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Brownian loop measure is a sigma-finite measure on unrooted planar Brownian motion loops. A loop has a duration $t>0$; changing its starting time cyclically represents the same unrooted loop. Let ${\mathbb{P}}_{z,t}^{\mathrm{br}}$ be the probability law of a planar Brownian bridge from $z$ to $z$ of duration $t$. For a nonnegative measurable function $F$ of an unrooted loop,Here $dA$ is planar Lebesgue measure. The weight combines the planar Brownian transition density $1/(2\pi t)$ with a further factor $1/t$ for unrooting. Restricting to loops contained in a domain gives its loop measure. A conformal map, with the conformal Brownian clock on each loop, transports this restricted measure to that of the image domain.^[ref]

A Poisson point process with intensity $\lambda {\mu }^{\mathrm{loop}}$ is a Brownian loop soup. This construction connects Brownian motion to conformal restriction measure on simple loops through loop boundaries. The measure is infinite, so drawing one loop requires a finite restriction and normalization.

The Feynman-Kac formula connects a partial differential equation to an expected value over paths of a diffusion process. Let $X$ be a nonexplosive Itô diffusion with infinitesimal generator $L$, let $V,g$ be bounded with $V\ge 0$, and let $u$ be a bounded classical solution ofThenThe Itô formula and Itô product rule make $e^{-{\int }_0^{s}V(X_r)dr}u(t-s,X_s)$ a local martingale; boundedness makes it a true martingale on the finite horizon. Its equal endpoint expectations prove the identity and uniqueness within the bounded solution class.

The coefficient $V$ acts as a position-dependent killing rate. For constant $V=\lambda$ and $g=1$, the formula gives $u(t,x)=e^{-\lambda t}$. For $V=0$ and Brownian motion, it gives the heat equation solution by averaging $g$ against the Brownian transition density. Sources, terminal data and boundary stopping lead to the parabolic Feynman-Kac formula with a source and elliptic Feynman-Kac formula.

Let $D$ be the unit disc and normalize its positive Poisson kernel at the boundary point $1$ byIt is harmonic in $D$. Let $p_D(t,z,w)$ be the killed Brownian transition density, for planar Brownian motion with generator $\frac{1}{2}\Delta$. The Doob h-transform has sub-Markov transition densityIts missing mass represents paths already absorbed at the distinguished boundary point. Equivalently, for a stopping time $\sigma$ before exit from a compact subdomain, its law is weighted relative to ordinary killed Brownian motion by $h(B_{\sigma })/h(B_0)$. These stopped laws are consistent and define a diffusion up to its lifetime. Its generator isThis is a rigorous Brownian motion conditioned to exit at a boundary point, not conditioning on an event of positive probability. Exhaustion of $D$ shows that its terminal boundary limit is $1$; its lifetime is finite. For example its expected lifetime from $0$ is ${\int }_D{G}_D^{\mathrm{BM}}(0,w)h(w)dw$, which is finite: the Green function vanishes linearly near the smooth boundary and cancels the Poisson-kernel singularity there, while its logarithmic singularity at $0$ is integrable.

One can also condition ordinary Brownian motion on exiting through an arc $I_{ϵ}$ about $1$, and then let its length decrease to zero. The conditional harmonic functions ${\omega }_D(z,I_{ϵ})/{\omega }_D(0,I_{ϵ})$ converge locally uniformly to $h(z)$. Thus their stopped conditional laws converge to this same diffusion. Both constructions specify the conditioning unambiguously.

By the Tonelli theorem and the planar Brownian transition density,For $s\ge 1$,while for $0\le s\le 1$ it is at most $\Vert f{\Vert }_{\infty }$. Hencefor $t\ge 1$, and consequently

For $s<t<1$, independence and the Gaussian Brownian transition density yield the Chapman-Kolmogorov equationThe numerator is positive and its expectation is $p(1,0,y)$, soThese statements hold for every $t<1$, in particular on the specified closed interval $[0,T]$.

The function $h(t,z)=p(1-t,z,y)$ solves the backward heat equation $h_t+\frac{1}{2}{h}_{zz}=0$. Itô's formula therefore givesOn each interval ending before one this adapted continuous integrand is locally square integrable. Thusgives $M_t=\exp (N_t-⟨N{⟩}_t/2)$. This also follows by applying Itô's formula to $\log M_t$. The source's notation $t\ge 0$ for this density process is restricted by its defining formula to $t<1$.