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This question (continuation of Probability of a particle surviving forever) is related to the supercooled Stefan problem (models for the evolution of the interface between two phases of a substance undergoing a phase transition) :

$$Y_t = y + t + W_t - \alpha\big(1-\Lambda_t\big),\quad \mbox{for all } t\ge 0,$$

where $y>0$, $(W_t)_{t\ge 0}$ is a standard Brownian motion,

$$\Lambda_t:=\mathbb P(\tau>t)\quad \mbox{and}\quad \tau:=\inf\{t\ge 0: Y_t\le 0\}.$$

Is there anyway to compute the probability $\Lambda_{\infty}=\mathbb P(Y_t>0, \forall t\ge 0)$?

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