Integral in Gardiner's book on stochastic methods

[eoghan]

Dear all,
I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral
$\sum_i\int d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)]$
where $p_1$ and $p_2$ are two solutions of the Chapman-Kolmogorov equation and $\vec A$ is a function of $\vec x$. Then Gardiner says, suppose that we take $p_2$ as a stationary distribution $p_s(\vec x)$ which is nonzero everywhere, except at infinity, where it and its first derivative vanish. The integral can be integrated to give surface terms which vanish at infinity.
I don't know how to prove this! I used the Gauss theorem to obtain:
$\sum_i\int_D d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)]=<br /> -\int_{D} d\vec x \nabla[\vec A p_1\log(p_1/p_2)]=<br /> -\int_{\partial D} dS \:\:\hat n\cdot[\vec A p_1\log(p_1/p_2)]$

and this is a surface term, where the surface extends to infinity. Now I should conclude that $p_1\log(p_1/p_2)$ is zero at infinity, but I don't know how to proof that. I mean, I only know that $p_2$ is zero at infinity and this would make the integral to diverge! Maybe I can say that since $p_1$ it's solution to the Chapman-Kolmogorov equation, it is itself a distribution and so also $p_1$ vanishes at infinity, but I'm not sure about this.

 

[Stephen Tashi]

Dear all,
I have troubles in one proof of the book Handbook of stochastic methods by Gardiner.

Not that I can answer your question, but which edition of Handbook of Stochastic Methods is involved? There is a "section" 3.7.3 in the second edition, but I don't see the equation you mention.

 

[eoghan]

Not that I can answer your question, but which edition of Handbook of Stochastic Methods is involved? There is a "section" 3.7.3 in the second edition, but I don't see the equation you mention.

Hi Stephen!
The book is the third edition. Chapter 3 = Markov Processes, section 3.7=Stationary and Homogeneous Markov Processes, subsection 3.7.3=Approach to a Stationary Process