Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes
Mathis Guéneau (LPTHE) and Léo Touzo (LPENS)
Active particles are systems — typically living organisms — capable of converting energy from their environment into directed motion. On the one hand, there has been a lot of interest recently in the first-passage properties of these models, partly motivated by applications to biology. On the other hand, active particles are known to exhibit exotic behaviors in the presence of confinement, such as hard walls, due to their non-equilibrium nature.
Surprisingly, the first-passage properties of many stochastic processes happen to be closely related to their spatial distribution in the presence of hard walls. This type of connection has a long history in mathematics, where it is known as Siegmund duality. However, it remains relatively unknown in the physics community and had never been shown for active particles.
In this talk, we will show how to extend this duality relation to a large class of stochastic processes which are particularly relevant in physics, including the most well-known models of active particles, but also diffusing diffusivity models, and particles subjected to stochastic resetting.