اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn 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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