arXiv subjects

Raluca M. Balan

Publications and source records attributed to Raluca M. Balan.

At least 19 recordsLinked to original sources

arXiv2026

Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise

In this article, we consider the parabolic Anderson model driven by a L\'evy white noise with finite variance in dimension 1, and we study the asymptotic behaviour of the spatial average of the solution. The main result shows that, with appropriate normalization and centering, the spatial integral converges in distribution to the standard normal distribution, and gives an estimate for the rate of this convergence in the Wasserstein distance, the Kolmogorov distance, and the Fortet-Mourier distance. We also prove the functional limit theorem corresponding to this result.

math.PR↗

arXiv2026

Itô integral for a two-sided Lévy process

In this article, we construct an Itô integral with respect to a two-sided finite-variance Lévy process $\{L(x)\}_{x\in \mathbb{R}}$, without a Gaussian component. Using Rosenthal inequality for discrete-time martingales, we give an estimate for the $p$-th moment of this integral, for any even integer $p\geq 2$. Then, using Poisson-Malliavin calculus, we show that the Itô integral is an extension of the Hitsuda-Skorohod integral with respect to the compensated Poisson random measure associated to the Lévy process.

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arXiv2026

SPDEs with time-independent Lévy colored noise

In this article, we introduce a time-independent version of the Lévy colored noise considered in Balan (2015) and Balan and Jiménez (2026). We study the existence of the solution of a linear stochastic partial differential equation with this type of noise, and we identify some necessary conditions which guarantee that the solution has finite $p$-th order moments. Using tools from Malliavin calculus, we investigate the existence of the solution for the equation with multiplicative noise. As examples, we consider the stochastic heat and wave equations in any dimension $d \geq 1$.

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arXiv2026

Almost sure CLT for hyperbolic Anderson model with Lévy colored noise

In this note, we prove the Almost Sure Central Limit Theorem (ASCLT) for the spatial integral of the solution of the hyperbolic Anderson model driven by the Lévy colored noise introduced in Balan (2015). For this, we use the central limit theorem for the normalized spatial integral, and an estimate for the Malliavin derivative of the solution, both derived in the recent preprint Balan and Stephenson (2026). We assume that the spatial correlation kernel of the noise is either integrable, or it is given by the Riesz kernel.

math.PR↗

arXiv2026

Gaussian fluctuations for hyperbolic Anderson model with Lévy colored noise

In this article, we study the asymptotic behaviour of the spatial integral $F_R(t)$ of the solution to the hyperbolic Anderson model in dimension $d=1$, driven by the Lévy colored noise introduced in Balan and Jiménez (2026). We assume that the spatial coloration kernel of the noise is either integrable on $\mathbb{R}$, or is the Riesz kernel of order $α\in (0,1)$, and the Lévy measure of the noise has finite moments of order $p$ and $2p$ for some $p \in (1,2]$. By applying a recent result of Trauthwein (2025), we prove that $F_R(t)/\sqrt{{\rm Var}\big(F_R(t)\big)}$ converges to the standard normal distribution as $R \to \infty$, and we give an estimate for the rate of this convergence in the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance. We also provide the corresponding functional limit result.

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arXiv2026

Moment estimates for solutions of SPDEs with Lévy colored noise

In this article, we continue the investigations initiated by the first author in Balan (2015) related to the study of stochastic partial differential equations (SPDEs) with Lévy colored noise on $\mathbb{R}_{+} \times \mathbb{R}^d$. This noise is constructed from a Lévy white noise (which is in turn built from a Poisson random measure with intensity $dtdx ν(dz)$), using the convolution with a suitable spatial kernel $κ$. We assume that the Lévy measure $ν$ has finite variance. Therefore, the stochastic integral with respect to this noise is constructed similarly to the integral with respect to the spatially-homogeneous Gaussian case considered in Dalang (1999). Using Rosenthal's inequality, we provide an upper bound for the $p$-th moment of the stochastic integral with respect to the Lévy colored noise, which allows us to identify sufficient conditions for the solution of an SPDE driven by this noise to have higher order moments. We first analyze this question for the linear SPDE, considering as examples the stochastic heat and wave equations in any dimension $d$, for three examples of kernels $κ$: the heat kernel, the Riesz kernel, and the Bessel kernel. Then, we present a general theory for a non-linear SPDE with Lipschitz coefficients, and perform a detailed analysis in the case of the heat equation (in dimension $d\geq 1$), and wave equation (in dimension $d\leq 3$), for the same kernels $κ$. We show that the solution of each of these equations has a finite upper Lyapounov exponent of order $p\geq 2$, and in some cases, is weakly intermittent (in the sense of Foondun and Khoshnevisan, 2013). In the case of the parabolic/hyperbolic Anderson model with Lévy colored noise, we provide the Poisson chaos expansion of the solution and the explicit form of the second-order Lyapounov exponent.

