We solve a selection problem for multidimensional SDE dX(t) = a(X^𝜖(t)) dt + 𝜖σ(X^𝜖(t)) dW(t), where the drift and diffusion are locally Lipschitz continuous outside a fixed hyperplane H. It is assumed that X^𝜖 (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.

我们解决了多维SDE dX（t）= a（X ^𝜖（t））dt + 𝜖σ（X ^𝜖（t））dW（t）的选择问题，其中漂移和扩散在固定超平面外局部为Lipschitz连续H.假设X ^ε（0）= X ⁰ ε H，漂移一个（X）具有保持作为渐近X接近H，极限ODE dX（t）= a（X（t））dt没有唯一解。结果表明，如果漂移使解远离H，则具有一定概率的极限过程会选择极限ODE的一些极限解。如果漂移将解吸引到H，则极限过程满足具有某些平均系数的ODE。为了证明最后的结果，我们制定了一个平均的原理，这是相当普遍和新颖的。