tokens: this study as the approach leads to reasonable results for the limited energy range considered. Fig. 2 Sketch of the simulation setup as detailed in Dunzlaff et al. (2015) . The four different exit possibilities are colour-coded: the Sun (yellow), the Jovian magnetosphere (orange), the Heliopause (blue), as well as the option that the phase space trajectory is terminated by reaching a predefined end time (green). We note that the figure is not to scale. As described in great detail by e.g. Kloeden Platen (2011) , the stochastic nature of diffusion is treated within the SDE method as a Wiener process with ζ being a vector of Gaussian distributed random numbers. The resulting set of four integral equations, as they are derived by Strauss et al. (2011b ), is solved iteratively by applying the Euler-Maruyama scheme. This leads to a random walk type solution which is terminated if a spatial or temporal boundary is reached, as shown in Fig. 2 for various possible exit positions. A time-backward phase-space trajectory can either terminate at the assumed position of the heliopause (blue line), the Jovian magnetosphere (orange line), or after a pre-specified number of steps have been made without an encounter with either of the formerly mentioned structures (green line). As discussed in Sect. 2.2 , in the low-MeV energy range Jovian electrons dominate the spectrum. The solar population therefore can be assumed to be negligible during quiettimes, and thus trajectories exiting at the Sun are discarded. Galactic electrons, on the other hand are considered via the local interstellar spectrum according to Potgieter Vos (2017) , although the transport of the Galactic population into the inner heliosphere is much more sensitive to the drift effect, which is not considered here. Instead of solving Parker’s TPE directly to obtain a time-dependent distribution function f ( r , t ) covering thewhole phase space, the SDE method provides a chain of point-like solutions both in space and time. The solutions must be sampled at a large number of phase-space points to approximate a spatial solution covering the phase space of interest. In order for the computational results to be comparable with spacecraft data, the time-backward setup is solved as derived and discussed by e.g. Kopp et al. (2012) . A comparison between the time-forward and time-backward setup, made using a simpler 1D model, can be found in the review by Strauss Effenberger (2017) . The left panel of Fig. 5 shows histograms of the so called exit energy 1 (i.e. the energy of which pseudo-particles leave the computational domain) at times of good magnetic connection between the Jovian magnetosphere and the observational point, in contrast to Fig. 5 b, which illustrates the energy distribution for an observational point in opposition to the Jovian source. The distributions in Fig. 5 b are much broader. This, of course, is in agreement with Fig. 5 a being dominated by the much more efficient parallel
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