A Penalization Limit Theorem for the Boundary Local Time of a Reflected Diffusion

A. Bencherif, N. Boufelgha

2023, v.29, Issue 3, 347-365

ABSTRACT

Let $X$ be a diffusion, with drift $b$ and matrix $\sigma$, reflecting inside a (not necessarily convex) smooth bounded domain $D\subset\mathbb{R}^{d}$, $d\geq\: 1$, in the direction of the normal unit vector field $\gamma$, with boundary local time $L(t)$. Let $S(x)=\sum_{j}(\gamma(x),\sigma_{\cdot j}(x))^{2}$ near the boundary (and suitably defined elsewhere) and $\beta(x)$ a vector function which vanishes in $D$ and is equal to $x-\pi(x)$ near the boundary $\Gamma$, $\pi(x)$ is the unique projection of $x$ on the (close-by) boundary and let
\begin{equation*}
X^{n}_{t}=x+\int_{0}^{t}b(X^{n}_{s})ds
+\int_{0}^{t}\sigma (X^{n}_{s})dB_{s}-n\int_{0}^{t}S\beta(
X^{n}_s)ds
\end{equation*}
be a penalized diffusion that approximates $X$; it is proved that under the non-degeneracy of $S$ we have in probability as $n\rightarrow\infty$
\begin{equation*}
\sqrt{n}\int_{0}^{t}\mathbb{I}_{D^{c}}(X^{n}_{s})S(X^{n}_{s})ds\rightarrow \sqrt{\pi}L(t)
\end{equation*}
where $\mathbb{I}_{\cdot}$ is the indicator function.

Keywords: Reflecting diffusion, Boundary local time, Limit theorem