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Register a new userOn the chaotic character of the stochastic heat equation, II
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Consider the stochastic heat equation $partial_t u = (frac{varkappa}{2})Delta u+sigma(u)dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)in (0, infty)timesR^d$, and $dot{F}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-$|x|$ fixed-$t$ behavior of the solution $u$ in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function $f$ of the noise is of Riesz type, that is $f(x)propto |x|^{-alpha}$, then the fluctuation exponents of the solution are $psi$ for the spatial variable and $2psi-1$ for the time variable, where $psi:=2/(4-alpha)$. Moreover, these exponent relations hold as long as $alphain(0, dwedge 2)$; that is precisely when Dalangs theory implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.
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We consider a nonlinear stochastic heat equation $partial_tu=frac{1}{2}partial_{xx}u+sigma(u)partial_{xt}W$, where $partial_{xt}W$ denotes space-time white noise and $sigma:mathbf {R}to mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $sigma$, $sup_{xin mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $limsup_{|x|toinfty}u_t(x)/({log}|x|)^{1/6}>0$ when $u_0$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.
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 $t>0$.
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 $Lambda:= lim_{|x|toinfty} |log u_0(x)|/(log|x|)^{2/3}$.
We investigate the compact interface property in a recently introduced variant of the stochastic heat equation that incorporates dormancy, or equivalently seed banks. There individuals can enter a dormant state during which they are no longer subject to spatial dispersal and genetic drift. This models a state of low metabolic activity as found in microbial species. Mathematically, one obtains a memory effect since mass accumulated by the active population will be retained for all times in the seed bank. This raises the question whether the introduction of a seed bank into the system leads to a qualitatively different behaviour of a possible interface. Here, we aim to show that nevertheless in the stochastic heat equation with seed bank compact interfaces are retained through all times in both the active and dormant population. We use duality and a comparison argument with partial functional differential equations to tackle technical difficulties that emerge due to the lack of the martingale property of our solutions which was crucial in the classical non seed bank case.
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.
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