Transformation from a Wiener process to another stochastic process

Question

Consider a functional of Wiener process $W_{t}$, for example $$ Z_{t}\equiv F[W_{t}]=\sin[W_{t}], $$ how to calculate the probability measure of this new process. In other words, is there a simple formula for the transformation between random processes like the transformation between random variables $$ P(X){\rm d}X=P(X)\dfrac{{\rm d}X}{{\rm d}Y}{\rm d}Y\equiv \hat{P}(Y){\rm d}Y, $$ where $X$ and $Y$ are random variables. If you could provide me with specific steps and references, I would greatly appreciate it.