Stability of transition semigroups and applications to parabolic equations
Moritz Gerlach, Jochen Glück, Markus Kunze
备注：
25 pages. This article originally emerged from a subsection of arXiv:1705.01587v1 that was split into a separate paper. The final results in the present paper are, however, stronger and more general than those from the removed subsection of arXiv:1705.01587v1. This is version 4; various changes in the presentation compared to v3
The paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochastic analysis.
The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup.
We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.
This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation. In this paper, infinite-dimensional generalizations of the Euclidean analogue of the Schrödinger equation for anharmonic oscillator are considered in the class of measures. The Cauchy problem for these equations is solved. In particu…
We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation ıs obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
We study polynomial and exponential stability for $C_{0}$-semigroups using the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability …
In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.
Stability of weighted composition strongly continuous semigroups acting on Lebesgue and Sobolev spaces is studied, without the use of spectral conditions on the generator of the semigroup. Applications to the generalized von Foerster - Lasota semigroup and a comparison with hypercylicity conditions are presented.\
In this paper we study the robustness of strong stability of a discrete semigroup on a Hilbert space under bounded finite rank perturbations. As the main result we characterize classes of perturbations preserving the strong stability of the semigroup.
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 prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G. Smolyanov concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space $C_c^{2,α}(\R^n)$. Based on these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article we show that the assumptions are fulfilled for domains with $C^…
We investigate uniform, strong, weak and almost weak stability of multiplication semigroups on Banach space valued $L^p$-spaces. We show that, under certain conditions, these properties can be characterized by analogous ones of the pointwise semigroups. Using techniques from selector theory, we prove a spectral mapping theorem for the point spectra of the pointwise and global semigroups and apply this as a major tool for determining almost weak stability.
Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compact…
In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the imaginary axis and the norm of its resolvent operator is polynomially bounded near these points. We characterize classes of finite rank perturbations preserving the strong stability of the semigroup. In addition, we improve recent results on preservation of polynomial stability of a semigroup under finite rank perturbations of its generator. Theoretic results are il…
In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincaré inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Hölder regularity of parabolic functions for symmetric non-local Dirichlet forms.
In this paper, we give a proof of the Hyers-Ulam stability of the Jensen functional equation $$f(xy)+f(xσ(y))=2f(x),\phantom{+} x,y\in{G},$$ where $G$ is an amenable semigroup and $σ$ is an involution of $G.$
In this paper we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretic results are applied to deriving conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equations.
A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup $(e^{tA})_{t \ge 0}$ on $L_p$ may be determined by the quantity $s(A)$. We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum $σ(A)$.
We show that noncommutative analog of additive functional equation has Hyers-Ulam stability on amenable discrete quantum (semi)groups. This generalizes an old classical result.
In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-sp…
Chernoff approximations to strongly continuous one-parameter semigroups give solutions to a wide class of differential equations. This paper studies the rate of convergence of the Chernoff approximations. We provide simple natural examples for which the convergence is arbitrary fast, is arbitrary slow, and holds in the strong operator topology but does not hold in the norm operator topology. We also prove a general theorem that gives an upper estimate for the speed of decay of the norm of the residual term of the Chernoff approximations. The result is applied to one-dimensional parabolic diffe…
A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent coefficients, which may depend on $x$. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity…
The purpose of this paper is to revisit previous works of the author with J. Sjöstrand (2010--2021) proved in the Hilbert case by considering the Banach case at the light of a paper by Y.~Latushkin and V.~Yurov (2013).
We give a relation between the exponential stability of $ C_{0}- $semigroup $ \textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 0} $ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $ B^{+}(X,X^{*}) $, with $ X $ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $ R(λ,A) $ inside and outside of the left half-plane $ \Reλ\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup T.
An asymptotic stability result for parabolic semilinear problems in $L_2(Ω)$ and interpolation spaces is shown. Some known results about stability in $W^{1,2}(Ω)$ are improved for semilinear parabolic mixed boundary value problems. The approach is based on Amann's power extrapolation scales. In a Hilbert space setting, a better understanding of this approach is provided for operators satisfying Kato's square root problem; as a side result some equivalent characterizations of these operators are obtained.
We present a new and very short proof of the fact that, for positive $C_0$-semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying space completely transparent and also works if the space does not contain the constant functions - a situation in which all earlier proofs become technically quite involved. We also show how the argument can be adapted to yield the same result for semigroups that are only eventually positive rather than positive.
