How I can find the weak solution and transition distribution of Orstein-Uhlenbeck Process using Girsanov's Theorem. More specifically, suppose that $$ dx_t = -\lambda \:x_t\:dt + \sigma\:d\beta_t, $$ where $\beta_t$ is a wiener process. I want to compute the transition probability $P(x_t|x_s),~t\geq s$ based on the Girsanov's theorem.
As such, I define another process $y_t$ with $dy_t=d\beta_t$, and apply the Girsanov's theorem. I then need to compute the likelihood ratio: $$ z(t)= \exp\bigg(\int_0^t (y_0+\beta_t)d\beta_t - \frac12\int_0^t(y_0+\beta_t)^2dt \bigg) $$ I can compute the integral $\int_0^t (y_0+\beta_t)d\beta_t$, but cannot compute $\int_0^t(y_0+\beta_t)^2dt$. As such, I cannot simplify $z(t)$. If I were able to have $z(t)$, then I could compute $P(x_t)$ using $P(x_t) = z(t)P(y_t)$ with $P(y_t)$ following a Normal distribution.