For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\varepsilon >0$ we have $$\Bbb P (\vert B_s - f(s) \vert < \varepsilon \;\forall s \in [0,1]) > 0. \tag{1}$$

If we define for a continuous function $g:[0,\infty) \to \Bbb R$

$$T_g^X := \inf\{ t> 0 : X_t \geq g(t) \}$$ the statement for Brownian motion will imply that $$\text{support}( \Bbb P (B_t \in \cdot , \tau_g^B > t)) = (-\infty , g(t)) \tag{2}$$

Question: Are there a similar statements as (1) and (2) for Lévy processes (under appropriate conditions but even for subordinators)?

Remarks:

1. If one looks into the proof of (1) the joint distribution of $B_t $ and $\sup_{s\leq t} \vert B_s \vert$ plays an important role in the sense that it is used that

$$\text{support}\left( \Bbb P \left(B_t \in \cdot , \sup_{s\leq t} \vert B_s \vert < \varepsilon \right)\right) = [-\varepsilon , \varepsilon] $$ which is in turn related to (2).

2. Regarding (2) in general it depends on the Lévy tripel of what structure even $\text{support}(\Bbb P (X_t \in \cdot))$ is, which can be found in [Chapter 5.24 Supports, Ken-Iti Sato, Lévy processes and infinitely divisible distributions]. By following his case distinction it is possible to make a good analysis related to (2), but it took me several pages and I rather would like to cite it.