Let’s just cut to the chase and say that You’re Welcome is the correct way to write it. You’re Welcome is a contraction, or a combination of two words, You and Are, thus making You’re Welcome the shortened version of You Are Welcome.

Saying Your Welcome would imply that the welcome belonged to you or somebody, which could potentially make sense in the following sentence: You were offensive last night and wore out your welcome at their house. In this case, it obviously has an entirely different meaning than You Are Welcome or You’re Welcome.

WikiHow.com has a nice little article on How to Use Apostrophes, here’s a brief exerpt:

Use apostrophes in contractions. Sometimes, especially in informal writing, apostrophes are used to indicate one or more missing letters. For example, the word “don’t” is short for “do not”; other examples include “isn’t,” “wouldn’t,” and “can’t.” Contractions can also be made with the verbs “is,” “has,” and “have.” For example, we can write “She’s going to school” instead of “She is going to school”; or “He’s lost the game” instead of “He has lost the game.” A similar usage can be found in the notation of calendar years, as in ‘07. In this case, the apostrophe appears in the spot where the missing numbers would have been (before the number, not after as in 07′).