论文标题
谐波函数和线性椭圆形的dirichlet问题与随机边界值 - 某些经典定理和估计值的结构扩展
Harmonic Functions And Linear Elliptic Dirichlet Problems With Random Boundary Values--Stochastic Extensions Of Some Classical Theorems And Estimates
论文作者
论文摘要
令$ψ：{\ mathcal {d}} \ rightArow {\ mathbf {r}} $是一个谐波函数，使所有$ x \ in \ mathcal {D} \ subset {d} \ subset {\ subset {\ subbf {\ mathbf {r} {r}} $ y mathcal {d} = 0 $ for $Δψ（x）= 0 $。然后有许多良好的经典结果：迪利奇问题和泊松公式，harnack不平等，最大原则，平均价值属性等。在这里，一个“嘈杂”或随机域是一个经典的标量量表随机字段（grf）$ {\ nathscr {j}（x} $ x $ {或$ x \ in \ partial {\ mathcal {d}} $相对于概率空间$ [ω，\ mathcal {f}，{\ mathrm {i \！p}}] $。 GRF具有消失的平均值$ \ MATHBF {E} [\！[\ MATHSCR {J}（x）] \！] = 0 $和一个受调节的协方差$ {\ MathBf {e}}}}} [\！[\！所有$（x，y;ξ）$ for {\ Mathcal {d}} $和/或$（x，y）\ in {\ partial \ Mathcal {d}} $，具有相关长度$ξ$ and $ { \ otimes {\ Mathscr {J}}（x）] \！] =α<\ infty $。梯度$ \ nabla {\ Mathscr {J}（x）} $和Integral $ \ int _ {\ Mathcal {d}} {\ Mathscr {\ Mathscr {J}}（x）dμ（x）dμ（x）$也存在于$ {\ Mathcal {d}}}} $ aft $ {谐波函数和电势可以随机变为$ \ overline {ψ（x）} =ψ（x）+λ{\ mathscr {j}}}（x）$的随机扰动的GRF。从物理上讲，这种情况来自噪音源或质量/电荷密度，嘈杂或随机边界/表面数据的随机波动。并将湍流/随机性引入光滑的流体流，稳态扩散或热流。这导致了对随机扰动的谐波功能以及Riesz和Newtonian潜力的经典定理的随机修改。以及稳定性的估计和界限，以使其波动和时刻的增长和衰减。
Let $ψ:{\mathcal{D}}\rightarrow{\mathbf{R}}$ be a harmonic function such that $Δψ(x)=0$ for all $x\in\mathcal{D}\subset{\mathbf{R}}^{n}$. There are then many well-established classical results:the Dirichlet problem and Poisson formula, Harnack inequality, the Maximum Principle, the Mean Value Property etc. Here, a 'noisy' or random domain is one for which there also exists a classical scalar Gaussian random field (GRF) ${\mathscr{J}(x)}$ defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ with respect to a probability space $[Ω,\mathcal{F},{\mathrm{I\!P}}]$. The GRF has vanishing mean value $\mathbf{E}[\![\mathscr{J}(x)]\!] = 0$ and a regulated covariance ${\mathbf{E}}[\![{\mathscr{J}(x)} \otimes {\mathscr{J}(y)}]\!] = αJ(x,y;ξ)$ for all $(x,y)\in{\mathcal{D}}$ and/or $(x,y)\in{\partial\mathcal{D}}$, with correlation length $ξ$ and ${\mathbf{E}}[\![{\mathscr{J}(x)} \otimes {\mathscr{J}}(x)]\!] = α<\infty$. The gradient $\nabla{\mathscr{J}(x)}$ and integral $\int_{\mathcal{D}}{\mathscr{J}}(x) dμ(x)$ also exist on ${\mathcal{D}}\bigcup\partial\mathcal{D}$. Harmonic functions and potentials can become randomly perturbed GRFs of the form $\overline{ψ(x)}=ψ(x)+λ{\mathscr{J}}(x)$. Physically, this scenario arises from noisy sources or random fluctuations in mass/charge density, noisy or random boundary/surface data; and introducing turbulence/randomness into smooth fluid flows, steady state diffusions or heat flow. This leads to stochastic modifications of classical theorems for randomly perturbed harmonic functions and Riesz and Newtonian potentials; and to stability estimates and bounds for the growth and decay of their volatility and moments.
