Answer
Your own captured Pokémon will not run away. The only way to lose it is transferring it by accident. Your journal is showing an instance of a wild Ponyta fleeing from battle, which can happen randomly if you fail to capture it too many times, probably more commonly with stronger Pokémon.
A good way to ensure a capture is using better Pokéballs and Razz Berries, but keep in mind that capture attempts can still fail and Pokémon can still flee even when using those items.
Attribution
Source : Link , Question Author : Dawny33 , Answer Author : fcm