Hausdorff dimension of the zero level set of alpha-Stochastic heat equation

Posted on March 23, 2020 by thomaskojar

Consider the equation

dz+\nu A^{\alpha}z=C^{1/2}dW, z(0)=0

where for x:=\sum_{k\in Z^{2}\setminus \{(0,0)\}} x_{k}e_{k}, with \sum |x_{k}|^{2}<\infty,

we set A^{\alpha}(x):=\sum_{k\in Z^{2}_{*}}|k|^{\alpha} x_{k}e_{k} for \alpha>1

and C(x):=\sum_{k\in Z^{2}_{*}}\sigma_{k}^{2}x_{k}e_{k} such that for \delta\in (1-\alpha,2-\alpha) and \frac{c_{2}}{|k|^{\delta}}\leq |\sigma_{k}|\leq \frac{c_{3}}{|k|^{\delta}}.

RO: The dimension of the zero level set is dim(\{x\in T_{2}: z_{t}(x)=0\})=3-\alpha-\delta.

Reference: Hausdorff dimension of the level sets of some stochastic PDEs from fluid dynamics

This entry was posted in Zero level sets and Stochastic PDEs. Bookmark the permalink.

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