Existence and uniqueness of reflecting diffusions in cusps

We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i.e. in a neighborhood of the origin has the form {(x1, x2): 0 < x1 ≤ δ0, ψ1(x1) < x2 < ψ2(x1)}, with ψ1(0) = ψ2(0) = 0, ψ1(0)′ = ψ2′(0) = 0. Given a vector field g of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin gi (0):= limx1→0+ g(x1, ψi(x1)), i = 1, 2, and assuming there exists a vector e∗ such that 〈e∗, gi (0)〉 > 0, i = 1, 2, and e∗ 1 > 0, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin. Our proof uses a new scaling result and a coupling argument. © 2018, University of Washington. All rights reserved.