A little mistake in your thesis. In Equation(5.25), the sign before (x-x') term should be plus. Also the Equation(5.11), I get the mean value 1/2*(2*log(sigma/omega)-log(2)-gamma), It is the same as yours except for the 1/2 term in the front. If it is true, the variance should be pi^2/8. Would you like to check your answer? I can send my code to you and see if it goes wrong somewhere. I would be happy to discuss with you in any details.
A little mistake in your thesis. In Equation(5.25), the sign before (x-x') term should be plus. Also the Equation(5.11), I get the mean value 1/2*(2*log(sigma/omega)-log(2)-gamma), It is the same as yours except for the 1/2 term in the front. If it is true, the variance should be pi^2/8. Would you like to check your answer? I can send my code to you and see if it goes wrong somewhere. I would be happy to discuss with you in any details.