  Time-irreversible stochastic processes are frequently used in natural
sciences to explain non-equilibrium phenomena and to design efficient
stochastic algorithms. Our main goal in this thesis is to analyse their
dynamics by means of large deviation theory. We focus on processes that become
deterministic in a certain limit, and characterize their fluctuations around
that deterministic limit by Lagrangian rate functions. Our main techniques for
establishing these characterizations rely on the connection between large
deviations and Hamilton-Jacobi equations. We sketch this connection with
examples in the introductory parts of this thesis.
  The second part of the thesis is devoted to irreversible processes that are
motivated from molecular motors, Markov chain Monte Carlo (MCMC) methods and
stochastic slow-fast systems. We characterize the asymptotic dynamics of
molecular motors by Hamiltonians defined in terms of principal-eigenvalue
problems. From our results about the zig-zag sampler used in MCMCs, we learn
that maximal irreversibility corresponds to an optimal rate of convergence. In
stochastic slow-fast systems, our main theoretical contributions are techniques
to work with the variational formulas of Hamiltonians that one encounters in
mean-field systems coupled to fast diffusions.
  In the final part of the thesis, we study a family of Fokker-Planck equations
whose solutions become singular in a certain limit. The associated
gradient-flow structures do not converge since the relative entropies diverge
in the limit. To remedy this, we propose to work with a different variational
formulation that takes fluxes into account, which is motivated by density-flux
large deviations.
