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.