Existence and uniqueness of reflecting diffusions in cusps

Abstract

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 $\{(x_1,x_2):0<x_1\le {\delta }_0,{\psi }_1(x_1)<x_2<{\psi }_2(x_1)\}$, with ${\psi }_1(0)={\psi }_2(0)=0$, ${\psi }_1^{\prime }(0)={\psi }_2^{\prime }(0)=0$. Given a vector field $\gamma$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin ${\gamma }^i(0):={\lim }_{x_1\to 0^{+}}\gamma (x_1,{\psi }_i(x_1))$, $i=1,2,$ and assuming there exists a vector $e^{\ast }$ such that $⟨e^{\ast },{\gamma }^i(0)⟩>0$, $i=1,2$, and $e_1^{\ast }>0$, 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.

References in corpus (1)

• Viscosity methods giving uniqueness for martingale problems

Cited by in corpus (4)

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