We prove that conservation of probability for the free heat semigroup on a Riemannian manifold M (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on M of the form $u_t=\mathrm{\Delta }\varphi (u)$, ϕ being an arbitrary concave, increasing, positive function, regular outside the origin and with $\varphi (0)=0$. Either property is also equivalent to nonexistence of nonnegative, nontrivial, bounded solutions to the elliptic equation $\mathrm{\Delta }W={\varphi }^{-1}(W)$, with ϕ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds are given, these being the first results on such issues.

我们证明，黎曼流形M上的自由热半群的概率守恒（即随机完备性），因此是线性性质，等于形式为M的快速扩散型非线性演化方程的正有界解的唯一性${ü}_{Ť}=\mathrm{\Delta }\varphi （ü）$，ϕ是一个任意的凹面，递增，正函数，在原点外规则并具有$\varphi （0）=0$。任一性质也等同于椭圆方程的非负，非平凡有界解的不存在$\mathrm{\Delta }\mathrm{w\; ^}={\varphi }^{-\mathrm{1个}}（\mathrm{w\; ^}）$，与ϕ相同。结果，给出了关于流形上快速扩散型方程的有界解的唯一性或非唯一性的明确标准，这些是关于此类问题的第一个结果。