Absolute continuity under time shift of trajectories and related stochastic calculus

The text 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) under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, m, and on A with A_0=0 we verify \frac{P(dX_{\cdot -t})}{P(dX_\cdot)}=\frac{m(X_{-t})}{m(X_0)}\cdot \prod_i\left|\nabla_{d,W_0}X_{-t}\right|_i i.e. where the product is taken over all coordinates. Here \sum_i \left(\nabla_{d,W_0}X_{-t}\right)_i is the divergence of X_{-t} with respect to the initial position. Crucial for this is the temporal homogeneity of X in the sense that X\left(W_{\cdot +v}+A_v \mathbf{1}\right)=X_{\cdot+v}(W), v\in {\mathbb R}, where A_v \mathbf{1} is the trajectory taking the constant value A_v(W). By means of such a density, partial integration relative to a generator type operator of the process X is established. Relative compactness of sequences of such processes is established.