Brownian Motion with a Singular Drift
We study the effect of a power law drift on Brownian motion in the positive half-line, where the order of the drift at 0 and infinity is different.
math.PR↗
arXiv subjects
Robert G. Smits
Publications and source records attributed to Robert G. Smits.
Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions
Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$
math.CA↗
The Expected Time to End the Tug-of-War in a Wedge
Using a solution of a nonhomogeneous partial differential equation involving the p- Laplacian, we study the finiteness of the expected time to end the tug-of-war in a wedge.
math.PR↗