Upanshu Sharma, École des Ponts Paris Tech
Estimating coarse-graining error beyond reversibility
Coarse-graining is the procedure of approximating a complex and high-dimensional system by a simpler and lower dimensional one. Typically, such an approximation is achieved by using a coarse-graining map F, which projects the full state of a system X (representing for instance the position of particles in the system), onto a lower-dimensional state space. Assuming that the state X evolves according to a stochastic differential equation (SDE), it is easy to identify the evolution of F(X) — however, this cannot be used in practice since the evolution still depends on the original state space. Legoll and Lelièvre (2010) addressed this by introducing a natural approximation, called the effective dynamics, and quantified the error of this approximation when X solves the overdamped Langevin equation. Since then a variety of results have been discussed in this direction, in the setting when X evolves according to a reversible SDE. In this talk, I will show that reversibility is not a limitation, and both the construction of the effective dynamics and the error estimates can be generalised to the setting of non-reversible SDEs. This is joint work with C. Hartmann and L. Neureither.