= Solution
The <Brownian motion> displacement has a <normal distribution> with <variance> $\Gamma t$. Its transition <probability density function> is therefore
$$
p(y_f,t\mid y_i,0)
=\frac1{\sqrt{2\pi\Gamma t}}
\exp\!\left[-\frac{(y_f-y_i)^2}{2\Gamma t}\right].
$$
Comparison with the <free particle> <heat kernel> obtained in part (c) gives the required matching:
$$
\boxed{\Gamma t=\frac{\beta\hbar^2}{m}.}
$$
This matches whole path distributions after a linear reparametrization, not merely endpoint <variances>. To see this, divide the Brownian duration $t$ into $M$ slices of size $\delta t$. The <independent increments> have <normal distributions> with <variance> $\Gamma\delta t$, so their joint density contributes
$$
\prod_{j=1}^M\frac1{\sqrt{2\pi\Gamma\delta t}}
\exp\!\left[-\sum_{j=1}^M\frac{(y_j-y_{j-1})^2}{2\Gamma\delta t}\right].
$$
For the <Euclidean path integral>, the corresponding expression is
$$
\prod_{j=1}^M\sqrt{\frac{m}{2\pi\hbar\delta\tau}}
\exp\!\left[-\sum_{j=1}^M\frac{m(x_j-x_{j-1})^2}{2\hbar\delta\tau}\right],
\qquad \delta\tau=\frac{\beta\hbar}{M}.
$$
The matching condition makes $\Gamma\delta t=\hbar\delta\tau/m$, so both the exponents and all normalization factors coincide. In particular, identifying Brownian time with <imaginary time>, $t=\beta\hbar$, requires $\Gamma=\hbar/m$, or <diffusion coefficient> $D=\hbar/(2m)$.
With both endpoints fixed, the normalized path law is a \b[<Brownian bridge>]. The <Euclidean path integral> itself is the endpoint density times that conditional path law. In the normalized time coordinate $s=\tau/(\beta\hbar)$, its <expected value> and <covariance> are
$$
\mathbb E[x(s)]=x_i+s(x_f-x_i),
\qquad
\operatorname{Cov}(x(s),x(s'))
=\frac{\beta\hbar^2}{m}\bigl[\min(s,s')-ss'\bigr].
$$
The <classical path> is therefore the bridge's <expected value>, while individual <Brownian bridge> paths fluctuate about it. Typical <Brownian motion> paths are continuous but nowhere differentiable: the kinetic <Euclidean action> is interpreted through the <time-sliced diffusion path integral>, rather than by assuming that a typical path has an ordinary velocity.
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