Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{P}(\forall t\in[0,T],W_t\geqslant x)=1-e^{\frac{2(x-a)(b-x)}{T}} $$ (see Probability of general Brownian (or non) bridge to be higher than given parameter? for a proof). My question is : Is there a generalization of this result for $d$-dimensional Brownian bridges ? That is, if $(W_t)_{t\in[0,T]}$ is a $d$-dimensional Brownian bridge such that $W_0=a\in\mathbb{R}^d$ and $W_T=b\in\mathbb{R}^d$, what is the probability that for all $t\in [0,T],\|W_t\|\geqslant x$ where $x>0$, for a convenient norm $\|\cdot\|$ whether it is $\|\cdot\|_2$, $\|\cdot\|_{\infty}$ or any norm that makes it possible to compute/approximate.
In the case where $\|\cdot\|$ is the euclidean norm of $\mathbb{R}^d$, $t\mapsto\|W_t\|$ is a Bessel process which is well known but I can't find any related paper tackling the above probability.
