**Memoirs of the American Mathematical Society**

2017;
135 pp;
Softcover

MSC: Primary 60;

Print ISBN: 978-1-4704-2603-3

Product Code: MEMO/249/1185

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-4137-1
Product Code: MEMO/249/1185.E**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

# Absolute Continuity Under Time Shift of Trajectories and Related Stochastic Calculus

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*Jörg-Uwe Löbus*

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.

#### Table of Contents

# Table of Contents

## Absolute Continuity Under Time Shift of Trajectories and Related Stochastic Calculus

- Cover Cover11
- Title page i2
- Chapter 1. Introduction, Basic Objects, and Main Result 18
- Chapter 2. Flows and Logarithmic Derivative Relative to 𝑋 under Orthogonal Projection 2532
- Chapter 3. The Density Formula 4552
- Chapter 4. Partial Integration 105112
- Chapter 5. Relative Compactness of Particle Systems 115122
- Appendix A. Basic Malliavin Calculus for Brownian Motion with Random Initial Data 125132
- References 133140
- Index 135142
- Back Cover Back Cover1148