Semigroups of Kernel Operators

Semigroups of Kernel Operators Moritz Gerlach Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultätfür Mathematik und Wirtschaftswissenschaften der UniversitätUlm. Vorgelegt von Moritz Gerlach aus Limburg an der Lahn im Jahr 2014. Tag der Prüfung: 28. Mai 2014 Gutachter: Prof. Dr. Wolfgang Arendt Prof. Dr. Günther Palm Prof. Dr. Rainer Nagel Amtierender Dekan: Prof. Dr. Dieter Rautenbach Contents Introduction 1 1 Preliminaries 7 2 Weakly continuous operators on the space of measures 13 2.1 Norming dual pairs . 14 2.2 The lattice of transition kernels . 15 2.3 The sublattice of weakly continuous operators . 20 3 Mean ergodic theorems on norming dual pairs 25 3.1 Average schemes . 27 3.2 An ergodic theorem on norming dual pairs . 31 3.3 An ergodic theorem on the space of measures . 36 3.4 Counterexamples . 43 4 Kernel and Harris operators 47 4.1 Definition and basic properties . 48 4.2 Characterization by star{order continuity . 53 4.3 Triviality of the peripheral point spectrum . 61 5 Stability of semigroups 71 5.1 Stability of semigroups of Harris operators . 72 5.2 Doob's theorem . 79 5.3 A Tauberian theorem for strong Feller semigroups . 81 Appendix 89 Bibliography 97 Index 103 Introduction Semigroups of operators describe the evolution of linear systems and processes. As an illustrating example, let us consider the heat equation 8 @ @2 < u(t; x) = u(t; x) t ≥ 0, x 2 R @t @2x : u(0; x) = u0(x) x 2 R: A solution u(t; x) of this equation describes the heat at a specific time t ≥ 0 at position x 2 R. Hence, this equation models the diffusion of heat on the real line, starting with initial distribution u0 at time t = 0. It is well-known that this equation has a unique bounded solution for every continuous and bounded initial value u0. This solution is given by 1 Z 1 (x − y)2 (T (t)u0)(x) = p exp − u0(y) dy: 4πt −∞ 4t The so defined operators T (t), which map an initial value u0 to the corresponding solution at time t ≥ 0, are linear and satisfy the semigroup law T (t)T (s) = T (t+s) for all t; s ≥ 0. The latter means that the state of the system at time t + s equals the state at time t when the system is considered to be initialized with T (s)u0 instead of u0. The family of operators (T (t))t≥0 is called the Gaussian Semigroup. One important property of this semigroup is that each operator T (t) maps positive functions to positive ones. That is to say that if the heat is everywhere larger or equal than zero, it remains so for all time. Such semigroups are called positive. Intuitively, this is a typical property of semigroups describing some kind of diffusion. Moreover, the operators of the Gaussian Semigroup have an additional special property. Each operator T (t) is a kernel operator, i.e. it is of the form Z T (t)f = ht( · ; y)f(y) dy 1 Introduction for some function ht. In general, one cannot describe the operators of a semigroup by an explicit formula as in the example above. If, for instance, one is interested in the diffusion of heat on a plate, it is hard to impossible to calculate the single operators T (t) in one way or another unless a very specific geometry of the plate is given. Instead, there are often abstract arguments at hand to ensure that a semigroup consists of kernel operators although one does not know the kernel functions ht explicitly. For example, one can show that basically all differential operators generate a semigroup of kernel operators, see [12]. In this thesis we study several aspects of semigroups of kernel operators. First, we provide abstract characterizations of kernel operators which we use in particular to identify kernel operators on spaces of measures. Secondly, we study the asymptotic behavior of such semigroups, namely convergence of the semigroups in mean and convergence of the semigroups itself. It is well-known that kernel operators can be described in the following way. A regular and order continuous operator is a kernel operator if and only if it belongs to the band generated by the finite rank operators. If the latter is used as definition, this generalizes the notion of a kernel operator to arbitrary Banach lattices. On the other hand, kernel operators can be characterized by a continuity condition. Bukhvalov showed in [16] that an operator on Lp is a kernel operator if and only if it maps order bounded norm convergent sequences to almost everywhere convergent ones. We provide a generalization of Bukhvalov's theorem for operators on Banach lattices and describe abstract kernel operators in terms of the so-called star{order continuity. This characterization is originally due to Grobler and van Eldik [35]. We give an alternative proof under different conditions on the spaces. The analysis of the asymptotic behavior of operator semigroups has a