The generalized Schrödinger–Langevin equation

• We generalize the Kostin equation for arbitrary system–bath coupling.
• This generalization is developed both in the Schrödinger and Bohmian formalisms.
• We write the generalized Kostin equation for two measurement problems.
• We reformulate the generalized uncertainty principle in terms of this equation.

In this work, for a Brownian particle interacting with a heat bath, we derive a generalization of the so-called Schrödinger–Langevin or Kostin equation. This generalization is based on a nonlinear interaction model providing a state-dependent dissipation process exhibiting multiplicative noise. Two straightforward applications to the measurement process are then analyzed, continuous and weak measurements in terms of the quantum Bohmian trajectory formalism. Finally, it is also shown that the generalized uncertainty principle, which appears in some approaches to quantum gravity, can be expressed in terms of this generalized equation.