Not sure about Europe, but in the 11th century, what is now considered "China" was split between the Song Dynasty, the Liao/Khitan Empire, and the Xi Xia or Western Xia Dynasty, with the Song Dynasty being the dominant force of the three.

Within the Song Dynasty in particular, they basically adopted the preexisting Tang Dynasty's legal code and used it as their own with minor changes. Primogeniture, or all the property passing down to the eldest son of the deceased, was illegal. This was initially made illegal in the Tang Dynasty to weaken the power of the aristocratic northern clans, thereby preventing the rise of a powerful landed noble class. Instead, equal distribution of the deceased's property between all his children was practiced - something similar to gavelkind succession, from a western perspective.

While this isn't explicitly regarding peasants - but it probably wasn't exclusive to the aristocratic class either. One could surmise that the more well-off peasants would be subject to this law of succession by authorities, while lesser or poorer families would be ignored and could split their property as they pleased or as dictated by their late parents, as per the recurring problem of China not really being able to assert centralized control over its entire expanse due to the great size of its territory.

Source: John K. Fairbank and Merle Goldman (2006), "China: A New History".