PhDPiecewise-smooth stochastic systems have attracted a lot of interest in the last
decades in engineering science and mathematics. Many investigations have focused
only on one-dimensional problems. This thesis deals with simple two-dimensional
piecewise-smooth stochastic systems in the absence of detailed balance. We investigate
the simplest example of such a system, which is a pure dry friction model
subjected to coloured Gaussian noise. The nite correlation time of the noise establishes
an additional dimension in the phase space and gives rise to a non-vanishing
probability current. Our investigation focuses on stick-slip transitions, which can be
related to a critical value of the noise correlation time. Analytical insight is provided
by applying the uni ed coloured noise approximation. Afterwards, we extend our
previous model by adding viscous friction and a constant force. Then we perform
a similar analysis as for the pure dry friction case. With parameter values close
to the deterministic stick-slip transition, we observe a non-monotonic behaviour of
the probability of sticking by increasing the correlation time of the noise. As the
eigenvalue spectrum is not accessible for the systems with coloured noise, we consider
the eigenvalue problem of a dry friction model with displacement, velocity and
Gaussian white noise. By imposing periodic boundary conditions on the displacement
and using a Fourier ansatz, we can derive an eigenvalue equation, which has a
similar form in comparison to the known one-dimensional problem for the velocity
only. The eigenvalue analysis is done for the case without a constant force and with
a constant force separately. Finally, we conclude our ndings and provide an outlook
on related open problems.Queen Mary University of London and the School of Mathematical Science