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Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as ‘superstatistics’ or ‘diffusing diffusivity’. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.

Supermassive black holes reside in the hearts of almost all massive galaxies. Their evolutionary path seems to be strongly linked to the evolution of their host galaxies, as implied by several empirical relations between the black hole mass (M BH ) and different host galaxy properties. The physical driver of this co-evolution is, however, still not understood. More mass measurements over homogeneous samples and a detailed understanding of systematic uncertainties are required to fathom the origin of the scaling relations.
In this thesis, I present the mass estimations of supermassive black holes in the nuclei of one late-type and thirteen early-type galaxies. Our SMASHING sample extends from the intermediate to the massive galaxy mass regime and was selected to fill in gaps in number of galaxies along the scaling relations. All galaxies were observed at high spatial resolution, making use of the adaptive-optics mode of integral field unit (IFU) instruments on state-of-the-art telescopes (SINFONI, NIFS, MUSE). I extracted the stellar kinematics from these observations and constructed dynamical Jeans and Schwarzschild models to estimate the mass of the central black holes robustly. My new mass estimates increase the number of early-type galaxies with measured black hole masses by 15%. The seven measured galaxies with nuclear light deficits (’cores’) augment the sample of cored galaxies with measured black holes by 40%. Next to determining massive black hole masses, evaluating the accuracy of black hole masses is crucial for understanding the intrinsic scatter of the black hole- host galaxy scaling relations. I tested various sources of systematic uncertainty on my derived mass estimates.
The M BH estimate of the single late-type galaxy of the sample yielded an upper limit, which I could constrain very robustly. I tested the effects of dust, mass-to-light ratio (M/L) variation, and dark matter on my measured M BH . Based on these tests, the typically assumed constant M/L ratio can be an adequate assumption to account for the small amounts of dark matter in the center of that galaxy. I also tested the effect of a variable M/L variation on the M BH measurement on a second galaxy. By considering stellar M/L variations in the dynamical modeling, the measured M BH decreased by 30%. In the future, this test should be performed on additional galaxies to learn how an as constant assumed M/L flaws the estimated black hole masses.
Based on our upper limit mass measurement, I confirm previous suggestions that resolving the predicted BH sphere-of-influence is not a strict condition to measure black hole masses. Instead, it is only a rough guide for the detection of the black hole if high-quality, and high signal-to-noise IFU data are used for the measurement. About half of our sample consists of massive early-type galaxies which show nuclear surface brightness cores and signs of triaxiality. While these types of galaxies are typically modeled with axisymmetric modeling methods, the effects on M BH are not well studied yet. The massive galaxies of our presented galaxy sample are well suited to test the effect of different stellar dynamical models on the measured black hole mass in evidently triaxial galaxies. I have compared spherical Jeans and axisymmetric Schwarzschild models and will add triaxial Schwarzschild models to this comparison in the future. The constructed Jeans and Schwarzschild models mostly disagree with each other and cannot reproduce many of the triaxial features of the galaxies (e.g., nuclear sub-components, prolate rotation). The consequence of the axisymmetric-triaxial assumption on the accuracy of M BH and its impact on the black hole - host galaxy relation needs to be carefully examined in the future.
In the sample of galaxies with published M BH , we find measurements based on different dynamical tracers, requiring different observations, assumptions, and methods. Crucially, different tracers do not always give consistent results. I have used two independent tracers (cold molecular gas and stars) to estimate M BH in a regular galaxy of our sample. While the two estimates are consistent within their errors, the stellar-based measurement is twice as high as the gas-based. Similar trends have also been found in the literature. Therefore, a rigorous test of the systematics associated with the different modeling methods is required in the future. I caution to take the effects of different tracers (and methods) into account when discussing the scaling relations.
I conclude this thesis by comparing my galaxy sample with the compilation of galaxies with measured black holes from the literature, also adding six SMASHING galaxies, which were published outside of this thesis. None of the SMASHING galaxies deviates significantly from the literature measurements. Their inclusion to the published early-type galaxies causes a change towards a shallower slope for the M BH - effective velocity dispersion relation, which is mainly driven by the massive galaxies of our sample. More unbiased and homogenous measurements are needed in the future to determine the shape of the relation and understand its physical origin.

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

In this paper we report on photoswitchable polymer surfaces with dynamically and reversibly fluctuating topographies. It is well known that when azobenzene containing polymer films are irradiated with optical interference patterns the film topography changes to form a surface relief grating. In the simplest case, the film shape mimics the intensity distribution and deforms into a wave like, sinusoidal manner with amplitude that may be as large as the film thickness. This process takes place in the glassy state without photo-induced softening. Here we report on an intriguing discovery regarding the formation of reliefs under special illumination conditions. We have developed a novel setup combining the optical part for creating interference patterns, an AFM for in situ acquisition of topography changes and diffraction efficiency signal measurements. In this way we demonstrate that these gratings can be “set in motion” like water waves or dunes in the desert. We achieve this by applying repetitive polarization changes to the incoming interference pattern. Such light responsive surfaces represent the prerequisite for providing practical applications ranging from conveyer or transport systems for adsorbed liquid objects and colloidal particles to generation of adaptive and dynamic optical devices.