Brownian bridge $Z_t$ is a diffusion process distributed as Brownian motion $B_t$ conditioned on the event $B_1 = 0$. It is rather well-studied, and allows for a Markov-like SDE representation. I wonder whether such representations are known for Brownian motion conditioned on $B_1\in \{x_1,\dots,x_n\}$. Even if this is the case, are the probabilities $P(Z_1 = x_i)$ fixed?
A similar question from a different perspective: if I would like to construct a continuous Markovian diffusion $Z_t$ with a property $Z_1 \in \{x_1,\dots,x_n\}$, are there known examples of such processes in the literature? I am particularly interested in the case $n = 2$.
One representation of $B_t$ knowing that $B_1$ can take value $x_0$ with probability $p$ and $x_1$ with probaiblity $1-p$ is the following.
Take 2 Brownian Bridges $B_1(t)=B_t^{0\to x_0}$ and $B_2(t)=B_t^{0\to x_1}$ that take respectively the value $x_0$ and $x_1$ at time 1 and a Bernoulli random variable $\beta$ with parameter $p$, then the process you are looking for is the following :
$$B^{x_0,x_1}(t)= \beta.B_1(t)+(1-\beta).B_2(t)$$
with $\beta$ being $\mathcal{F}_0$-measurable (i.e. measurable for the initial $\sigma$- algebra).
In a way you can view this process as a $\mathcal{F}_0$-measurable mixture of Brownian Bridges.
You can add refinements for example the time where the final value is decided could be dependent on a stopping time, this would only add a step for the Bernoulli random variable and the two Bridges at that time.
Best regards.