Cylindrical Wiener processes

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well-known stochastic integrals in literature

Heterogeneous perturbations in the Doppler-free S1 ← S0 two-photon spectrum of benzene: Evidence for intrastate coupling

Rotational perturbations are identified in Doppler-free two-photon spectra of the 1410 and 1410110 vibronic bands in C6H6. Evidence is found that Coriolis coupling between some of the rotational levels of two distinct vibrational states within S1 is the mechanism responsible. This coupling mechanism is thought to be responsible for irreversible intramolecular relaxation at higher excess energies and higher vibrational state densities

Ornstein-Uhlenbeck processes driven by cylindrical L\'evy processes

In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical L\'evy processes in separable Banach spaces. Here, a cylindrical L\'evy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical L\'evy processes and several examples are considered.Comment: This version is significantly revised and correcte

Sub-doppler two-photon spectrum of asymmetric rotor molecules

The Doppler-free two-photon excitation spectrum of the qqQ branch of the 1410 vibrational band of the S1(1B2u) ← S0(1A1g) transition of benzene-d1 has been recorded using a cw single-mode dye laser coupled to an external concentric resonator. The spectrum has been analysed using a non-rigid Watson Hamiltonian. More than 200 lines with J up to 20 have been assigned and the rotational constants which best reproduce the spectrum are A1v = 0.181435, B1v = 0.169990, C1v = 0.089055 cm−1. The Ka = odd lines of the qqQ5(J) subbranch show small and quite regular perturbations of 60 ± 5 MHz which are probably due to a coupling to another vibrational state of the S1 manifold

Infinitely divisible cylindrical measures on Banach spaces

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Further properties of infinitely divisible cylindrical measures such as continuity are derived. Moreover, the result on the classification enables us to conclude new results on genuine Levy measures on Banach spaces

Mixing of the vibrational angular momentum components of multiply degenerate vibronic states of benzene by vibrational l-type resonance

The rotationally resolved spectra of the 6011002 and 6011602 vibronic transitions of benzene at low rotational temperat reported and analyzed in detail. Deperturbed spectroscopic constants for the 61102 and 61162 states are reported which reproduce the observed line positions to within experimental accuracy. The splitting of 17.9 cm−1 between the two subbands of the 6011002 transition and of 6.2 cm−1 for the 6011602 transition is found to be due to vibrational l-type resonances with matrix element 8.91 and 2.65 cm−1, respectively. These large resonances cause strong distortions of the rotational structure and mix the vibrational angular momentum substates ν6minus-or-plus sign + 2ν100 and ν6± + 2ν10±2 nearly completely and the substates ν6minus-or-plus sign + 2ν160 and ν6± + 2ν16minus-or-plus sign2 substantially. The importance of the mixing for the intramolecular vibrational redistribution (IVR) and the decay behaviour of S1 benzene is discussed

Radonifying operators and infinitely divisible Wiener integrals

In this article we illustrate the relation between the existence of Wiener integrals with respect to a Levy process in a separable Banach space and radonifying operators. For this purpose, we introduce the class of theta-radonifying operators, i.e. operators which map a cylindrical measure theta to a genuine Radon measure. We study this class of operators for various examples of infinitely divisible cylindrical measures theta and highlight the differences from the Gaussian case.Comment: 16 page

Invariant measure for the stochastic Cauchy problem driven by a cylindrical L\'evy process

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical L\'evy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical L\'evy process and demonstrate their application in some examples