On a selection problem for small noise perturbation of unstable dynamical systems (Andrey Pilipenko,Kyiv)

19.03.2024 14:00

The well-known Peano existence theorem states that an ordinary differential equation (ODE) with continuous coefficients has a local solution, which, however, may not be unique. On the contrary, an addition of a non-degenerate noise term usually yields the existence of a unique solution to the corresponding stochastic differential equation, even if the drift term is discontinuous. We study the limit of the stochastic equations as the noise intensity converges to zero. This limit can be interpreted as a natural selection of a solution for the initial ODE.
The identification of the limit is closely related to a study of the exact growth rate of certain stochastic equations, as well as to an averaging principle.

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