Feynman–Kac path targets and normalizers

Assume AC and fix a finite horizon $T\in {\mathbb{N}}_0$. For each $0\le k\le T$, let $({\mathsf{X}}_k,{\mathcal{X}}_k)$ be a standard Borel space. Let ${\mu }_0$ be a probability measure on ${\mathsf{X}}_0$, and for $1\le k\le T$ let $M_k:{\mathsf{X}}_{k-1}⇝{\mathsf{X}}_k$ be a probability kernel. A potential at time $k$ is a finite-valued nonnegative measurable function $G_k:{\mathsf{X}}_k\to [0,\mathrm{\infty })$. Write ${\mathsf{X}}_{0:t}={\prod }_{k=0}^t{\mathsf{X}}_k$ with its finite-product sigma-algebra.

For each $0\le t\le T$, let $P_t^M$ be the finite path law with ordered kernel integral ${\mu }_0(d{x}_0){\prod }_{k=1}^t{M}_k(x_{k-1},d{x}_k)$, supplied by Ionescu–Tulcea (Ionescu-Tulcea construction of a Markov chain). The unnormalized Feynman–Kac path measure is ${\gamma }_t(A)={\int }_{{\mathsf{X}}_{0:t}}{\mathbf{1}}_A(x_{0:t}){\prod }_{k=0}^t{G}_k(x_k)P_t^M(d{x}_{0:t}),A\in {⨂}_{k=0}^t{\mathcal{X}}_k\mathrm{.}$ Its normalizer is $Z_t={\gamma }_t({\mathsf{X}}_{0:t})={\gamma }_t(1)\in [0,\mathrm{\infty }]$. Only when $0<Z_t<\mathrm{\infty }$ define the normalized path target ${\pi }_t^{\mathrm{F}\mathrm{K}}={\gamma }_t\mathrm{/}{Z}_t$. The sequence $({\gamma }_t)$ with its kernels and potentials is the Feynman–Kac flow.

State-space filtering kernels are conditional-law versions as in Filtering, predictive, and smoothing laws in a finite-horizon state-space model. A Feynman–Kac path target is not by itself an assertion that a pointwise conditional version has been selected at observation-null histories.

Each coordinate projection from the finite product is measurable by The product sigma-algebra and its finite iterates. Thus $V_t(x_{0:t})={\prod }_{k=0}^t{G}_k(x_k)$ is nonnegative product-measurable by composition and finite-product measurability (Composition with a Borel measurable outer map preserves measurability, Arithmetic and lattice operations preserve measurability whenever they are defined). Consequently, ${\gamma }_t(A)={\int }_A{V}_{t}d{P}_t^M$ is a measure, possibly of infinite mass, by The indefinite integral of a nonnegative measurable function is a measure.