For certain classes of negative definite symbols $q(x,ξ)$ and state space dependent Bernstein function $f(x,s)$ we prove that $-p(x,D)$, the pseudo-differential operator with symbol $-p(x,ξ)=-f(x,q(x,ξ))$, extends to the generator of a Feller semigroup. Our result extends previously known results related to operators of variable (fractional) order of differentiation, or variable order fractional powers. New concrete examples are given.
We prove that a weakly ergodic, strong Feller semigroup on the space of measures converges strongly to a projection onto its fixed space. In contrast to a recent result of Gerlach we do not assume the semigroup to be stochastically continuous.
We consider operators $A$ on a sequentially complete Hausdorff locally convex space $X$ such that $-A$ generates a (sequentially) equicontinuous equibounded $C_0$-semigroup. For every Bernstein function $f$ we show that $-f(A)$ generates a semigroup which is of the same `kind' as the one generated by $-A$. As a special case we obtain that fractional powers $-A^α$, where $α\in (0,1)$, are generators.
A suitable notion of hypercontractivity for a nonlinear semigroup $\{T_t\}$ is shown to imply Gagliardo--Nirenberg inequalities for its generator $H$, provided a subhomogeneity property holds for the energy functional $(u,Hu)$. We use this fact to prove that, for semigroups generated by operators of $p$--Laplacian--type, hypercontractivity implies ultracontractivity. We then introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is $f(x)=x^α$, and write down an explicit formula for the associated generator. It is shown that, in certain cases, hyperc…
We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions. The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's Lévy-Khinchine formula to a function of two variables. The Hille-Yosida-Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.
We prove that a weakly ergodic, eventually strong Feller semigroup on the space of measures on a Polish space converges strongly to a projection onto its fixed space.
For an increasing sequence $(T_n)$ of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps the constant 1 function to a continuous function. This is instrumental in the study of elliptic operators on $\mathbb{R}^d$ with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and …
This paper generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asymptotically sharp estimate for the observation constant and which is explicit in the model parameters. We give two types of applications where the extension of the growth/decay rates naturally appear. By exploiting subordination techniques we show how th…
We study perturbations of Feller generators under `lower order terms' with measurable coefficients. We investigate which properties of the original semigroup -- such as positivity, conservativeness and the Feller property -- are passed to the perturbed semigroup. We give several examples and discuss applications in the theory of martingale problems and stochastic differential equations with measurable coefficients.
We prove that for any Bernstein function $ψ$ the operator $-ψ(A)$ generates a holomorphic $C_0$-semigroup $(e^{-tψ(A)})_{t \ge 0}$ on a Banach space, whenever $-A$ does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that $(e^{-tψ(A)})_{t \ge 0}$ is holomorphic in the holomorphy sector of $(e^{-tA})_{t \ge 0},$ and if $(e^{-tA})_{t \ge 0}$ is sectorially bounded in this sector then $(e^{-tψ(A)})_{t \ge 0}$ has the same property. We also obtain new sufficient conditions on $ψ$ in order that…
A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated integrals of some elementary functions as $n\to\infty$. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman--Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae gi…
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial v…
In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund in [24]. We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in $C_b(\R^N)$ and to evolution operators associated with nonautonomous second-order differential operators in $C_b(\R^N)$ with time-periodic coefficients.
We show that a Krein-Feller operator is naturally associated to a fixed measure $μ$, assumed positive, $σ$-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, $L^{2}\left(μ\right)$ and $L^{2}\left(λ\right)$, where $λ$ denotes Lebesgue measure. An associated operator pair consists of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. This then yields a rigorous analysis of the corresponding $μ$-Krein-Feller operator as a closable quadratic form. As an application, for a given measure $μ$, including t…
We consider Feller semigroups of operators determinated by systems of jumps dominated by the rotation invariant stable Lévy measure. Using an approximation schema we prove the existence and obtain estimates of corresponding heat kernels.
