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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

[itex]\sum_i\int d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)][/itex]

where [itex]p_1[/itex] and [itex]p_2[/itex] are two solutions of the Chapman-Kolmogorov equation and [itex]\vec A[/itex] is a function of [itex]\vec x[/itex]. Then Gardiner says, suppose that we take [itex]p_2[/itex] as a stationary distribution [itex]p_s(\vec x)[/itex] 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:

[itex]\sum_i\int_D d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)]=

-\int_{D} d\vec x \nabla[\vec A p_1\log(p_1/p_2)]=

-\int_{\partial D} dS \:\:\hat n\cdot[\vec A p_1\log(p_1/p_2)][/itex]

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

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# Integral in Gardiner's book on stochastic methods

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