In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$
I know absolutely nothing about what Brownian bridges are, or about the conditional Wiener measure, where can I find material to learn how this measure works? Mainly the measure, it assigns a value to what?
Brownian bridges, heuristically, are Brownian motions that are conditioned to start at a given point and end at a given point.
In this case, the conditional Wiener measure they refer to is just the law of a Brownian bridge on $C[0, t]$, the space of continuous functions on $[0, t]$ (with starting value $x$ and terminal value $y$). This is a probability measure on $C[0, t]$ that assigns to each Borel subset of paths, the probability that a sample path of your Brownian bridge lies in it.
There are numerous ways to construct a Brownian bridge. I think a good reference is Karatzas and Shreve’s Brownian Motion and Stochastic Calculus. However for a start, you can look at the Wikipedia article, which gives a good overview.