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arXiv2025

Gaussian fluctuations for the nonlinear stochastic heat equation with drift

In this article, we prove the Quantitative Central Limit Theorem (QCLT) for the spatial average of the solution of the nonlinear stochastic heat equation with constant initial condition, driven by space-time Gaussian white noise in dimension 1. The novelty is that the equation contains a drift term. We assume that the drift and diffusion coefficients are twice differentiable with bounded first and second order derivatives. For the proof, we use Malliavin calculus, and the second-order Poincaré inequality due to Vidotto (2020). To estimate the moment of the second Malliavin derivative of the solution, we develop a novel estimate for the product of two heat kernels, which is of independent interest. Finally, we provide the functional result corresponding to this CLT.

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arXiv2025

Global well-posedness for hyperbolic SPDEs with non-Lipschitz coefficients driven by space-time Lévy white noise

In this article, we study the global well-posedness of hyperbolic SPDEs on a bounded domain in $\mathbb{R}^d$, driven by a space-time Lévy white noise, when the drift and diffusion coefficients are locally Lipschitz and have linear growth. The equations are driven by two types of space-time Lévy noise: (i) a finite-variance Lévy white noise; or (ii) a symmetric Lévy basis that may have infinite variance. A typical example of noise of the second type is the symmetric $α$-stable (S$α$S) random measure with $α\in (0,2)$.

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arXiv2025

Central limit theorem for stochastic nonlinear wave equation with pure-jump Lévy white noise

In this paper, we study the random field solution to the stochastic nonlinear wave equation (SNLW) with constant initial conditions and multiplicative noise $σ(u)\dot{L}$, where the nonlinearity is encoded in a Lipschitz function $σ: \mathbb{R}\to\mathbb{R}$ and $\dot{L}$ denotes a pure-jump Lévy white noise on $\mathbb{R}_+\times\mathbb{R}$ with finite variance. Combining tools from Itô calculus and Malliavin calculus, we are able to establish the Malliavin differentiability of the solution with sharp moment bounds for the Malliavin derivatives. As an easy consequence, we obtain the spatial ergodicity of the solution to SNLW that leads to a law of large number result for the spatial integrals of the solution over $[-R, R]$ as $R\to\infty$. One of the main results of this paper is the obtention of the corresponding Gaussian fluctuation with rate of convergence in Wasserstein distance. To achieve this goal, we adapt the discrete Malliavin-Stein bound from Peccati, Solé, Taqqu, and Utzet ({\it Ann. Probab.}, 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and Itô tools. Our work substantially improves our previous results (\textit{Trans.~Amer.~Math.~Soc.}, 2024) on the linear equation that heavily relied on the explicit chaos expansion of the solution. In current work, we also establish a functional version, an almost sure version of the central limit theorems, and the (quantitative) asymptotic independence of spatial integrals from the solution. The asymptotic independence result is established based on an observation of L. Pimentel (\textit{Ann.~Probab.}, 2022) and a further adaptation of Tudor's generalization (\textit{Trans.~Amer.~Math.~Soc.}, 2025) to the Poisson setting.

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arXiv2024

Series expansions for SPDEs with symmetric $α$-stable Lévy noise

In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $α$-stable (S$α$S) Lévy noise, that is multiplied by a linear function $σ(u)=u$ of the solution. The solution is interpreted in the mild sense. For this models, in the case of the Gaussian noise, the solution has an explicit Wiener chaos expansion, and is studied using tools from Malliavin calculus. These tools cannot be used for an infinite-variance Lévy noise. In this article, we provide sufficient conditions for the existence of a solution, and we give an explicit series expansion of this solution. To achieve this, we use the multiple stable integrals, which were developed in Samorodnitsky and Taqqu (1990, 1991), and originate from the LePage series representation of the noise. To give a meaning to the stochastic integral which appears in the definition of solution, we embed the space-time Lévy noise into a Lévy basis, and use the stochastic integration theory (Bichteler and Jacod 1983, Bichteler 2002) with respect to this object, as in other studies of SPDEs with heavy-tailed noise: Chong (2017a), Chong (2017b), Chong, Dalang and Humeau (2019). As applications, we consider the heat and wave equations with linear multiplicative noise, also called the parabolic/hyperbolic Anderson models.

