The generalized Ornstein-Uhlenbeck (GOU) process driven by a bivariate Lévy process is widely known for its applications e.g. in finance, insurance and physics. In this talk we present the generator of the GOU process and show that GOU processes are Feller processes. Using these results, we will then gather information on the stationary distributions of GOU processes, also known as exponential functionals of Lévy processes. In particular we can show that under some regularity conditions and for independent driving Lévy processes the mapping of (the law of) one Lévy process on the corresponding (law of the) exponential functional is injective and give conditions for continuity of this mapping in a distributional sense.
This is joint work with Alexander Lindner.

An overview of the mathematical and physical discoveries related to the moving interface between two fluids will be presented without technical details. A century and a half of progress in this field will be illustrated on a single, very relevant example: Hele-Shaw flows. The contributions of the British, French, Russian and American schools will be described, as a background for the fascinating discoveries made during the last decade. A leap into the quantum world will widen and simplify the horizon.

In this talk I will present a model for cell deformation and cell movement that couples the mechanical and biochemical properties of the cortical network of actin filaments with its concentration. Actin is a polymer that can exist either in filamentous form (F-actin) or in monometric form (G-actin) (Chen et al., in Trends Biochem Sci 25:19–23, 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrishnan et al., in Recent Res Devel Biophys 5:39–69, 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its biochemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al., in J Comput Phys 190:478–500, 2003). Furthermore, by assuming slow deformations of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the mathematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contraction, cell translation and cell relocation as well as cell protrusions in agreement with experimental observations. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter. Cell migration plays a critical and pivotal role in a variety of biological and biomedical disease processes and is important for emerging areas of biotechnology which focus on cellular transplantation and the manufacture of artificial tissues and surfaces, as well as for the development of new therapeutic strategies for controlling invasive tumor cells.

Simulating continuum reaction–diffusion processes on complex deforming surfaces requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We present a meshless Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries. We further show extensions of the employed meshless particle method to adaptive-resolutions simulations where the particles (colocation points) self-organize. In such a discretization, neither the total number of particles, nor their spatial distribution need to be specified by the user. The user only specifies the desired error level of the solution and the particles in the simulation self-organize to a distribution that reaches this.

Consider a rigid ball B partially immersed in equilibrium, in an unlimited liquid bath with surface S symmetric and horizontal at infinity and subject to surface tension in a downward gravity field g. B will create a local disturbance in S, which vanishes rapidly with distance from its center. Moving B vertically a small distance h leads to no substantive change in the configuration, and similar behavior will be observed. If gravity is removed, the problem changes basically. The only liquid surface under the conditions becomes a horizontal plane meeting the ball in a contact angle determined entirely by the materials. Thus if B is now moved vertically the distance h, S must move rigidly with it; consequently the entire liquid must also move, which conceivably would require an infinite amount of work, a seemingly absurd conclusion.
This talk will be devoted to my attempts to reformulate the conditions of the problem in terms of mathematically precise and physically reasonable statements, and to continuing efforts to obtain correct predictions as to what actually would occur under such circumstances.

Wohin mit dem Ersparten? Anlageberatung aus mathematischer Perspektive
Die Frage nach der vernünftigen Wahl eines Wertpapierportfolios zum Zweck des Vermögenserhalts beschäftigt nicht nur große wie kleine Anleger, sie ist auch aus wissenschaftlicher Sicht sehr facettenreich. In diesem Vortrag sollen einige mathematische Aspekte beleuchtet werden, wobei neben klassischen Resultaten auch neuere Resultate wie etwa der Einfluss von Transaktionskosten und -steuern zur Sprache kommen.