  The paper is concerned with a class of two-sided stochastic processes of the
form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data
at time zero and $A\equiv A(W)$ is a function of $W$. Elements of the related
stochastic calculus are introduced. In particular, the calculus is adjusted to
the case when $A$ is a jump process. Absolute continuity of $(X,P_{\sbnu})$
under time shift of trajectories is investigated. For example under various
conditions on the initial density with respect to the Lebesgue measure,
$m=d\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*}
\frac{P_{\sbnu}(dX_{\cdot -t})}{P_{\sbnu}(dX_\cdot)}=\frac{m(X_{-t})}
{m(X_0)}\cdot\prod_i|\nabla_{W_0}X_{-t}|_i {eqnarray*} % a.e. where the product
is taken over all coordinates. Here $\sum_i(\nabla_{W_0}X_{-t})_i$ is the
divergence of $X_{-t}$ with respect to the initial position. Crucial for this
is the {\it temporal homogeneity} in the sense that $X(W_{\cdot
+v}+A_v\1)=X_{\cdot+v}(W)$, $v\in {\Bbb R}$, where $A_v\1$ is the trajectory
taking the constant value $A_v (W)$.
  By means of such a density, partial integration relative to the generator of
the process $X$ is established. Relative compactness of sequences of such
processes is established.
