The stochastic two-body problem with dissipation

Abstract We consider the stochastic two-body problem, which has been introduced in Sharma and Parthasarathy (2007) to model the effect of randomness on the orbital motion of a particle in a stochastically fluctuating dust cloud. We give the equations of motion under the additional effect of dissipative, Stokes-type drag forces. It is known that the stochastic two-body problem does not admit the energy integral, even in the weak sense; on the other hand, the dissipative two-body problem by assumption does not preserve the energy. Motivated by this remark, the main problem addressed in this work consists in determining whether there exist parameter values and/or initial conditions for which the combined influence of noise and dissipation justifies the conclusion that energy and/or angular momentum behave as weak integrals of motion, namely their expectations are preserved over time. In this context, we provide an explicit condition on the parameters and the initial data under which the energy is a weak integral. We also show that a modified angular momentum that incorporates dissipative effects is a weak integral. The analytical results are further substantiated by numerical validation on sample cases.