We use the recently-developed multiparameter theory of additive Levy processes to establish novel connections between an arbitrary Levy process $X$ in $\mathbf{R}^d$, and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Levy processes. First, we compute the Hausdorff dimension of the image $X(G)$ of a nonrandom linear Borel set $G\subset \mathbf{R}_+$, where $X$ is an arbitrary Levy process in $\mathbf{R}^d$. Our work completes the various earlier efforts of Taylor [Proc. Cambridge Phil. Soc. 49 (1953) 31-39], McKean [Duke Math. J. 22 (1955) 229-234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370-375, J. Math. Mech. 10 (1961) 493-516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53-73], Pruitt [J. Math. Mech. 19 (1969) 371-378], Pruitt and Taylor [Z. Wahrsch. Verw. Gebiete 12 (1969) 267-289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90-102, J. London Math. Soc. (2) 17 (1978) 567-576, Probab. Theory Related Fields 112 (1998) 1-11], Hendricks [Ann. Math. Stat. 43 (1972) 690-694, Ann. Probab. 1 (1973) 849-853], Kahane [Publ. Math. Orsay (83-02) (1983) 74-105, Recent Progress in Fourier Analysis (1985b) 65-121], Becker-Kern, Meerschaert and Scheffler [Monatsh. Math. 14 (2003) 91-101] and Khoshnevisan, Xiao and Zhong [Ann. Probab. 31 (2003a) 1097-1141], where $\dim X(G)$ is computed under various conditions on $G$, $X$ or both.
Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of Statistics and Probability, Michigan State University（密歇根州立大学统计与概率系）
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise
考虑了在d维时空白噪声驱动下的空间维1的d非线性随机热方程组。非线性既表现为加性漂移项，又表现为噪声的乘数。利用Malliavin演算的技巧，我们建立了解u（t，x）的单点密度的上界和下界，以及（u（s，y），u（t，x））的两点密度的高斯型上界.特别地，该估计量化了该密度如何随着（s，y）收敛到（t，x）而退化。由这些结果，我们分别根据Hausdorff测度和牛顿容量，导出了过程{u（t，x）}_{tPDMATHABMM \mathbb{R}_+，x \in [0，1]}的命中概率的上、下界.这些估计使得证明当d> 6时点是极坐标，而当d 6时点不是极坐标成为可能，并对过程t \mapsto u（t，x）和x \mapsto u（t，x）给出类似的结果.<6. We also show that the Hausdorff dimension of the range of the process is 6 when d>最后，我们得到了这些过程的水平集的豪斯多夫维数值.
我们给出了一般的充分条件，这意味着一个多参数过程击中一个给定的集合E的概率的上界和下界的能力E有关的过程。这延伸的结果Khoshnevisan和施[安。27（1999）1135-1159]，其中获得了（N，d）布朗单的命中概率关于（d-2N）牛顿容量的估计，并且容易地应用于广泛的一类高斯过程。使用Malliavin演算，特别是Kohatsu-Higa [Probab. Theory Related Fields 126（2003）421-457]中，我们将这些一般结果应用于具有两个变量的d非线性双曲随机偏微分方程系统的解。我们表明，在标准的假设下的系数，这个解决方案的命中概率是有界的上下常数倍（d-4）牛顿容量。因此，我们刻画了这一过程的极集，并证明了它的值域的豪斯多夫维数为min（d，4）a.s.
