Explore dynamical phase transitions in Markov processes through this informative colloquium talk by Hugo Touchette from Stellenbosch University. Delve into the similarities and differences between dynamical and equilibrium phase transitions, and examine a simple Brownian motion model demonstrating a dynamical phase transition in the long-time limit. Learn about the physical interpretation of this transition as a localisation-delocalisation phenomenon. The talk covers key concepts including equilibrium phase transitions, dynamical phase transitions, rate function calculations, and large deviation theory. Gain insights into the Garner-Ellis Theorem, Penon-Frobenino approach, and spectral problems associated with these processes. Conclude with a comprehensive Q&A session to deepen your understanding of this fascinating topic in statistical physics.