  The close-in planet orbiting GJ 436 presents a puzzling orbital eccentricity
considering its very short orbital period. Given the age of the system, this
planet should have been tidally circularized a long time ago. Many attempts to
explain this were proposed in recent years, either involving abnormally weak
tides, or the perturbing action of a distant companion. We address here the
latter issue based on Kozai migration. We propose that GJ 436b was formerly
located further away from the star and that it underwent a migration induced by
a massive, inclined perturber via Kozai mechanism. In this context, the
perturbations by the companion trigger high amplitude variations to GJ 436b
that cause tides to act at periastron. Then the orbit tidally shrinks to reach
its present day location. We numerically integrate the 3-body system including
tides and General Relativity correction. We first show that starting from the
present-day location of GJ 436b inevitably leads to damping the Kozai
oscillations and to rapidly circularizing the planet. Conversely, starting from
5-10 times further away allows the onset of Kozai cycles. The tides act in peak
eccentricity phases and reduce the semi-major axis of the planet. The net
result is an evolution characterized by two phases: a first one with Kozai
cycles and a slowly shrinking semi-major axis, and a second one once the planet
gets out of the Kozai resonance characterized by a more rapid decrease. The
timescale of this process appears in most cases much longer than the standard
circularization time of the planet by a factor larger than 50. This model can
provide a solution to the eccentricity paradox of GJ 436b. Depending on the
various orbital configurations, it can take several Gyrs to GJ 436b to achieve
a full orbital decrease and circularization. According to this scenario, we
could be witnessing today the second phase of the scenario where the semi-major
axis is already reduced while the eccentricity is still significant. We then
explore the parameter space and derive in which conditions this model can be
realistic given the age of the system. This yields constraints on the
characteristics of the putative companion.
