The rapid advancement of Large Language Models (LLMs) has revolutionized the generation of emotional support conversations (ESC), offering scalable solutions with reduced costs and enhanced data privacy. This paper explores the role of personas in the creation of ESC by LLMs. Our research utilizes established psychological frameworks to measure and infuse persona traits into LLMs, which then generate dialogues in the emotional support scenario. We conduct extensive evaluations to understand the stability of persona traits in dialogues, examining shifts in traits post-generation and their impact on dialogue quality and strategy distribution. Experimental results reveal several notable findings: 1) LLMs can infer core persona traits, 2) subtle shifts in emotionality and extraversion occur, influencing the dialogue dynamics, and 3) the application of persona traits modifies the distribution of emotional support strategies, enhancing the relevance and empathetic quality of the responses. These findings highlight the potential of persona-driven LLMs in crafting more personalized, empathetic, and effective emotional support dialogues, which has significant implications for the future design of AI-driven emotional support systems.