Conference Agenda

We develop and analyze a random field model for the reconstruction of inhomogeneous turbulence from characteristic flow quantities provided by $k$-$\varepsilon$ simulations. The model is based on stochastic integrals that combine moving average and Fourier-type representations in time and space, respectively, where both the time integration kernel and the spatial energy spectrum depend on the macroscopically varying characteristic quantities. The structure of the model is derived from standard spectral representations of homogeneous fields by means of a two-scale approach in combination with specific stochastic integral transformations. Our approach allows for a rigorous analytical verification of the desired statistical properties and is accessible to numerical simulation.

We investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with Hölder continuous paths, covering in particular fractional Wiener processes with general Hurst parameter. In the latter example, we obtain explicit error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the set of travelling wave fronts in terms of the Hurst parameter and the spatial regularity for small noise amplitude. Our bounds can be optimised for short times in terms of the Hurst parameter and for large times in terms of the spatial regularity of the noise covariance of the driving fractional Wiener process.

In this talk we discuss several results about anomalous diffusion processes with random paramaters, which are inspired by recent single particle tracking biological experiments. We focus on three processes, namely: fractional Brownian motion with random Hurst exponent (FBMRE), Riemann-Liouville fractional Brownian motion with random Hurst exponent (RLFBMRE), and scaled Brownian motion with random anomalous diffusion exponent (SBMRE). In all cases, we present the basic probabilistic properties like transition density, q-th moment of absolute value of the process, autocovariance function, and expectation of time-averaged mean squared displacement. Moreover, we analyze ergodic properties of all three processes. Additionally, for SBMRE we analyze its martingale properties and law of large numbers. Together with theoretical analysis, we provide the numerical anlysis of obtained results. The talk is based on [1] and [2].

[1] H. Woszczek, A. Chechkin, A. Wyłomańska, Scaled Brownian motion with random anomalous diffusion exponent, Communications in Nonlinear Science and Numerical Simulation, 2025, vol. 140, pt. 1, art. 108388, s. 1-27

[2] H. Woszczek, A. Wyłomańska, A. Chechkin, Riemann-Liouville fractional Brownian motion with random Hurst exponent, preprint, arXiv:2410.11546

Stochastic elliptic problems arise mainly by substituting deterministic parameters in elliptic problems by certain random parameters. Then one issue in the consideration of random equations is the measurability of desired solutions. Based on the fact that there exist different measurability concepts it is important to use the appropriate measurability concept for each problem. Hereby mainly the Borel, weak and strong measurability concepts are of interest.

In the talk these measurability concepts are presented and some of the relations between them are discussed. This is important, because in elliptic problems also non-separable Banach spaces play a certain role and in these spaces the measurability concepts mentioned above do not coincide necessarily. Based on these findings measurability properties of solutions of elliptic problems are investigated.

Furthermore it will be shown exemplarily, which stochastic elliptic inverse problems can be treated as abstract elliptic inverse problems and which such stochastic inverse problems require a specific stochastic investigation.