We solve a selection problem for multidimensional SDE dX(t) = a(X𝜖(t)) dt + 𝜖σ(X𝜖(t)) dW(t), where the drift and diffusion are locally Lipschitz continuous outside a fixed hyperplane H. It is assumed that X𝜖 (0) = x0 𝜖 H, the drift a(x) has a Hölder asymptotics as x approaches H, and the limit ODE dX(t) = a(X(t)) dt does not have a unique solution. It is shown that if the drift pushes the solution away from H, then the limit process with certain probabilities selects some extreme solutions of the limit ODE. If the drift attracts the solution to H, then the limit process satisfies an ODE with certain averaged coefficients. To prove the last result, we formulate an averaging principle, which is quite general and new.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 9, pp. 1254–1285, September, 2020. Ukrainian DOI: 10.37863/umzh.v72i9.6292.
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Kulik, A., Pilipenko, A. On Regularization by a Small Noise of Multidimensional Odes with Non-Lipschitz Coefficients. Ukr Math J 72, 1445–1481 (2021). https://doi.org/10.1007/s11253-021-01865-7
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DOI: https://doi.org/10.1007/s11253-021-01865-7