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Kunwoo Kim

Publications and source records attributed to Kunwoo Kim.

27 records · Page 2Linked to original sources
arXiv2015

A Boundedness Trichotomy for the Stochastic Heat Equation

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 $x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $Λ:= \lim_{|x|\to\infty} |\log u_0(x)|/(\log|x|)^{2/3}$.

math.PR↗
arXiv2015

On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian

We consider stochastic heat equations with fractional Laplacian on $\mathbb{R}^d$. Here, the driving noise is generalized Gaussian which is white in time but spatially homogenous and the spatial covariance is given by the Riesz kernels. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the tall peaks and compute the macroscopic Hausdorff dimensions of the tall peaks for both (i) and (ii). These results imply that both (i) and (ii) exhibit multi-fractal behavior in a macroscopic scale even though (i) is not intermittent and (ii) is intermittent. This is an extension of a recent result by Khoshnevisan et al to a wider class of stochastic heat equations.

math.PR↗
arXiv2015

Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

Consider the stochastic heat equation $\partial_tu=\mathscr{L}u+λσ(u)ξ$, where $\mathscr{L}$ denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group $G$, $σ:\mathbf{R}\to\mathbf{R}$ is Lipschitz continuous, $λ\gg1$ is a large parameter, and $ξ$ denotes space-time white noise on $\mathbf{R}_+\times G$. The main result of this paper contains a near-dichotomy for the (expected squared) energy $\mathrm{E}(\|u_t\|_{L^2(G)}^2)$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $u$ is intermittent, the energy of the solution behaves generically as $\exp\{\operatorname {const}\cdot\,λ^2\}$ when $G$ is discrete and $\ge\exp\{\operatorname {const}\cdot\,λ^4\}$ when $G$ is connected.

math.PR↗
arXiv2015

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Let $ξ$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + uξ$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + ξ$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $\mathbf{R}_+\times\mathbf{R}^d$ with $d\ge 2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $\mathbf{R}$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.

math.PR↗
arXiv2014

Dissipation and high disorder

Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function $σ:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter $λ$ on the behavior of the system, \begin{align*} \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + λσ(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0δ_0(x). \end{align*} We show that for any $λ>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for dimensions greater than two there is a phase transition point $λ_c\in(0,\infty)$ such that for $λ>λ_c,\, \sum_{\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for $λ<λ_c,\,\sum_{\mathbf{Z}^d}u_t(x)\not\to 0$ as $t\to\infty.$

math.PR↗
arXiv2014

Decorrelation of total mass via energy

The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.

math.PR↗
arXiv2014

On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations

In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on $\mathbb{R}$ with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller's comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at $\pm\infty$. These results generalize a recent work by Moreno Flores [25], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [6].

math.PR↗
arXiv2013

Non-linear noise excitation and intermittency under high disorder

Consider the semilinear heat equation $\partial_t u = \partial^2_x u + λσ(u)ξ$ on the interval $[0\,,1]$ with Dirichlet zero boundary condition and a nice non-random initial function, where the forcing $ξ$ is space-time white noise and $λ>0$ denotes the level of the noise. We show that, when the solution is intermittent [that is, when $\inf_z|σ(z)/z|>0$], the expected $L^2$-energy of the solution grows at least as $\exp\{cλ^2\}$ and at most as $\exp\{cλ^4\}$ as $λ\to\infty$. In the case that the Dirichlet boundary condition is replaced by a Neumann boundary condition, we prove that the $L^2$-energy of the solution is in fact of sharp exponential order $\exp\{cλ^4\}$. We show also that, for a large family of one-dimensional randomly-forced wave equations, the energy of the solution grows as $\exp\{cλ\}$ as $λ\to\infty$. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.

math.PR↗