Khoa Lê

Stochastic heat equation with rough dependence in space

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4<H<1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.

A multiparameter Garsia–Rodemich–Rumsey inequality and some applications

We extend the classical Garsia–Rodemich–Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint Holder continuity along the rectangles for fractional Brownian fields W(t,x) and for the solution u(t,y) of the stochastic heat equation with additive white noise.

Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise

In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter

Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

H\in (\frac 14, \frac 12]

Nonlinear Young Integrals via Fractional Calculus

in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model.

Laws of large numbers for supercritical branching Gaussian processes

We consider the linear stochastic heat equation on

Stochastic differential equation for Brox diffusion

\mathbb {R}^\ell

Joint H\"older continuity of parabolic Anderson model

Rℓ, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter