The Brownian transition semigroup

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion Brownian motion. For $t>0$ define the Brownian transition kernel $p_t(x,y):=(2\pi t{)}^{-1\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{(y-x{)}^2}{2t}),x,y\in \mathbb{R},$ and for every bounded Borel function $f:\mathbb{R}\to \mathbb{R}$ define $P_{t}f(x):={\int }_{\mathbb{R}}f(y)p_t(x,y)dy(t>0),P_{0}f:=f\mathrm{.}$ The family $(P_t{)}_{t\ge 0}$ is the Brownian transition semigroup, and each $P_t$ is a transition operator.

Two equivalent descriptions are part of the definition and are used below.

1. Expectation form. For a standard Brownian motion $B$ as in Brownian motion and $t\ge 0$, $P_{t}f(x)=E[f(x+B_t)]\mathrm{.}$ Under AC, $N(0,t)$ is the law of $\sqrt{t}Z$ for a standard normal $Z$, whose density $\phi (z)=e^{-z^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$ is fixed in Standard normal and normal laws; the density of $N(x,t)$ is the translate $y\mapsto p_t(x,y)$, and the agreement of the two displayed expressions is proved as the first assertion of the semigroup lemma later on this page.
2. Basic regularity. For $t>0$ the map $(x,y)\mapsto p_t(x,y)$ is continuous, hence Borel; consequently $P_{t}f$ is Borel for bounded Borel $f$, $P_t$ is linear, and $\mathrm{\parallel }{P}_{t}f{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, with equality for $f\equiv 1$ once the kernel is known to be a probability density. At $t=0$ the convention is $P_{0}f=f$, so $P_0$ is the identity.

The cases $t=0$ and $s=0$ of every later identity are the identity operator and are recorded separately rather than derived from the $t>0$ formula. No choice beyond the declared AC is made by the kernel: the integral is a Lebesgue integral of a fixed continuous density.

Lawler, Section 2.6, and Durrett, Section 7.3, define the Brownian transition density and the operator $P_{t}f$. The expectation form is the definition of $P_t$ in Lawler's Markov-viewpoint treatment; here it is stated as an equivalent description and proved in the following lemma, so that no step of the later arguments has to treat it as an extra hypothesis.