# Solving the hyperbolic Anderson model 1: Skorohod setting

@inproceedings{Chen2021SolvingTH, title={Solving the hyperbolic Anderson model 1: Skorohod setting}, author={Xia Chen and Aur'elien Deya and Jian Song and Samy Tindel}, year={2021} }

This paper is concerned with a wave equation in dimension d ∈ {1, 2, 3}, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

## 3 Citations

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

- Mathematics
- 2022

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 ≥ 1, as the domain of the…

### Necessary and sufficient conditions to solve parabolic Anderson model with rough noise

- Mathematics
- 2022

We obtain necessary and sufficient conditions for the existence of n-th chaos of the solution to the parabolic Anderson model ∂ ∂t u(t, x) = 1 2 ∆u(t, x) + u(t, x)Ẇ (t, x), where Ẇ (t, x) is a…

### Hyperbolic Anderson model 2: Strichartz estimates and Stratonovich setting

- Mathematics
- 2022

. We study a wave equation in dimension d ∈ { 1 , 2 } with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions…

## References

SHOWING 1-10 OF 17 REFERENCES

### Hyperbolic Anderson Model with space-time homogeneous Gaussian noise

- Mathematics
- 2016

In this article, we study the stochastic wave equation in arbitrary spatial dimension $d$, with a multiplicative term of the form $\sigma(u)=u$, also known in the literature as the Hyperbolic…

### The Non-Linear Stochastic Wave Equation in High Dimensions

- Mathematics
- 2008

We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental…

### Fractional stochastic wave equation driven by a Gaussian noise rough in space

- MathematicsBernoulli
- 2020

In this article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional…

### The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

- Mathematics
- 2010

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of…

### Exact asymptotics of the stochastic wave equation with time-independent noise

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2022

In this article, we study the stochastic wave equation in all dimensions $d\leq 3$, driven by a Gaussian noise $\dot{W}$ which does not depend on time. We assume that either the noise is white, or…

### Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

- Mathematics
- 2014

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of…

### On a non-linear 2D fractional wave equation

- Mathematics, Physics
- 2017

We pursue the investigations initiated in [Aur{\'e}lien Deya: A non-linear wave equation with fractional perturbation (2017)] about a wave-equation model with quadratic perturbation and stochastic…

### Parabolic Anderson model with rough or critical Gaussian noise

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2019

This paper considers the parabolic Anderson equation ∂u ∂t = 1 2 u + u ∂ d+1WH ∂t∂x1 · · · ∂xd generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H0,H1, . . . ,Hd). The…

### Moment estimates for some renormalized parabolic Anderson models

- MathematicsThe Annals of Probability
- 2021

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 \Delta u_{t}(x) + u_{t}(x) \, \dot…