a transition kernel
A transition kernel is the bookkeeping device that records, for a Markov process, the probability of where you end up given where you started and how long you wait. Write p_t(x, B) for the probability that the process started at x lies in the (measurable) set B after time t. As a function it is a map (t, x, B) -> p_t(x, B) that is, for each fixed t and x, a probability measure in B, and for each fixed t and B, a measurable function in x. It is the continuous-time analogue of the one-step transition matrix of a discrete chain.
Formally, for each t >= 0, p_t is a Markov kernel: p_t(x, .) is a probability measure on the state space E for every x, and x -> p_t(x, B) is measurable for every Borel B. The transition function of the process is the collection {p_t : t >= 0} together with p_0(x, .) = delta_x (the point mass at x, encoding 'no time, no motion'). The process and the kernel are linked by P( X_t in B given X_0 = x ) = p_t(x, B); densities, when they exist, are written p_t(x, y) so that p_t(x, B) = integral over B of p_t(x, y) dy. The kernel ACTS on functions to give the transition operator, (P_t f)(x) = integral f(y) p_t(x, dy) = E_x[ f(X_t) ], and on measures to push them forward, (mu P_t)(B) = integral p_t(x, B) mu(dx).
Why it matters: every quantity of interest — the law of X_t, expectations, hitting probabilities, the generator, invariant measures — is read off the kernel. The single algebraic constraint a transition function must satisfy, beyond being a kernel, is the Chapman-Kolmogorov equation, which says that splitting a journey of length t+s through any intermediate time is consistent. A caveat: a kernel is the LAW of the process, not the process itself; many different path-constructions (and even different sample-path regularities) can share one transition function, which is why one separately demands càdlàg paths and the strong Markov property.
For Brownian motion on the real line the transition kernel has the Gaussian density p_t(x, y) = (1 / sqrt(2 pi t)) exp( -(y - x)^2 / (2 t) ). Then P( B_t in B given B_0 = x ) integrates this density over B, and (P_t f)(x) = E[ f(x + B_t) ] is the heat-equation solution at time t.
The Gaussian heat kernel is the prototypical transition kernel; it spreads a point mass into a widening bell.
A transition kernel encodes the law, not the trajectory: extra path regularity (càdlàg) and the strong Markov property must be supplied separately.