eBook ISBN:  9781470441371 
Product Code:  MEMO/249/1185.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470441371 
Product Code:  MEMO/249/1185.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 249; 2017; 135 ppMSC: Primary 60
The text is concerned with a class of twosided stochastic processes of the form \(X=W+A\). Here \(W\) is a twosided 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

Chapters

1. Introduction, Basic Objects, and Main Result

2. Flows and Logarithmic Derivative Relative to $X$ under Orthogonal Projection

3. The Density Formula

4. Partial Integration

5. Relative Compactness of Particle Systems

A. Basic Malliavin Calculus for Brownian Motion with Random Initial Data


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The text is concerned with a class of twosided stochastic processes of the form \(X=W+A\). Here \(W\) is a twosided 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.

Chapters

1. Introduction, Basic Objects, and Main Result

2. Flows and Logarithmic Derivative Relative to $X$ under Orthogonal Projection

3. The Density Formula

4. Partial Integration

5. Relative Compactness of Particle Systems

A. Basic Malliavin Calculus for Brownian Motion with Random Initial Data