Radial drift gives weighted geometric contraction of Langevin diffusion

Fix a deterministic starting point $x$ and horizon $t$. Stop the continuous solution at its first exit ${\tau }_n$ from the radius-$n$ ball about $x_{\star }$, taking $n>\mathrm{\parallel }x-x_{\star }\mathrm{\parallel }$. Apply [F4] to $e^{a_{s}u}{H}_s(X_u)$ up to $t\wedge {\tau }_n$. Its drift is at most $b_s{e}^{a_{s}u}$ by [F2]. Its stochastic integrand is bounded on this stopped finite cylinder, so its expectation is zero. Hence ${\mathbb{E}}_x[e^{a_s(t\wedge {\tau }_n)}{H}_s(X_{t\wedge {\tau }_n})]\le H_s(x)+\frac{b_s}{a_s}(e^{a_{s}t}-1)\mathrm{.}$ The globally defined continuous paths in [F1] are bounded on $[0,t]$, so ${\tau }_n>t$ eventually on each such path. Fatou [F5] and division by $e^{a_{s}t}$ give the claimed semigroup moment bound.

In particular $P_1{H}_s\le a{H}_s+b$ with $a=e^{-a_s}\in (0,1)$ and $b=(b_s\mathrm{/}{a}_s)(1-e^{-a_s})<\mathrm{\infty }$. Any sufficiently large level set $\{H_s\le R\}$ is contained in a ball by [F3], and inherits the common minorization for $P_1$ from [F1]. The weighted drift-minorization result [F6] therefore gives an invariant law ${\xi }_s$ with finite $H_s$ moment and constants $M_s<\mathrm{\infty }$, $r_s\in (0,1)$ such that $\mathrm{\parallel }\mu P_n-\nu P_n{\mathrm{\parallel }}_{H_s}\le M_s{r}_s^n\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{H_s}(n\in \mathbb{N})\mathrm{.}$ Since [F1] already makes $\pi$ invariant and [F6] gives uniqueness among every invariant probability, ${\xi }_s=\pi$ and $\pi H_s<\mathrm{\infty }$.

For $0\le u<1$, step 1.1 gives $P_u{H}_s\le H_s+b_s\mathrm{/}{a}_s\le (1+b_s\mathrm{/}{a}_s)H_s$, since $H_s\ge 1$. For every Borel $f$ with $\mathrm{\mid }f\mathrm{\mid }\le H_s$, therefore $\mathrm{\mid }{P}_{u}f\mathrm{\mid }\le (1+b_s\mathrm{/}{a}_s)H_s$. Taking the test-function supremum shows $\mathrm{\parallel }\mu P_u-\nu P_u{\mathrm{\parallel }}_{H_s}\le (1+b_s\mathrm{/}{a}_s)\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{H_s}\mathrm{.}$ All integrals are finite by step 1.1 and [F5]. Write $t=n+u$ with $n=\lfloor t\rfloor$ and apply this estimate after the integer contraction in step 2.1. With $v_s=-\log r_s>0$ and $C_s=M_s(1+b_s\mathrm{/}{a}_s)r_s^{-1}$, the result follows from $r_s^n\le r_s^{-1}{e}^{-v_{s}t}$. This includes $t=0$ and all stated finite-moment initial laws.