arXiv subjects

Juan J. Jiménez

Publications and source records attributed to Juan J. Jiménez.

7 recordsLinked to original sources

arXiv2026

Stochastic heat equations with non-Lipschitz coefficients driven by symmetric $α$-stable Lévy white noise

We study the stochastic heat equation on a bounded interval with zero Dirichlet boundary conditions, driven by symmetric $α$-stable space--time Lévy white noise, where $1<α<2$. Both the drift and noise coefficients are continuous and have at most linear growth. For bounded deterministic initial data and every $r>1/2$, we construct a weak solution with càdlàg paths in the negative Sobolev space $\mathbb{H}_{-r}$. We also construct a predictable mild solution on the same stochastic basis and with the same noise. At each fixed deterministic time, this mild solution is a density of the distribution-valued weak solution.

math.PR↗

arXiv2026

Spatial covariance of KPZ from flat initial profile

We study the fixed-time spatial covariance of the KPZ equation with flat initial profile. Using Malliavin calculus and a Clark-Ocone representation, we show that as $|x|\to\infty$, $\mathrm{Cov}[h(t,x),h(t,0)]$ is governed by a boundary-layer regime near the initial time and satisfies $\mathrm{Cov}[h(t,x),h(t,0)] \sim κ(t) \int_0^t p_{2r}(x) dr = \frac{2κ(t)}{\sqrtπ}t^{3/2}|x|^{-2}\exp\left(-\frac{x^2}{4t}\right),$ as $|x|\to\infty$, where $κ(t) = \big(\mathbb{E}[Z(t,0)^{-1}]\big)^2$, $Z$ is the flat stochastic heat equation solution, and $p_t$ is the one-dimensional heat kernel. In sharp contrast with the narrow-wedge regime, where Gu-Pu (2025, Theorem 1.1) proved that for each fixed $t>0$, $\mathrm{Cov}\left[h^{\mathrm{nw}}(t,x),h^{\mathrm{nw}}(t,0)\right]\sim \frac{t}{|x|},$ as $|x|\to\infty$, the flat initial profile exhibits Gaussian decay, yielding, to the best of our knowledge, the first exact spatial covariance asymptotic for the KPZ equation under flat initial data. We also establish an explicit closed-form formula for the second moment of the continuum directed random polymer partition function.

math.PR↗

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.

math.PR↗

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 $α\in (0,2)$.

math.PR↗

arXiv2025

Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise

In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dotξ$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric Lévy noise $\dotξ$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power $γ\in (0,\infty)$. In particular, whenever $γ>\tfrac{d}{2}$, the solution admits a continuous modification.

math.PR↗

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

Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions and past light-cone property

In this article, we study the stochastic wave equation in spatial dimensions $d \le 2$ with multiplicative Lévy noise that can have infinite $p$-th moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the $p$-integrability of the Lévy measure $ν$ for the region corresponding to the small jumps of the noise. For $d=1$, there are no restrictions on $ν$. For $d=2$, we assume that there exists a value $p \in (0,2)$ for which $\int_{ \{|z|\le 1 \} } |z|^p ν(dz) < + \infty$.

math.PR↗