论文标题
随机完整性和$ l^1 $ -liouville物业，用于二阶椭圆操作员
Stochastic completeness and $L^1$-Liouville property for second-order elliptic operators
论文作者
论文摘要
令$ p $为线性，二阶，椭圆运算符，其实际系数在非划分的riemannian歧管上定义，并满足$ p1 = 0 $ in $ m $。进一步假设$ p $承认$ m $中的最小绿色功能。我们证明，在$ m $上定义的平稳正函数$ρ$，因此$ m $与操作员$p_ρ：=ρ\，p $是随机不完整的，也就是说 \ [ \ int_ {m} k_ {p_ρ}^{m}（x，y，y，t）\ {\ rm d} y <1 <1 \ qquad \ qquad \ forall（x，t）\ in m \ times（0，\ infty），，\ infty）， \] 其中$ k_ {p_ρ}^{m} $表示与$p_ρ$相关的最小正加热内核。 此外，相对于$p_ρ$，$ m $是$ l^1 $ -liouville，并且仅当$ m $是$ l^1 $ -liouville相对于$ p $。此外，我们研究了随机完整性与两个二阶椭圆操作员偏斜产物的$ l^1 $ -liouville属性之间的相互作用。
Let $P$ be a linear, second-order, elliptic operator with real coefficients defined on a noncompact Riemannian manifold $M$ and satisfies $P1=0$ in $M$. Assume further that $P$ admits a minimal positive Green function in $M$. We prove that there exists a smooth positive function $ρ$ defined on $M$ such that $M$ is stochastically incomplete with respect to the operator $ P_ρ := ρ\, P $, that is, \[ \int_{M} k_{P_ρ}^{M}(x, y, t) \ {\rm d}y < 1 \qquad \forall (x, t) \in M \times (0, \infty), \] where $k_{P_ρ}^{M}$ denotes the minimal positive heat kernel associated with $P_ρ$. Moreover, $M$ is $L^1$-Liouville with respect to $P_ρ$ if and only if $M$ is $L^1$-Liouville with respect to $P$. In addition, we study the interplay between stochastic completeness and the $L^1$-Liouville property of the skew product of two second-order elliptic operators.
