7 research outputs found
Stochastic switching in infinite dimensions with applications to random parabolic PDE
© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides
Smooth invariant densities for random switching on the torus
We consider a random dynamical system obtained by switching between the flows
generated by two smooth vector fields on the 2d-torus, with the random
switchings happening according to a Poisson process. Assuming that the driving
vector fields are transversal to each other at all points of the torus and that
each of them allows for a smooth invariant density and no periodic orbits, we
prove that the switched system also has a smooth invariant density, for every
switching rate. Our approach is based on an integration by parts formula
inspired by techniques from Malliavin calculus.Comment: 19 page