论文标题
限制定理的非线性随机热方程的时间相关平均值
Limit theorems for time-dependent averages of nonlinear stochastic heat equations
论文作者
论文摘要
我们的研究限制了$ x_t的时间相关的平均定理：= \ frac {1} {2l（t）} \ int _ { - l（t）}^{l（t）}^{l（t）} u（t，x，x，x）\，dx $ dx $到$ \ mathbb {r} _+\ times \ mathbb {r} $由时空白噪声驱动的$ u_0（x）= 1 $ for ALL $ x \ in \ MATHBB {r} $驱动的随机加热方程。我们以$ x_t $的方式表明 （i）当$λ>λ_1$时，大数字的薄弱定律， （ii）当$λ>λ_2$时，大数字的强大定律 （iii）当$λ>λ_3$时，中央限制定理成立，但是当$λ<λ_4\leqλ_3$， （iv）$λ>λ_5$时，定量中心限制定理成立 其中$λ_i$的s是正常数，具体取决于$ u（t，x）$的时刻Lyapunov指数。
We study limit theorems for time-dependent averages of the form $X_t:=\frac{1}{2L(t)}\int_{-L(t)}^{L(t)} u(t, x) \, dx$, as $t\to \infty$, where $L(t)=\exp(λt)$ and $u(t, x)$ is the solution to a stochastic heat equation on $\mathbb{R}_+\times \mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $x\in \mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $λ>λ_1$, (ii) the strong law of large numbers holds when $λ>λ_2$, (iii) the central limit theorem holds when $λ>λ_3$, but fails when $λ<λ_4\leq λ_3$, (iv) the quantitative central limit theorem holds when $λ>λ_5$, where $λ_i$'s are positive constants depending on the moment Lyapunov exponents of $u(t, x)$.
