No Arabic abstract
The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $varepsilon$ and its strength is weakened as $beta_varepsilon=hat{beta} sqrt{frac{2pi varepsilon}{-log varepsilon}}$, and they prove that when $hat{beta}in (0,1)$, $frac{1}{beta_varepsilon}(log u_varepsilon-mathbb{E}[log u_varepsilon])$ converges in distribution to a solution of Edward-Wilkinson model as a random field. In this paper, we consider a stochastic heat equation $u_varepsilon$ with general initial condition $u_0$ and its transformation $F(u_varepsilon)$ for $F$ in a class of functions $mathfrak{F}$, which contains $F(x)=x^p$ ($0<pleq 1$) and $F(x)=log x$. Then, we prove that $frac{1}{beta_varepsilon}(F(u_varepsilon(t,x))-mathbb{E}[F(u_varepsilon(t,x))])$ converges in distribution to Gaussian random variables jointly in finitely many $Fin mathfrak{F}$, $t$, and $u_0$. In particular, we obtain the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depends on $u_0$. Our main tools are It^os formula, the martingale central limit theorem, and the homogenization argument as in the works by Cosco and the authors. To this end, we also prove the local limit theorem for the partition function of intermediate $2d$-directed polymers
We consider a family of nonlinear stochastic heat equations of the form $partial_t u=mathcal{L}u + sigma(u)dot{W}$, where $dot{W}$ denotes space-time white noise, $mathcal{L}$ the generator of a symmetric Levy process on $R$, and $sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $mathcal{L}f=cf$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.
The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirich-let problem for Laplaces equation, its implementation is rather easy. The definition of the random walk is based on a new mean value formula for the heat equation. The convergence results and numerical examples permit to emphasize the efficiency and accuracy of the algorithm.
We consider the stochastic heat equation with a multiplicative white noise forcing term under standard intermitency conditions. The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $xmapsto u(t,,x)$ can be characterized generically by the decay rate, at $pminfty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $Lambda:= lim_{|x|toinfty} |log u_0(x)|/(log|x|)^{2/3}$.
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 1/4textless{}Htextless{}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.
In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.