Moment densities of super Brownian motion, and a Harnack estimate for a class of \(X\)-harmonic functions
Let \(D\) be a bounded and smooth domain in \({\mathbb R^d},d \geqslant 2\), and let \(\Delta = \sum\nolimits_{i = 1}^d {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \). Let \({g_D}(x,y)\) and \({k_D}(x,y)\) be the Green and the Poisson kernel, respectively, of \(D\) for the operator \(\Delta\). That is, for \(x \in D\), \({g_D}(x, \cdot ):D - \{ x\} \to [0,\infty )\) is the unique \({C^2}(D - \{ x\} ) \cap {C^1}(D - \{ x\} )\) solution of \(\Delta u = - 2{\delta _x}\) on \(D\) with \(u=0\) on \(\partial D\). Then, for \(z \in \partial D\), \({k_D}(x,y) \in [0,\infty )\) is the derivative of \({g_D}(x,z)\), considered as a function of \(z\), in the direction of the inward conormal to \(\partial D\) at the point \(y \in \partial D\). For fixed \(y \in \partial D\), the function \(u = {k_D}( \cdot ,y)\) satisfies \(\Delta u = 0\) on \(D\) and \(u=0\) on \(\partial D - \{ y\}\). Let \(n \geqslant 1\), and let \({z_1},\dots{},{z_n}\) be distinct points on \(\partial D\). Define a function \({\rho _D}(x,{z_1},\dots{},{z_n})\), \(x \in D\), recursively, by defining a family of functions \({\rho _D}(x,{z_A})\), where \(A = \{ {i_1},\dots{},{i_k}\} \subset \{ 1,\dots{},n\} \) and \({z_A} = ({z_{{i_1}}},\dots{},{z_{{i_k}}})\). Set \({\rho _D}(x,{z_A}) = {k_D}(x,{z_i})\) if \(A = \{ i\} \) and, for \(\left| A \right| > 1\), set \({\rho _D}(x,{z_A}) = \sum\limits_{B \subset A,B \neq \emptyset ,A} {\int\limits_D {{g_D}(x,y){\rho _D}(y,{z_B}){\rho _D}(y,{z_{A - B}})dy} } \). Let \({\rho _D}(x,{z_1},\dots{},{z_n}) = \rho (x,{z_A})\) for \(A=\{1,\dots,n\}\). The main estimate of the first section says that, for any compact \(C \subset D\), there exists a \(\lambda > 0\) depending only on \(C\) and \(D\) such that \(\rho (x,{z_1},\dots{},{z_n}) \leqslant {\lambda ^n}\rho ({x_0},{z_1},\dots{},{z_n})\) for all \(x\) and \({x_0} \in C\) and for all distinct \({z_1},\dots{},{z_n} \in \partial {D^n}\). This result is applied to a class of \(X\)-harmonic functions of super-Brownian motion.
scientific article; zbMATH DE number 6369356
scientific article