## Electronic Journal of Probability

### 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 $\{(x_1,x_2):0<x_1\leq \delta _0,\psi _1(x_1)<x_2<\psi _ 2(x_1)\}$, with $\psi _1(0)=\psi _2(0)=0$, $\psi _1'(0)=\psi _2'(0)=0$.

Given a vector field $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 84, 21 pp.

Dates
Accepted: 24 July 2018
First available in Project Euclid: 12 September 2018

https://projecteuclid.org/euclid.ejp/1536717743

Digital Object Identifier
doi:10.1214/18-EJP204

Mathematical Reviews number (MathSciNet)
MR3858912

Zentralblatt MATH identifier
06964778

#### Citation

Costantini, Cristina; Kurtz, Thomas G. Existence and uniqueness of reflecting diffusions in cusps. Electron. J. Probab. 23 (2018), paper no. 84, 21 pp. doi:10.1214/18-EJP204. https://projecteuclid.org/euclid.ejp/1536717743

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