Pathwise non-uniqueness for Brownian motion in a quadrant with oblique reflection
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| Format: | Preprint |
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2024
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| _version_ | 1866911872773521408 |
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| author | Bass, Richard F. Burdzy, Krzysztof |
| author_facet | Bass, Richard F. Burdzy, Krzysztof |
| contents | Consider the Skorokhod equation in the closed first quadrant: \[ X_t=x_0+ B_t+\int_0^t{\bf v}(X_s)\, dL_s,\] where $B_t$ is standard 2-dimensional Brownian motion, $X_t$ takes values in the quadrant for all $t$, and $L_t$ is a process that starts at 0, is non-decreasing and continuous, and increases only at those times when $X_t$ is on the boundary of the quadrant. Suppose ${\bf v}$ equals $(-a_1,1)$ on the positive $x$ axis, equals $(1,-a_2)$ on the positive $y$ axis, and ${\bf v}(0)$ points into the closed first quadrant. Let $θ_i=\arctan a_i$, $i=1,2$. It is known that there exists a solution to the Skorokhod equation for all $t\geq 0$ if and only if $θ_1+θ_2<π/2$ and moreover the solution is unique if $|a_1a_2|<1$.
Suppose now that $θ_1+θ_2<π/2$, $θ_2<0$, $θ_1>-θ_2>0$ and $|a_1a_2|>1$. We prove that for a large class of $(a_1,a_2)$, namely those for which \[\frac{\log|a_1|+\log|a_2|}{a_1+a_2}>π/2,\] pathwise uniqueness for the Skorokhod equation fails to hold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pathwise non-uniqueness for Brownian motion in a quadrant with oblique reflection Bass, Richard F. Burdzy, Krzysztof Probability 60J65 Consider the Skorokhod equation in the closed first quadrant: \[ X_t=x_0+ B_t+\int_0^t{\bf v}(X_s)\, dL_s,\] where $B_t$ is standard 2-dimensional Brownian motion, $X_t$ takes values in the quadrant for all $t$, and $L_t$ is a process that starts at 0, is non-decreasing and continuous, and increases only at those times when $X_t$ is on the boundary of the quadrant. Suppose ${\bf v}$ equals $(-a_1,1)$ on the positive $x$ axis, equals $(1,-a_2)$ on the positive $y$ axis, and ${\bf v}(0)$ points into the closed first quadrant. Let $θ_i=\arctan a_i$, $i=1,2$. It is known that there exists a solution to the Skorokhod equation for all $t\geq 0$ if and only if $θ_1+θ_2<π/2$ and moreover the solution is unique if $|a_1a_2|<1$. Suppose now that $θ_1+θ_2<π/2$, $θ_2<0$, $θ_1>-θ_2>0$ and $|a_1a_2|>1$. We prove that for a large class of $(a_1,a_2)$, namely those for which \[\frac{\log|a_1|+\log|a_2|}{a_1+a_2}>π/2,\] pathwise uniqueness for the Skorokhod equation fails to hold. |
| title | Pathwise non-uniqueness for Brownian motion in a quadrant with oblique reflection |
| topic | Probability 60J65 |
| url | https://arxiv.org/abs/2405.06144 |