We will begin with a brief introduction to the study of chaotic and ergodic properties of dynamical systems (streams) determined by partial differential equations. The main topic will be to show the construction of invariant measures for flows determined by first-order semilinear partial differential equations [1]. Such equations generate a flow on certain function spaces. The construction of such a measure uses a random field called a d-parameter Brownian-Lévy process and the isomorphism of the aforementioned flow with translation along the radii of a sphere. We will give examples of applications in biological models: the spatial distribution of population size and flow with random jumps from the edge to the interior of the area, used to describe the evolution of the maturity distribution of erythrocyte precursors when there is no external control of their production. At the end of the lecture, we will present the results concerning invariant measures for the flow generated by the heat equation [2]. The construction of such a measure will include, among other things, a stationary Gaussian process whose trajectories have rather surprising properties.

1. R. Rudnicki, Ergodic properties of a semilinear partial differential equation. Journal of Differential Equations (2023), 372, 235-253.
2. R. Rudnicki, Ergodic and chaotic properties of the heat equation. Journal of Dynamics and Differential Equations (2025), 37, 2719-2730.