Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences

In this article, we consider the Benes process with drift $\mu(x)=\alpha \tanh(\alpha x + \beta)$, with $\alpha > 0$, $\beta \in \mathbb{R}$, that is, the diffusion defined by the stochastic differential equation $dX(t)=\alpha \tanh(\alpha X(t)+\beta)\,dt + dW(t)$, with an absorbing barrier at $x=a$. After deriving the propagator and key associated quantities--the first-passage-time distribution and the survival probability--we then condition this process to have various prescribed first-passage-time distributions. When the conditioning is imposed at an infinite time horizon, this procedure reveals the existence of different processes that share the same first-passage-time distribution as the Benes process, a phenomenon recently observed in the case of Brownian motion with drift. When the conditioning is imposed at a finite time horizon, the procedure shows that the conditioned Benes process and the Brownian motion with drift under the same conditioning exhibit identical behaviors. This strengthens an elegant result of Benjamini and Lee stating that Brownian motion and the Benes process share the same Brownian bridge, and it also connects with more recent findings obtained by conditioning two independent identical Brownian motions with drift, or two independent Benes processes that annihilate upon meeting. Moreover, we show that several conditioned Benes drifts converge near the absorbing boundary to the drift of the taboo diffusion, i.e., the diffusion conditioned to never reach the absorbing boundary, which motivates a parallel analysis of the taboo process itself. Using Girsanov's theorem, we derive its propagator, first-passage-time distribution, and conditioned versions, thereby further clarifying the structural relationships between Benes, Brownian, and taboo dynamics.