This is a very interesting question to consider. The heart of this search for identity lies with Hester's refusal to let others determine her fate and her own self. She has been free to leave her town and take up another life elsewhere where she might live her life free from the social stigma of the scarlet letter, yet she refuses to do so. For Hester, were she to run away, she would be admitting that society is a force that has power over her and controls her. Such an action would indicate that the scarlet letter she bears is something that she wants to flee from. Hester's decision to stay in the town allows her to slowly but surely change the meaning of the symbol of the scarlet letter, imbuing it with her own character and actions and also recognising that the letter is a symbol both of what she is becoming and what she once was. To deny the sin that first caused her to wear it would be to deny part of the complex humanity that sums up her own character. Hester's determination to wear the scarlet letter therefore indicates a determination to fix her own identity rather than have it fixed for her by society. Note how she is successful in changing the way that the scarlet letter is viewed:
...the scarlet letter ceased to be a stigma which attracted the world's scorn and bitterness, and became a type of something to be sorrowed over, and looked upon with awe, and yet reverence too.
This quote, above all else perhaps, shows the success of Hester's search for her own identity. She steadfastly refuses to bow to how others perceive her and bravely struggles for the right to determine who she is herself.