Large deviation of diffusion processes with discontinuous drift and their occupation times

Abstract

For the system of $d$-dim stochastic differential equations, dX^{\varepsilon} (t) = b(X^{\varepsilon}(t)) dt + \varepsilon dW(t), \quad t \in [0, 1] X^{\varepsilon} (0) = x^0 \in R^d where $b$ is smooth except possibly along the hyperplane $x_1 = 0$, we shall consider the large deviation principle for the lawof the solution diffusion process and its occupation time as $\varepsilon\rightarrow0$. In other words, we consider $P(\|X^\varepsilon-\varphi\|<\delta,\|u^{\varepsilon}-\psi\|\<\delta)$ where $u^\varepsilon(t)$ and $\psi(t)$ are the occupation times of $X^\varepsilon$ and $\varphi$ in the positive half space $\{x\in R^d: x_1>0\}$, respectively. As a consequence, an unified approach of the lower level large deviation principle for the law of $X^\varepsilon(\cdot)$ can be obtained.