# What is the distribution of the risk-free asset?

If the risk-free asset has a volatility of $$0$$, therefore making its mean equal to the risk-free rate, $$r_f$$, does this mean that it has no probability distribution, and therefore there is no reason to model it parametrically (i.e. with $$\mathcal{N}(\cdot)$$ or other)?

How does this situation change when we drop the usual assumption of a constant $$r_f$$, since, empirically, central banks actually make it time-varying?

The standard way to think about this is that at time $$t$$ the riskless asset gives you known return of $$r_{f,t}$$ over a short time period. However, this rate may itself be time-varying and stochastic so that we don't know its futures values, say $$r_{f,t+s}$$. E .g. a common assumption is that the rate follows an Ornstein Uhlenbeck process (implying that the conditional distribution is normal).
In case you assume that $$r_{f,t}$$ is actually constant, say $$c$$, it still has a probability distribution. Here you need to define $$r_{f,t}$$ as a random variable that takes the value $$c$$ for all outcomes of the sample space. Naturally the distribution of $$r_{f,t}$$ is then such that all the probability mass lies at this single point: $$P(r_{f,t}=c)=1$$.
$$dB_t =r_t B_t \; dt, \; B_0 =1,$$ or, equivalently, $$B_t = \exp \left( \int_0^t r_u\; du \right),$$
where $$r_t$$ is a stochastic progressively measurable process with locally integrable paths, making $$B_t$$ a finite variation process with null quadratic variation. Intuitively, this means that it has a smaller degree of randomness with respect to the other risky assets. (In FX or equity option pricing, based on such intuition, one even assumes time-dependent deterministic interest rate.)