AI 中文总结
该研究针对加权图，通过非线性抛物型方程建立了无穷远处随机完备性的刻画，推导了对应质量平衡关系，还验证了特殊情形下与经典随机完备性的关联。
AI 中文摘要
我们证明了加权图上无穷远处随机完备性的一种非线性抛物型刻画。对于过滤方程 $(\\_t + \Delta \Phi)u =0$，其中 $\Delta$ 是非负形式图拉普拉斯算子，$\Phi u = \phi \circ u$，且 $\phi:\mathbb{R}\to\mathbb{R}$ 为非常数、连续且递增的函数，无穷远处随机完备性等价于对每个有界初值，存在有界逐点解的唯一性。当 $\Phi=\id$ 时，这可恢复无穷远处随机完备性的经典热方程刻画，且当杀伤项平凡即 $\kappa=0$ 时，也可恢复随机完备性。若无穷远处随机完备性不成立，则每个有界初值都有无穷多个过滤方程的有界逐点解。可允许的非线性项包括带符号的多孔介质和快速扩散幂函数 $\phi(s)=s|s|^{m-1}$（对所有 $m>0$）及其他多种形式。无穷远处随机完备性还可通过广义质量平衡刻画：正逐点解在时刻 $t$ 的总质量，加上杀伤项 $\kappa$ 耗散的质量 $\int_0^t\sum_{x}\kappa(x)\phi(u(s,x)) \dd s$，等于初始质量。该平衡对有限测度图上的每个有界正解成立，且对任意测度图上满足极限条件 $\limsup_{r\to0^+}\phi(r)/r<\infty$ 的有界有限质量初值也成立，还可推广到每个正时间区间上有界的 $\ell^1$ 中正逐点解。当杀伤项平凡时，无穷远处随机完备性简化为随机完备性，广义平衡简化为质量守恒。
英文摘要
We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + ΔΦ)u =0 \] where $Δ$ is the non-negative formal graph Laplacian and $Φu =ϕ\circ u$ with $ϕ\colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(Φ=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(κ=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $ϕ(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}κ(x)ϕ(u(s,x)) \dd s$ dissipated by the killing term $κ$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}ϕ(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.