math.PR↗

arXiv2024

Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise

In this paper, we present an almost sure central limit theorem (ASCLT) for the hyperbolic Anderson model (HAM) with a Lévy white noise in a finite-variance setting, complementing a recent work by Balan and Zheng (\emph{Trans.~Amer.~Math.~Soc.}, 2024) on the (quantitative) central limit theorems for the solution to the HAM. We provide two different proofs: one uses the Clark-Ocone formula and takes advantage of the martingale structure of the white-in-time noise, while the other is obtained by combining the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits' method of characteristic functions. Both approaches are different from the one developed in the PhD thesis of C. Zheng (2011), allowing us to establish the ASCLT without lengthy computations of star contractions. Moreover, the second approach is expected to be useful for similar studies on SPDEs with colored-in-time noises, whereas the former, based on Itô calculus, is not applicable.

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arXiv2024

Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional central limit theorem by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality.

math.PR↗

arXiv2023

Gaussian fluctuations for the wave equation under rough random perturbations

In this article, we consider the stochastic wave equation in spatial dimension $d=1$, with linear term $σ(u)=u$ multiplying the noise. This equation is driven by a Gaussian noise which is white in time and fractional in space with Hurst index $H \in (\frac{1}{4},\frac{1}{2})$. First, we prove that the solution is strictly stationary and ergodic in the spatial variable. Then, we show that with proper normalization and centering, the spatial average of the solution converges to the standard normal distribution, and we estimate the rate of this convergence in the total variation distance. We also prove the corresponding functional convergence result.

math.PR↗

arXiv2023

Continuity in law for solutions of SPDEs with space-time homogeneous Gaussian noise

In this article, we study the continuity in law of the solutions of two linear multiplicative SPDEs (the parabolic Anderson model and the hyperbolic Anderson model) with respect to the spatial parameter of the noise. The solution is interpreted in the Skorohod sense, using Malliavin calculus. We consider two cases: (i) the regular noise, whose spatial covariance is given by the Riesz kernel of order $α\in (0,d)$, in spatial dimension $d\geq 1$; (ii) the rough noise, which is fractional in space with Hurst index $H<1/2$, in spatial dimension $d=1$. We assume that the noise is colored in time. The similar problem for the white noise in time was considered in Bezdek (2016) and Giordano, Jolis and Quer-Sardanyons (2020).

math.PR↗

arXiv2023

Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations

In this article, we study the hyperbolic Anderson model in dimension 1, driven by a time-independent rough noise, i.e. the noise associated with the fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that, with appropriate normalization and centering, the spatial integral of the solution converges in distribution to the standard normal distribution, and we estimate the speed of this convergence in the total variation distance. We also prove the corresponding functional limit result. Our method is based on a version of the second-order Gaussian Poincaré inequality developed recently in [27], and relies on delicate moment estimates for the increments of the first and second Malliavin derivatives of the solution. These estimates are obtained using a connection with the wave equation with delta initial velocity, a method which is different than the one used in [27] for the parabolic Anderson model.

math.PR↗

arXiv2023

Stochastic wave equation with Lévy white noise

In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time Lévy white noise with possibly infinite variance (such as the $α$-stable Lévy noise). In this equation, the noise is multiplied by a Lipschitz function $σ(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the Lévy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a càdlàg modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.

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arXiv2022

Parabolic Anderson model with rough noise in space and rough initial conditions

In this note, we consider the parabolic Anderson model on $\mathbb{R}_{+} \times \mathbb{R}$, driven by a Gaussian noise which is fractional in time with index $H_0>1/2$ and fractional in space with index $0 3/4$. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all $p$-th moments with $p\ge 2$.

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arXiv2022

Central limit theorems for heat equation with time-independent noise: the regular and rough cases

In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension $d\geq 1$, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H \in (\frac{1}{4},\frac{1}{2})$ in dimension $d=1$. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

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