Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\varepsilon ,\mathcal{D}(\varepsilon ))$ on ${{L}^{2}}(E;m)$. For $u\,\in \,\mathcal{D}{{(\varepsilon )}_{e}}$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\hat{X}$ be the dual process of $X$ and $\hat{Y}$ the Girsanov transformed process of $\hat{X} $. We give a necessary and sufficient condition for $(Y,\hat{Y})$ to be in duality with respect to the measure ${{e}^{2u}}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\hat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.