Generator Of Semigroup Research Articles

VMO Spaces Associated with Operators with Gaussian Upper Bounds on Product Domains

In this paper, we extend the well-known result “the predual of Hardy space \(H^1\) is VMO” to the product setting, associated with differential operators. Let \(L_i\), \(i = 1, 2\), be the infinitesimal generators of the analytic semigroups \(\{e^{-tL_i}\}\) on \(L^2({\mathbb {R}})\). Assume that the kernels of the semigroups \(\{e^{-tL_i}\}\) satisfy the Gaussian upper bounds. We introduce the VMO spaces VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) associated with operators \(L_1\) and \(L_2\) on the product domain \(\mathbb {R}\times \mathbb {R}\), then show that the dual space of VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) is the Hardy space \(H^1_{L_1^*, L_2^*}(\mathbb {R}\times \mathbb {R})\) associated with the adjoint operators \(L^*_1\) and \(L^*_2\).

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

AbstractRecently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset $$\Gamma \subset G$$ that contains a semi-simple subgroup $$G_{1}$$ of G. If one can show that $$ \Gamma $$ does not leave invariant a contractible subset on any flag manifold of G, then $$\Gamma $$ generates G if $$\mathrm {Ad}\left( \Gamma \right) $$ generates a Zariski dense subgroup of the algebraic group $$\mathrm {Ad}\left( G\right) $$ . The proof is reduced to check that some specific closed orbits of $$G_{1}$$ in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups $$G_{1}\subset G$$ .

Erratum to: Application of the Laplace Transform of Tempered Distributions to the Construction of Functional Calculus

We use the generalized n-dimensional Laplace transform of tempered distributions whose supports are located in a positive n-dimensional cone to construct functional calculus for the commutative collections of injective generators of n-parameter analytic semigroups of operators acting in a Banach space.

Ordinal sum construction for uninorms and generalized uninorms

The ordinal sum construction yielding uninorms is studied. A special case when all summands in the ordinal sum are isomorphic to uninorms is discussed and the most general semigroups that yield a uninorm via the ordinal sum construction, called generalized uninorms, are studied.

On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

We investigate the boundedness of the H∞-calculus by estimating the bound b(ε) of the mapping H∞→B(X): f↦f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H∞ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε)=O(|log⁡ε|) in general, whereas b(ε)=O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε)=O(|log⁡ε|).

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.

Infinitesimal generators of evolution families of nonexpansive mappings

The nonlinear resolvent, ρ-monotonicity and initial value problems were used to study infinitesimal generators of one-parameter semigroups in Nρ(D), the set of all ρ-nonexpansivemappings onD, the unit disk. In this paper, the notions of an algebraic evolution family in Nρ(D) and the infinitesimal generator of such a family are introduced. Under a locally uniformly Lipschitz condition and using the nonlinear resolvent and ρ-monotonicity, a characterization of the infinitesimal generators of evolution families in Nρ(D) is obtained.

Regularity properties of degenerate convolution-elliptic equations

The coercive properties of degenerate abstract convolution-elliptic equations are investigated. Here we find sufficient conditions that guarantee the separability of these problems in $L_{p}$ spaces. It is established that the corresponding convolution-elliptic operator is positive and is also a generator of an analytic semigroup. Finally, these results are applied to obtain the maximal regularity properties of the Cauchy problem for a degenerate abstract parabolic equation in mixed $L_{\mathbf{p}}$ norms, boundary value problems for degenerate integro-differential equations, and infinite systems of degenerate elliptic integro-differential equations.

Optimal energy decay in a one-dimensional coupled wave–heat system

We study a simple one-dimensional coupled wave-heat system and obtain a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (Math. Ann., 2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations.

Feedback theory extended for proving generation of contraction semigroups

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with 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 equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.

Heat–wave interaction in 2–3 dimensions: Optimal rational decay rate

In this paper, we consider a simplified version of a fluid–structure PDE model which has been of longstanding interest within the mathematical and biological sciences. In it, a n-dimensional heat equation replaces the original Stokes system, so as to ultimately have a vector-valued heat equation and vector-valued wave equation compose the coupled PDE system under study. The coupling between the two disparate PDE components occurs across a boundary interface. As such, the entire PDE dynamics manifests features of both hyperbolicity and parabolicity. For this heat–structure system, our main result of uniform stability is as follows: Given smooth initial data – i.e., data in the domain of the underlying semigroup generator of the coupled PDE system – the corresponding solutions decay at the rate of o(t−1). This establishes the long-conjectured optimal rate. The problem of obtaining sharp rational decay rates for the heat–wave PDE, under present consideration, has been a much considered problem, with the modus operandi of earlier efforts taking place within the time domain. By contrast, we adopt here a frequency domain approach which is based upon a recent resolvent criterion, and which was initiated in our prior effort, wherein we obtained the rate of decay o(1t). The present optimal improvement o(t−1) is achieved by employing an additional tool in our analysis – a microlocal analysis argument – to estimate a critical term involving two problematic boundary traces.

Quasi-Hyperbolicity and Delay Semigroups

We study quasi-hyperbolicity of the delay semigroup associated with the equation u′(t)=Bu(t)+Φut, where ut is the history function and (B,D(B)) is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup.

Generating Set of the Complete Semigroups of Binary Relations

Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup , which are defined by semilattice D, can be generated by the set .

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l1(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ℓ1(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ℓ1(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ℓ1(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.

On the range of the generator of a quantum Markov semigroup

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.

Stochastic calculus over symmetric Markov processes with time reversal

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

On joint spectra of families of unbounded operators

We consider several types of joint spectra of a finite set of commuting closed operators in a Banach space. We establish new relations between these spectra (it was previously known only that the Taylor spectrum is contained in the commutant spectrum) and prove spectral mapping theorems in the case of generators of semigroups. Some of these theorems generalize previous results of the author. The results obtained are applied to stability issues for multi- parameter semigroups.

Stochastic calculus over symmetric Markov processes with time reversal

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations

We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: the dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator The eigenvalues of are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining non-Hermitian degeneracies emerge, i.e. exceptional points (EP). The dynamical signature of these EPs corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the EPs and thereby to determine the intrinsic system parameters with a high accuracy. This way we turn the disadvantage of the dissipation into an advantage. We demonstrate this method in the open system dynamics of a two-level system described by the Bloch equation, which has become the paradigm of diverse fields in physics, from NMR to quantum information and elementary particles.