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The Brownian transition semigroup identity proves that this positive density process has mean one for $t<1$. The backward heat equation and Itô's formula give $d{M}_t=M_t(y-B_t)(1-t{)}^{-1}d{B}_t$. On every fixed horizon $T<1$, Girsanov theorem under $dQ=M_{T}dP$ makes $B_t-{\int }_0^t(y-B_s)(1-s{)}^{-1}ds$ Brownian motion. Thus the original coordinate acquires the bridge drift coefficient toward $y$ at time one, without defining an equivalent endpoint density at the singular time.

The gradient bound makes $f$ a Lipschitz function, with $|f(x)|\le |f(0)|+K\Vert x\Vert$, so its value at a Gaussian random variable is integrable and square integrable. The future Brownian increment $X_1-X_t$ is independent of ${\mathcal{F}}_t$ and has distribution $N(0,(1-t)I_d)$. Conditioning on the current location therefore givesThis is the Markov property expressed through the Brownian transition semigroup. In particular $M_0=\mu$ and $M_1=f(X_1)$.

Let $u(t,x)=P_{1-t}f(x)$ for $t<1$. Differentiating the Brownian transition semigroup in the spatial variables is justified by the bounded gradient and givesThe heat equation for this semigroup is the backward equation ${\partial }_{t}u+\frac{1}{2}\Delta u=0$. Applying the multidimensional Itô formula yieldsThe finite-variation term vanishes. ThusInitially this calculation holds on $[0,1-\epsilon ]$. The vector of stochastic integral coefficients has norm at most $K$, so the Itô isometry lets the integrals extend to time one. Also $P_{1-t}f(X_t)\to f(X_1)$ in $L^2$ by the Lipschitz function bound and continuity of Brownian motion. The same representation therefore holds on the entire interval.

For $t>0$, differentiateThe growth assumptions justify differentiation under the expectation, and Gaussian integration by parts givesSince $u(0)=f(0)$, integration from $0$ to $t$ yields the Brownian transition semigroup identityContinuity at zero covers $t=0$.

The Weierstrass transform is Gaussian convolution, $Wf(x)=(4\pi {)}^{-1/2}{\int }_{\mathbb{R}}{e}^{-(x-y{)}^2/4}f(y)dy$. It smooths appropriate functions and is the heat semigroup at unit time. The Brownian transition semigroup uses variance-$t$ Gaussian kernels and therefore supplies the scaled time family.