Absolute Continuity under Time Shift and Related Stochastic Calculus

The talk 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 role of finite dimensional projections arising from the L\'{e}vy-Ciesielski representation of the Brownian motion are discussed. The calculus is adjusted to the case when $A$ is a jump process.

Absolute continuity of $(X,P_{\nu \nu })$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\nu \nu /dx$, and on $A$ with $A_0=0$ we verify

\begin{equation*} \frac{P_{\scriptscriptstyle\pmb{\nu}}(dX_{\cdot -t})}{P_{\scriptscriptstyle an \pmb{\nu}}(dX_\cdot)}=\frac{m(X_{-t})}{m(X_0)}\cdot\prod_i\left|\nabla_{W_0} X_{-t}\right|_i \end{equation*} a.e. where the product is taken over all coordinates. Here $\sum_i\left(\nabla_{W_0}X_{-t}\right)_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the {\it temporal homogeneity} in the sense that $X\left(W_{\cdot +v}+A_v{\, {\rm I}\mskip-10mu 1}\right) =X_{\cdot+v}(W)$, $v\in {\mathbb R}$, where $A_v{\, {\rm I}\mskip-10mu 1}$ is the trajectory taking the constant value $A_v(W)$.

By means of a such a density, partial integration relative to the generator of the process $X$ is established. A further application is relative compactness of sequences of processes of the form $X^n=W+A^n$, $n\in {\mathbb N}$.

References
J.-U. Löbus (2017). Absolute Continuity under Time Shift of Trajectories and Stochastic Calculus, Memoirs of the American Mathematical Society, 249, No. 1185 (2017).