• Corpus ID: 220496430

# Lyapunov exponents of the SHE for general initial data

@article{Ghosal2020LyapunovEO,
title={Lyapunov exponents of the SHE for general initial data},
author={Promit Ghosal and Yier Lin},
journal={arXiv: Probability},
year={2020}
}
• Published 13 July 2020
• Mathematics
• arXiv: Probability
We consider the $(1+1)$-dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar-Parisi-Zhang (KPZ) equation. We show an exact way of computing the Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial…

### Temporal increments of the KPZ equation with general initial data

. We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation H f ( t, x ) started with initial data f . In this article, we study the sample path properties of the KPZ temporal process

### Lyapunov Exponents of the Half-Line SHE

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer)

### Dissipation in Parabolic SPDEs II: Oscillation and decay of the solution

• Mathematics
• 2021
We consider a stochastic heat equation of the type, ∂ t u = ∂ 2 x u + σ ( u ) ˙ W on (0 , ∞ ) × [ − 1 , 1] with periodic boundary conditions and non-degenerate positive initial data, where σ : R → R

### Fractal Geometry of the Valleys of the Parabolic Anderson Equation

• Mathematics
• 2021
where Ẇ is the time-space white noise and 0 < infx∈R u0(x) ≤ supx∈R u0(x) < ∞. Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are

### Integrability in the weak noise theory

. WeconsiderthevariationalproblemassociatedwiththeFreidlin–WentzellLargeDeviationPrinciple (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show

### Law of Iterated Logarithms and Fractal Properties of the KPZ Equation

• Mathematics
• 2021
We consider the Cole-Hopf solution of the (1 + 1)-dimensional KPZ equation started from the narrow wedge initial condition. In this article, we ask how the peaks and valleys of the KPZ height

### Large deviations of the KPZ equation via the Stochastic Airy Operator

In this article we review the ideas in [Tsa18] toward proving the one-point, lower-tail large deviation principle for the Kardar–Parisi– Zhang equation. §

### KPZ equation with a small noise, deep upper tail and limit shape

• Mathematics
• 2021
In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter √ ε in front of the noise and let ε → 0. We prove that the one-point large deviation

### Upper-tail large deviation principle for the ASEP

• Mathematics
• 2021
We consider the asymmetric simple exclusion process (ASEP) on Z started from step initial data and obtain the exact Lyapunov exponents for H0(t), the integrated current of ASEP. As a corollary, we

### High Temperature Behaviors of the Directed Polymer on a Cylinder

• Mathematics
Journal of Statistical Physics
• 2022
In this paper, we study the free energy of the directed polymer on a cylinder of radius L with the inverse temperature $$\beta$$ β . Assuming the random environment is given by a Gaussian process

## References

SHOWING 1-10 OF 88 REFERENCES

### Moments and Lyapunov exponents for the parabolic Anderson model

• Mathematics
• 2014
We study the parabolic Anderson model in $(1+1)$ dimensions with nearest neighbor jumps and space-time white noise (discrete space/continuous time). We prove a contour integral formula for the second

### KPZ equation tails for general initial data

• Mathematics, Computer Science
Electronic Journal of Probability
• 2020
The upper and lower tail probabilities for the centered and scaled one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class are considered.

### On the chaotic character of the stochastic heat equation, before the onset of intermitttency

• Mathematics
• 2013
We consider a nonlinear stochastic heat equation @tu = 1 @xxu + (u)@xtW , where @xtW denotes space-time white noise and : R ! R is Lipschitz continuous. We establish that, at every xed time t > 0,

### Parabolic problems for the Anderson model

• Mathematics
• 1990
Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤd associated with the Anderson Hamiltonian H=κΔ+ξ(·) for

### Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise

• Mathematics
• 2016
This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $$H\in (0,1/2)$$H∈(0,1/2). We establish the Feynman–Kac representation of the solution

### The stochastic Burgers Equation

• Mathematics
• 1994
We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The

### Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions

• Mathematics
• 2015
We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be

### Lyapunov Exponents of the Half-Line SHE

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer)

### A Riemann-Hilbert approach to the lower tail of the KPZ equation

• Mathematics
• 2019
Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also

• Mathematics
• 2017
We analyze the one-dimensional periodic Kardar–Parisi–Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from