long history. Of particular interest is the characterization of stability, i.e. convergence to an equilibrium, by properties of the semigroup or its generator. Most results obtained in this area establish a connection between the asymptotic behavior and properties of the spectrum of the generator. Our approach is completely different and based on a technique developed by Greiner. He showed in [33] that a positive, contractive and irreducible strongly continuous semigroup on Lp converges to a projection onto its fixed space under the following two conditions: The peripheral point spectrum of its generator is f0g { in particular, the semigroup is assumed to admit a non-zero fixed point { and the semigroup contains two operators which are not disjoint. Due to a theorem of Axmann [14], the second condition holds automatically if the operators of the semigroup dominate kernel operators. Such operators are called Harris operators. Greiner proved in [33] that the first condition is satisfied for semigroups of kernel operators. Combining these result, Greiner 2 showed that a positive, contractive and irreducible strongly continuous semigroup on Lp converges to a projection onto its fixed space if it consists of kernel operators and admits a non-zero fixed point. This is an extraordinary theorem as it does not follow from one of the known spectral descriptions of stability. We generalize this theorem in several respects. First, we consider time-discrete semigroups in addition to strongly continuous ones and this not only on Lp but also on arbitrary Banach lattices. Secondly, we show that the peripheral point spectrum of the generator of a semigroup of Harris operators is trivial. As a consequence, we obtain a stability theorem for semigroups whose operators are merely assumed to dominate kernel operators. Moreover, we present applications of Greiner's technique to a different but related class of positive semigroups. In the study of Markov processes one is interested in certain operators on spaces of measures which describe the evolution of probability distributions and are in general not kernel operators. In such a process, a state x is not transported deterministically to a certain state y. Instead, the new state is chosen by chance. The measure k(x; · ), the so-called transition kernel, is the distribution of states one is moved to from state x in one time step. In general, a Markov process carries a probability measure µ over to Z T µ = k(x; · ) dµ(x): Ω This defines a linear operator T on the space of measures which is also called kernel operator in the probabilistic literature. To distinguish an operator of this form from kernel operators as mentioned before, we call them weakly continuous operators as they are precisely those operators which are continuous with respect to the weak topology induced by the bounded measurable functions. Here, we study probably for the first time order properties of weakly continuous operators. In doing so, we determine differences and similarities of weakly continuous operators and kernel operators. It turns out that the space of weakly continuous operators is a countably order complete sublattice of the bounded operators on the space of measures. However, in contrast to kernel operators which form a band, weakly continuous operators are not even an ideal. While it is nearly impossible to calculate the infimum of arbitrary regular operators in a practical way, the situation improves for weakly continuous operators. As for kernel operators, the computation of lattice operations of weakly continuous operators reduces to the corresponding lattice operations of their transition kernels. The infimum T1 ^T2 of two weakly continuous operators is again weakly continuous and given by the infimum of their transition kernels k1 and k2 defined pointwise as the infimum of the measures k1(x; · ) and k2(x; · ). 3 Introduction Therefore, it is now possible to impose conditions on the transition kernels to ensure that the corresponding operators are not disjoint. For example, it suffices to assume the measures k(x; · ) and k(y; · ) to be mutually absolutely continuous, which is a form of irreducibility. This enables us to apply Greiner's theorem in this context. By this method, we obtain a purely analytic proof of a version of Doob's theorem on stability of Markov processes and uniqueness of invariant measures. The same arguments also yield a version for time-discrete semigroups, i.e. for Markov chains. A necessary condition for stability is convergence of the semigroup in mean, a property called mean ergodicity. This is in general strictly weaker than convergence of the semigroup itself and can typically be checked easily by the well-known mean ergodic theorem.

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