论文标题
混合拓扑和马尔可夫流程的无限描述中的操作员半群
Operator semigroups in the mixed topology and the infinitesimal description of Markov processes
论文作者
论文摘要
我们定义一类不一定是线性$ c_0 $ -semigroups $（p_t）_ {{t \ geq0} $上的$ c_b（e）$（更一般而言，在$c_κ（e）上：= \frac1k1κC_B（e）$，对于某些有界的功能$κ$κ$κ$κ$，连续的连续限制，是连续的，是连续的序列，是连续的，是连续的， $τ_1^{\ Mathscr M} $用于一类大类拓扑状态空间$ e $。如果这些半群是线性的，则在局部凸空间上的操作员半群的经典理论以及双连续的半群的理论适用于它们。特别是，它们是由其发电机$（l，d（l））$的无限生成的，因此可以通过欧拉公式从其强派中的Euler公式重建，以$（c_b（e），τ_1^{\ Mathscr m}）$（c_b（e））$。在线性情况下，我们将这种$（p_t）_ {t \ geq0} $描述为通过满足某些紧密度属性的测量内核给出的积分运算符。结果，马尔可夫流程的过渡半群为$ c_0 $ -semigroups $（c_b（e），τ_1^{\ mathscr m}）$，如果它们留下$ c_b（e）$不变，并且它们在空间和时间上共同连续较弱。因此，它们可以从零以零的强导数中重建，因此具有完全无限的描述。此外，我们介绍了Markov核心运算符$（L_0，D（L_0））$的概念，用于上述发电机$（l，d（l））$，并证明了fokker-planck-kolmogorov方程的独特性，对应于$（l_0，d（l_0）$）$（l_0，d（l_0）$）$（l_0，d a marks $ is）$（l_ is dirac $ is）（l_ is d dirac（l_）（l_ $（l，d（l））$的运营商。如果每个$ p_t $仅是凸，我们证明$（p_t）_ {t \ geq0} $为其相关（非线性）Infinitessimal Generator给出的Cauchy问题提供了粘度解决方案。我们还表明，最佳控制问题的价值函数在有限和无限维度上都是$（C_κ（E），τ_κ^{\ Mathscr m}）$的凸的特定实例。
We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_κ(E):=\frac1κC_b(E)$, for some bounded function $κ$, which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology $τ_1^{\mathscr M}$ for a large class of topological state spaces $E$. If these semigroups are linear, classical theory of operator semigroups on locally convex spaces as well as the theory of bicontinuous semigroups apply to them. In particular, they are infinitesimally generated by their generator $(L,D(L))$ and thus reconstructable through an Euler formula from their strong derivative at zero in $(C_b(E),τ_1^{\mathscr M})$. In the linear case, we characterize such $(P_t)_{t\geq0}$ as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are $C_0$-semigroups on $(C_b(E),τ_1^{\mathscr M})$, if they leave $C_b(E)$ invariant and they are jointly weakly continuous in space and time. Hence, they can be reconstructed from their strong derivative at zero and thus have a fully infinitesimal description. Furthermore, we introduce the notion of a Markov core operator $(L_0,D(L_0))$ for the above generators $(L,D(L))$ and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to $(L_0,D(L_0))$ for all Dirac initial conditions implies that $(L_0,D(L_0))$ is a Markov core operator for $(L,D(L))$. If each $P_t$ is merely convex, we prove that $(P_t)_{t\geq0}$ gives rise to viscosity solutions to the Cauchy problem given by its associated (nonlinear) infinitesimal generator. We also show that value functions of optimal control problems, both, in finite and infinite dimensions are particular instances of convex $C_0$-semigroups on $(C_κ(E),τ_κ^{\mathscr M})$.
