arXiv 1206.70032012-06-29math.PR

空间维$k\geq 1$中非线性随机heat equations系统的命中概率

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $k\geq 1$

Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart

摘要：

考虑空间维上的$d$非线性随机热方程组，其解是一个$\R^d$值随机场$u= \{u(t\,,x),\, (t,x) \in \R_+ \times \R^k\}$ . $d$维驱动噪声在时间上是白色的，并且具有定义为具有指数$β$的Riesz核的空间均匀协方差，其中$0<β<(2 \wedge k)$。非线性既表现为加性漂移项，又表现为噪声的乘数。使用Malliavin演算的技术，我们建立了一个上界的两点密度，相对于勒贝格措施，$\R^{2d}$值的随机向量$(u(s,y),u(t,x))$，特别是，量化这个密度如何退化为$(s,y) \to (t,x)$。从这个结果，我们推导出一个下界的过程$u$的命中概率，在牛顿容量。我们还建立了一个上界的命中概率的过程中的Hausdorff措施。这些估计值可以显示点在$d > \frac{4+2k}{2-β}$时为极坐标，在$d < \frac{4+2k}{2-β}$时为非极坐标。%（如果$\frac{4+2k}{2-β}$是整数，则情况$d=\frac{4+2k}{2-β}$是开放的）。在第一种情况下，我们还表明，Hausdorff维数的范围的过程是$\frac{4+2k}{2-β}$ a.s.

英文摘要：

We consider a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$, whose solution is an $\R^d$-valued random field $u= \{u(t\,,x),\, (t,x) \in \R_+ \times \R^k\}$. The $d$-dimensional driving noise is white in time and with a spatially homogeneous covariance defined as a Riesz kernel with exponent $β$, where $0<β<(2 \wedge k)$. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish an upper bound on the two-point density, with respect to Lebesgue measure, of the $\R^{2d}$-valued random vector $(u(s,y),u(t,x))$, that, in particular, quantifies how this density degenerates as $(s,y) \to (t,x)$. From this result, we deduce a lower bound on hitting probabilities of the process $u$, in terms of Newtonian capacity. We also establish an upper bound on hitting probabilities of the process in terms of Hausdorff measure. These estimates make it possible to show that points are polar when $d > \frac{4+2k}{2-β}$ and are not polar when $d < \frac{4+2k}{2-β}$. %(if $\frac{4+2k}{2-β}$ is an integer, then the case $d=\frac{4+2k}{2-β}$ is open). In the first case, we also show that the Hausdorff dimension of the range of the process is $\frac{4+2k}{2-β}$ a.s.

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摘要：

英文摘要：

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