scales [2409.07358]. For Lévy colored noise, the ASCLT holds for \[ \widetilde F_R(t)=\frac{F_R(t)}{\sigma_R(t)} \] under the same two kernel classes, \(k\in L^1(\mathbb R)\) or \(k=R_{1,\alpha/2}\), and is proved by adapting the Clark–Ocone covariance-decay method from the white-noise paper [2602.24189]. 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 [2310.10784][2602.24189]. ## 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, \[ H=-\frac{1}{2m}\Delta+V(x), \] with Gaussian white-noise potential \[ \langle V(x)V(y)\rangle= \left(\frac{W}{k_0}\right)^2 |g(x)|^{-1/2}\delta(x-y), \] posed on \(\mathbb H^2\) of curvature radius \(L\) and scalar curvature \(R=-2L^{-2}\) [2604.24917]. The key geometric fact is that \(\mathbb H^2\) is locally Euclidean but has exponential volume growth at large distance, so returns are suppressed and the Anderson problem is no longer the flat-\(2d\) one. The resulting nonlinear sigma-model analysis yields a two-parameter renormalization flow \[ \frac{d\sigma}{d\ell} = -\frac{1}{2\pi}\frac{u^2\tanh(\pi u)}{u^2+\frac14}, \qquad \frac{du}{d\ell}=-u, \] with \(u(\ell)=\Lambda L e^{-\ell}\), and, after matching to lattice strong-disorder physics at the curvature scale, an extended critical line separating metallic and insulating phases [2604.24917]. On hyperbolic lattices, the tight-binding Anderson Hamiltonian \[ H=\sum_{\langle i,j\rangle} t(c_i^\dagger c_j+\mathrm{h.c.})+\sum_j \epsilon_j c_j^\dagger c_j, \qquad \epsilon_j\in[-W/2,W/2], \] shows a finite-disorder localization transition and mobility edges, in sharp contrast with ordinary \(2d\) Euclidean lattices [2312.11857][2310.07978]. On randomly boundary-connected \(\{3,8\}\) and \(\{4,8\}\) lattices, finite-size scaling of the adjacent-gap ratio and inverse participation ratio gives band-center estimates \[ W_c=77.2\pm 3 \ \text{or}\ 83.0\pm 4 \quad \text{for }\{3,8\}, \] and \[ W_c=88.2\pm 3 \ \text{or}\ 94\pm 4 \quad \text{for }\{4,8\}, \] with \(\nu\approx 1\) and multifractal critical states of dimension \(D_2\approx 0.3\) [2312.11857]. A complementary periodic-boundary construction on \(\{8,3\}\) and \(\{8,8\}\) likewise finds finite critical disorder, approximately \[ W_c^{\{8,3\}}\approx 15t, \qquad W_c^{\{8,8\}}\approx 100t, \] together with strong finite-size drift in level statistics [2310.07978]. This dual usage of the phrase “hyperbolic Anderson model” marks a genuine split in current literature. In one branch, hyperbolicity refers to the operator \(\partial_t^2-\Delta\); 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. Source: https://www.emergentmind.com/topics/hyperbolic-anderson-model