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,…
We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and an operator $L$ being the generator of a strongly continuous semigroup. In particular, $L$ may be the generator $L_0$ of a Markov process $ξ$ on some state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift $b$, or $L$ generating subordinate semigroups or Schrödinger type groups. This class of evolution equations includes in particular time- and space- fractional heat and Schrödinger type equations. We show that a subordinat…
We use Bochner's subordination technique to obtain caloric smoothing estimates in Besov- and Triebel--Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß--Weierstraß, Cauchy--Poisson and higher-order generalized Gauß--Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched.
We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doob's theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domain…
In J. Math. Anal. App. 305. (2005), we have considered the Gribov operator\\ $H_{λ'} = λ' S + H_{μ,λ}$ acting on Bargmann space where $ S = a^{*2} a^{2}$ and $ H_{μ,λ} =μa^* a + \ıλa^* (a+a^*)a$ with $i^{2} = -1$.\\ Here $a$ and $a^{*}$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a, a^{*}] = I$. In Reggeon field theory, the real parameters $λ{'}$ is the four coupling of Pomeron, $μ$ is Pomeron intercept, $λ$ is the triple coupling of Pomeron and $i^{2} = -1$.\\ We have given an approximation of the semigroup $e^{-tH_{λ'}}$ generated by the op…
Motivated by models of signaling pathways in B lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin $2D$ domains may be approximated by diffusion equations in regions of smaller dimensions. In particular, we study how transmission conditions featuring in the approximating equations become integral parts of the limit master equation. We device a scheme which, by appropriate rescaling of coefficients and finding a common reference space for all Feller semigroups involved, allows deriving the form of the limit equation formally. The res…
We formulate a criterion for the existence of an invariant measure for a Feller semigroup defined on a metric space with the e-property for bounded continuous functions and use it to prove the asymptotic stability of a semigroup satisfying a lower bound condition. Our results complement those of A. Lasota and J. A. Yorke in proper metric spaces and of T. Szarek in Polish spaces.
It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. The construction is possible provided the generator satisfies an invariance property for some dense subalgebra A_0 of the C* algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A_0, must satisfy a growth condition. Furtherm…
Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + <Ax, Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $μ$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then $$\n P(t)-P(s)\n_{L^1(E,μ)} = 2$$ for all $t,s\ge 0$ with $t\not=s$. This result is new even when $E = \R^n$. We also study the behaviour…
We show that if $-A$ generates a bounded $α$-times resolvent family for some $α\in (0,2]$, then $-A^β$ generates an analytic $γ$-times resolvent family for $β\in(0,\frac{2π-πγ}{2π-πα})$ and $γ\in (0,2)$. And a generalized subordination principle is derived. In particular, if $-A$ generates a bounded $α$-times resolvent family for some $α\in (1,2]$, then $-A^{1/α}$ generates an analytic $C_0$-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.
We obtain integral representations for the resolvent of $ψ(A)$, where $ψ$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $ψ$, we show that the operator $-ψ(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $ψ$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a pe…
We investigate a second order elliptic differential operator $A_{β, μ}$ on a bounded, open set $Ω\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq β\in L^{\infty}(\partialΩ)$ and $μ\colon \partial Ω\to \mathscr{M}(\overlineΩ)$, and boundary conditions of the form \[ \partial_ν^{\mathscr{A}}u(z)+β(z)u(z)=\int_{\overlineΩ}u(x)μ(z)(dx),\ z\in\partialΩ, \] where $\partial_ν^{\mathscr{A}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of …
We consider second order differential operators $A_μ$ on a bounded, Dirichlet regular set $Ω\subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_Ωu(x)\, μ(z, dx)\quad \mbox{for } z \in \partial Ω. \] Here the function $μ: \partialΩ\to \mathscr{M}^+(Ω)$ is $σ(\mathscr{M} (Ω), C_b(Ω))$-continuous with $0\leq μ(z,Ω) \leq 1$ for all $z\in \partial Ω$. Under suitable assumptions on the coefficients in $A_μ$, we prove that $A_μ$ generates a holomorphic positive contraction semigroup $T_μ$ on $L^\infty(Ω)$. The semigroup $T_μ$ is never strongly continuous, but it enjoys t…
