Deviations of the intersection of Brownian Motions in dimension four with general kernel

In this paper, we find a natural four dimensional analog of the moderate deviation results of Chen (2004) for the mutual intersection of two independent Brownian motions $B$ and $B^{\mathrm{\prime }}$. In this work, we focus on understanding the following quantity, for a specific family of kernels $H$, \begin{equation*} \int_0^1 \int_0^1 H (B_s - B’t) $\text{d}$t $\text{d}$s . \end{equation*} Given $H(z)\propto \frac{1}{\mathrm{\mid }z{\mathrm{\mid }}^{\gamma }}$ with $0<\gamma \le 2$, we find that the deviation statistics of the above quantity can be related to the following family of inequalities from analysis, \begin{equation} \label{eq:maxineq} \inf{f: |\nabla f|{L^2}<\infty} \frac{|f|^{(1-γ/4)}{L^2} |\nabla f|^{γ/4}{L^2}}{ [\int{($\mathbb{R}$^4)^2} f^2(x) H(x-y) f^2(y) $\text{d}$x $\text{d}$y]^{1/4}}. \end{equation} Furthermore, in the case that $H$ is the Green’s function, the above will correspond to the generalized Gagliardo-Nirenberg inequality; this is used to analyze the Hartree equation in the field of partial differential equations. Thus, in this paper, we find a new and deep link between the statistics of the Brownian motion and a family of relevant inequalities in analysis.