Interactions of noise and discontinuities: transitions and qualitative changes

While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context.  Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present.  We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance,  relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics.  The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention.  Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective
approaches.