论文标题
通过Lévy系数的固定扩散方程的溶液的可集成性和近似性
Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient
论文作者
论文摘要
我们使用（转化的）Lévy随机场给出的系数研究了固定扩散方程。 Lévy随机场是通过使用Matérn类的内核来平滑Lévy噪声场来构建的。我们表明，Lévy噪声自然会在Minlos的广义随机场理论中扩展高斯白噪声。得出了Lévy噪声的分布路径空间以及平滑量的结果，以确保此类分布成为连续路径。鉴于此，我们得出了对路径的存在和解决随机边界值问题（BVP）的测量性的结果。对于BVP的解决方案，我们证明在适当的生长条件下，在噪声场的levy量度上存在力矩（在$ h^1 $ norm中）。最后，引入了平滑的Lévy噪声场的内核扩展，并以明确的速率证明了与近似随机系数相关的解决方案的$ l^n $（$ n \ geq 1 $）。
We investigate the stationary diffusion equation with a coefficient given by a (transformed) Lévy random field. Lévy random fields are constructed by smoothing Lévy noise fields with kernels from the Matérn class. We show that Lévy noise naturally extends Gaussian white noise within Minlos' theory of generalized random fields. Results on the distributional path spaces of Lévy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the $H^1$-norm) under adequate growth conditions on the Lévy measure of the noise field. Finally, a kernel expansion of the smoothed Lévy noise fields is introduced and convergence in $L^n$ ($n\geq 1$) of the solutions associated with the approximate random coefficients is proven with an explicit rate.
