We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
If $(T_t)$ is a semigroup of Markov operators on an $L^1$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t \to \infty$. In this article we generalise and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a…
In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R} ^d$ under the assumption that its Lévy--Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.
In this paper we study the chaotic behaviour of the heat semigroup generated by the Dunkl-Laplacian on weighted $ L^p$ spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces $ L^p(\R^n, (φ_{iρ}(x))^2 dx) $ where $ φ_{iρ} $ is the Euclidean spherical function. The behaviour is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.
The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as $t \to \infty$. (ii) The more recent phenomenon that positivity itself sometimes …
An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) on a uniformly convex space X is holomorphic iff limsup_{t \downarrow 0} ||T(t)-Id|| < 2. We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent θ in (0,1). As application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to R-secto…
Let $Δ$ be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of $Δ$ with a multiple of the identity on $L^p$ spaces.
This paper considers discrete and continuous semigroups of (weighted) composition operators on the Fock space. For discrete semigroups consisting of powers of a single operator, the asymptotic behaviour of the semigroups is analysed. For continuous semigroups and groups, a full classification of possible semigroups is given, and the generator is calculated.
Similar to the theory of finite Markov chains it is shown that in a Banach space $X$ ordered by a closed cone $K$ with nonempty interior int($K$) a power bounded positive operator $A$ with compact power such that its trajectories for positive vectors eventually flow into int($K$), defines a "limit distribution", i.e. its adjoint operator has a unique fixed point in the dual cone. Moreover, the sequence (A^n) converges with respect to the strong operator topology and for each functional $f\in X'$ the sequence $((A^*)^n(f))$ converges with respect to the weak*-topology (Theorem 5). If a positive…
We discuss the behaviour at infinity of $n$-times integrated semigroups with nonquasianalytic growth and invertible generator. The results obtained extend in this setting a theorem of O. El Mennaoui on stability of bounded once integrated semigroups, and (partially) a theorem of Q. P. V$\tilde{\rm u}$ on stability of $C_0$-semigroups.
We study continuity and Hölder continuity of $t\mapsto P_tf$, where $P_t$ is a generalized Mehler semigroup in $C_b(X)$, the space of the continuous and bounded functions from a Banach space $X$ to $R$, and $f\in C_b(X)$. The generators $L$ of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators $L$ to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups,…
Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.
We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $\\mathrm{L}^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.
The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson ($2018$) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We esta…
With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.
We consider differential-algebraic equations in infinite dimensional state spaces and study, under which conditions we can associate a $C_{0}$-semigroup with such equations. We determine the right space of initial values and characterise the existence of a $C_{0}$-semigroup in the case of operator pencils with polynomially bounded resolvents.
The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given $C_0$-semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept - even for $C_0$-semigroups on Banach spaces - we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we …
We consider positive operator semigroups on ordered Banach spac\-es and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups which are asymptotically dominated by $\mathcal{T}$. Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time…
The main subject in this paper are degenerate $C$-ultradistribution semigroups in barreled sequentially complete locally convex spaces. Here, the regularizing operator $C$ is not necessarily injective and the infinitesimal generator is multivalued linear operator. We also consider exponential degenerate $C$-ultradistribution semigroups.
We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of $p$-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.
The main purpose of this paper is to investigate degenerate $C$-distribution semigroups in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate $C$-distribution semigroup is a multivalued linear operator and the regularizing operator $C$ is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate $C$-distribution semigroups.
We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in \{1,…
Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The…
In this paper we extend the Lumer-Phillips theorem to the context of two--parameter C_0-semigroup of contractions. That is, we characterize the infinitesimal generators of two--parameter C_0-semigroups of contractions. Conditions on the behavior of the resolvent of operators, which are necessary and sufficient for the pair of operators to be the infinitesimal generator of a C_0-semigroup of contractions are given.
The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. Another tool is a Drazin-like inverse operator $B$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This oper…
In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra $\mathcal{T}^*$ is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between $\mathcal{T}$ and the structure of weak Hopf algebra.
We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may…
We identify the semigroups consisting of bounded composition operators on the Hardy spaces $H^p(\U)$ of the upper half-plane. We show that any such semigroup is strongly continuous on $H^p(\U)$ but not uniformly continuous and we identify the infinitesimal generator.
We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak*-closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification.
This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral po…
We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 853-929, 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal est…
We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_\infty$ operator norm of the semigroup. As an application we prove a new $L_\infty$-bound for the torsion function of a Euclidean domain that is close to optimal.
Let $P(\partial/\partial x)$ be an $m\times n$ matrix whose entries are PDO on $\bbR^n$ with constant coefficients, and let $\calS(\bbR^n)$ be the space of infinitely differentiable rapidly decreasing functions on $\bbR^n$. It is proved that $P(\partial/\partial x)|_{(\calS(\bbR^n))^m}$ is the infinitesimal generator of a $(C_0)$-semigroup $(S_t)_{t\ge0}\subset L((\calS(\bbR^n))^m)$ if and only if $P(\partial/\partial x)$ satisfies the Petrovski\uıcorrectness condition. Moreover, if it is the case, then $(S_t)_{t\ge0}$ is an exponential semigroup whose characteristic exponent is equal to the s…
Consider a $C_0$-semigroup $(e^{tA})_{t \ge 0}$ on a function space or, more generally, on a Banach lattice $E$. We prove a sufficient criterion for the operators $e^{tA}$ to be positive for all sufficiently large times $t$, while the semigroup itself might not be positive. This complements recently established criteria for the individual orbits of the semigroup to become eventually positive for all positive initial values. We apply our main result to study the qualitative behaviour of the solutions to various partial differential equations.
We investigate the transition semigroup of the solution to a stochastic evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is the generator of a $C_0$-semigroup $S$ on a separable real Banach space $E$ and $W_H$ is cylindrical white noise with values in a real Hilbert space $H$ which is continuously embedded in $E$. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of $S$ in $H$. In particular we investigate the interplay between analyticity of the transition semigr…
