# Moment Intermittency in the PAM with Asymptotically Singular Noise

@inproceedings{Lamarre2022MomentII, title={Moment Intermittency in the PAM with Asymptotically Singular Noise}, author={Pierre Yves Gaudreau Lamarre and Promit Ghosal and Yuchen Liao}, year={2022} }

Let ξ be a singular Gaussian noise on R that is either white, fractional, or with the Riesz covariance kernel; in particular, there exists a scaling parameter ω > 0 such that cξ(c·) is equal in distribution to ξ for all c > 0. Let (ξε)ε>0 be a sequence of smooth mollifications such that ξε → ξ as ε → 0. We study the asymptotics of the moments of the parabolic Anderson model (PAM) with noise ξε as ε → 0, both for large (i.e., t → ∞) and fixed times t. This approach makes it possible to study the…

## References

SHOWING 1-10 OF 43 REFERENCES

### Stationary parabolic Anderson model and intermittency

- Mathematics
- 1995

SummaryThis paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem:
$$\frac{{\partial u}}{{\partial t}} = \kappa \Delta u\xi (x)u$$
when the…

### Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime

- Mathematics
- 2017

. In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense deﬁned by Skorokhod…

### Almost sure asymptotics for the continuous parabolic Anderson model

- Mathematics
- 2000

Abstract. We consider the parabolic Anderson problem ∂tu = κΔu + ξ(x)u on ℝ+×ℝd with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We…

### Moment asymptotics for the continuous parabolic Anderson model

- Mathematics
- 2000

We consider the parabolic Anderson problem ∂ t u = κ(cid:5)u + ξ (cid:3) x (cid:4) u on R + × R d with initial condition u (cid:3) 0 (cid:9)x (cid:4) = 1. Here ξ (cid:3)·(cid:4) is a random shift-…

### 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…

### The Parabolic Anderson Model

- Mathematics
- 2016

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model…

### 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…

### 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…

### Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

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

We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time…