论文标题
一类抛物线随机PDES的分数Sobolev空间的功率变化
Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs
论文作者
论文摘要
我们考虑有界域上的一类抛物面随机PDE $ d \ subseteq \ mathbb {r}^d $，其中包括随机热方程，但带有laplacian的分数幂$γ$。将解决方案视为具有分数Sobolev Spaces $ H_R $的值的过程，并以$ R <γ-D/2 $的形式研究其功率变化，沿时间轴的常规分区中的$ H_R $。由于网格尺寸趋于零，因此我们发现$ r = -d/2 $的相转换：当$ r <-d/2 $和$ p $ p $ p $ p $ p $ p =2γ/（γ-d/2-r）的非平凡$ p $ p $ ph订单变化时，解决方案具有非平凡的二次变化。更普遍的是，任何顺序的适当归一化功率变化在第一种情况下满足了大数字的真实定律，而在第二种情况下，否定限制限制定理。当$ r <-d/2 $时，二次变化将通过涉及频谱Zeta函数的表达式给出，当$ d = 1 $ d $和$ d $是间隔时，该表达式会降低到Riemann Zeta函数。
We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $γ$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < γ- d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2γ/(γ-d/2-r)>2$ when $r>-d/2$. More generally, suitably normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d/2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and $D$ is an interval.