Let T be a C_0-semigroup on X with generator A. We prove that if the abscissa of uniform boundedness of the resolvent A is non-negative then for each a non-decreasing function h:[0,\infty] -> [0,\infty], there are x' in X' and x in X such that integral from 0 to \infty of h(|< x',T(t)x)>| is equal to \infty. If Sp(A) contained a number from iR then such x may be taken in D(A^\infty).
Input-to-state stability (ISS) for systems described by partial differential equations has seen intensified research activity recently, and in particular the class of boundary control systems, for which truly infinite-dimensional effects enter the situation. This note reviews input-to-state stability for parabolic equations with respect to general $L^{p}$-input-norms in the linear case and includes extensions of recent results on semilinear equations.
We stretch the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. For such semigroups, we obtain exponential estimates with the best stability constants. We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type…
In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $C_0$-semigroup. To estimate the decay rate of the Cayley transform, for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations.
In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutio…
Let $S$ be a semigroup and $X$ a Banach space. The functional equation $ϕ(xyz)+ ϕ(x) + ϕ(y) + ϕ(z) = ϕ(xy) + ϕ(yz) + ϕ(xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\to X$ satisfying $\| f(xyz)+f(x) + f(y) + f(z) - f(xy)- f(yz)-f(xz)\| \leq δ$ for some positive real number $δ$ and all $x, y, z \in S$, there is a solution $ϕ: S \to X$ such that $f-ϕ$ is bounded. In this paper, among others, we prove the following results: 1) this functional equation, in general, is not stable on an arbitrary semigroup; 2) this equation is stable on periodic semigroups; 3) this equation is …
Let $U\subset K$ be an open and dense subset of a compact metric space and let $\{Φ_t\}_{t\ge0}$ be a Markov semigroup on the space of bounded Borel measurable functions on $U$ with the strong Feller property. Suppose that for each $x\in\bdu$ there exists a barrier $h\in C(K)$ at $x$ such that $Φ_t(h)\ge h$ for all $t\ge0$. Suppose also that every real-valued $g\in C(K)$ with $Φ_t(g)\ge g$ for all $t\ge0$ and which attains its global maximum at a point inside $U$ is constant. Then for each $f\in C(K)$ there exists the uniform limit $F=\lim_{t\to\infty}Φ_t(f)$. Moreover $F$ is continuous on $K$…
Let $\mathcal{L}= \partial_t- Δ_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $α>0$ define the Parabolic Hermite-Zygmund spaces $$ Λ^α_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+α},\;\: {\rm with }\, k=[α]+1, y>0. \right\}, $$ with the obvious norm. It is shown that these spaces have a pointwise description of Hölder type. The fractional powers $\mathcal{L}^{\pm β}$ are well defined in these spaces and the following regularity propertie…
In this paper we survey results concerning the asymptotic properties of C_0-semigroups on Banach spaces with respect to the weak operator topology. The property "no eigenvalues of the generator on the imaginary axis" is equivalent to weak stability for most time values; a phenomenon called "almost weak stability". Further, sufficient conditions actually implying weak stability are also given. By several examples we explain weak and almost weak stability and illustrate the fundamental difference between them. Many historical and bibliographical remarks position the material in the literature. W…
In this work we construct certain general bundles $<\mathfrak{M},ρ,X>$ and $<\mathfrak{B},η,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},π,X>$. Then we present conditions ensuring the existence of bounded sections $\mathcal{U}\in Γ^{x_{\infty}}(ρ)$ and $\mathcal{P}\in Γ^{x_{\infty}}(η)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$.
Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated to a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equ…
The well-known Batty's theorem states that if a $C_0$-semigroup $T(t)$ is bounded and the spectrum of the generator $A$ is contained in the open left-half plane of $\mathbb{C}$, then $\|T(t)A^{-1}\|$ tends to $0$. This can be thought of as a particular case of a more general property that, for $ω_0>-\infty$ and $(ω_0+i\mathbb{R})\cap σ(A)=\emptyset$ it holds $\|T(t)(A-ω_0 I)^{-1}\|/\|T(t)\|$ tends to 0. We show that it is true for $\|T(t)\|$ regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-half plane of $\mathbb…
Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.
