### The Large Deviation Principle and Steady-state Fluctuation Theorem for the Entropy Production Rate of a Stochastic Process in Magnetic Fields

Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. Steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes have not been rigorously proved yet due to technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function for $\theta \in \left(-\frac{\pi}2, 0\right) \cup \left(0,\frac{\pi}2\right)$. The proof is based on the Karhunen-Lo\`{e}ve expansion of complex-valued Ornstein-Uhlenbeck process, in which the key step is solving a Sturm-Liouville problem to determine the free energy function.