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Operator semigroups in the mixed topology and the infinitesimal description of Markov processes

Ben Goldys, Max Nendel, Michael Röckner

Abstract

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 $(C_κ(E),τ_κ^{\mathscr M})$.

Operator semigroups in the mixed topology and the infinitesimal description of Markov processes

Abstract

We define a class of not necessarily linear -semigroups on (more generally, on , for some bounded function , which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology for a large class of topological state spaces . 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 and thus reconstructable through an Euler formula from their strong derivative at zero in . In the linear case, we characterize such as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are -semigroups on , if they leave 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 for the above generators and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to for all Dirac initial conditions implies that is a Markov core operator for . If each is merely convex, we prove that 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 -semigroups on .
Paper Structure (19 sections, 21 theorems, 302 equations)

This paper contains 19 sections, 21 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and setup
  3. Strongly continuous semigroups on spaces of continuous functions with mixed topology
  4. Examples for linear $C_0$-semigroups on $\left( C_\kappa(E),\tau_\kappa^{\mathscr M}\right)$
  5. Transition semigroups of solutions to SDEs on Hilbert spaces of locally monotone type
  6. Transition semigroups of mild solutions to SDEs on Hilbert spaces with bounded weak topology
  7. Generalized Mehler semigroups on Banach spaces with norm topology
  8. Generalized Mehler semigroups on Banach spaces with bounded weak topology
  9. Strong and Weak infinitesimal generators
  10. Generators and (Markov) core operators
  11. Applications to SDEs on ${\mathbb R}^d$
  12. Applications to generalized Mehler semigroups (or OU-processes with Levy noise) on Hilbert spaces
  13. Applications to SDEs of locally monotone type on Hilbert spaces
  14. Convex $C_0$-semigroups on $(C_\kappa(E), \tau_\kappa^{\mathscr M})$
  15. Examples: value functions of optimal control problems
  16. ...and 4 more sections

Key Result

Theorem 3.4

Let $P=(P_t)_{t\geq 0}$ be a semigroup of linear operators on $C_\kappa(E)$. Then, the following properties are equivalent:

Theorems & Definitions (55)

  • Remark 2.2
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Definition 3.6
  • Proposition 3.7
  • ...and 45 more