论文标题
在边界上，当地时间的超棕色运动
On the boundary local time measure of super-Brownian motion
论文作者
论文摘要
如果$ l^x $是当地时间的总职业时间，$ d $二维超级棕色运动，$ x $，$ d = 2 $ and $ d = 3 $，我们构建了一个随机度量$ \ Mathcal {l} $，称为边界本地时间量度，作为$ l^e^e^e^e^e^eftty a $ cundy of unifty of und^e^eftty of unftty of unfty of unftty of cund of confty of cund of cund of confty a iffty of cund ftty aftty unifty。 \ cite {mp17}并进一步表明，$ \ nathcal {l} $的支持等于$ x $，$ \ partial \ mathcal {r} $的拓扑边界。后者的结果使用了第二次结构的当地时间$ \ widetilde {\ Mathcal {l}} $根据退出措施给出的，我们证明$ \ widetilde {\ Mathcal {l}} = C \ Mathcal {c \ Mathcal {l} $ A.S.对于某些常数$ c> 0 $。在负尺寸的贝塞尔过程方面，我们对$ \ Mathcal {l} $进行了合理明确的第一和第二mmoment措施，并将其与能量方法一起使用，以更直接地证明$ \ partial \ Mathcal \ Mathcal {r} $ in \ cite in \ cite in \ cite {hmp18}的hausdorff尺寸的下限。构造需要在\ cite {mp17}中对$ l^2 $上限进行改进，并\ cite {hmp18}到精确的$ l^2 $渐近学。这些方法还将\ cite {mp17}中的$ l^x $的左尾范围优化到精确的渐近学。我们推测$ \ partial \ Mathcal {r} $的Minkowski内容等于边界本地时间$ \ MATHCAL {L} $的总质量，达到一定常数。
If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-λL^x} dx$ as $λ\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mathcal{L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal{L}$ in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial\mathcal{R}$ in \cite{HMP18}. The construction requires a refinement of the $L^2$ upper bounds in \cite{MP17} and \cite{HMP18} to exact $L^2$ asymptotics. The methods also refine the left tail bounds for $L^x$ in \cite{MP17} to exact asymptotics. We conjecture that the Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant.
