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2018 Existence and uniqueness of reflecting diffusions in cusps
Cristina Costantini, Thomas G. Kurtz
Electron. J. Probab. 23(none): 1-21 (2018). DOI: 10.1214/18-EJP204


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.


Download Citation

Cristina Costantini. Thomas G. Kurtz. "Existence and uniqueness of reflecting diffusions in cusps." Electron. J. Probab. 23 1 - 21, 2018.


Received: 17 November 2017; Accepted: 24 July 2018; Published: 2018
First available in Project Euclid: 12 September 2018

zbMATH: 06964778
MathSciNet: MR3858912
Digital Object Identifier: 10.1214/18-EJP204

Primary: 60H10, 60J60
Secondary: 60G17, 60J55


Vol.23 • 2018
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