Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W\_t$ usual Brownian motion (with $W\_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W\_t|^p$$
purely in terms of $\mathbb E|X|^p$ and $\mathbb E|X+W\_1|^p$ (which I can assume to exist).


I was able to show the following, which looks a bit similar, but this does not solve my problem: For $X,Y\in L^p$, I can bound
$$\begin{align\*}\mathbb E|X + tY|^p &= \mathbb E|(1-t)X + t(X+Y)|^p \leq \mathbb E(\max\{|X|^p, |X+Y|^p\}) \leq \mathbb E[|X|^p + |X+Y|^p] \\
&= \mathbb E|X|^p + \mathbb E|X+Y|^p\end{align\*}$$
(first inequality by quasiconvexity of the p-norm). This is essentially the kind of bound I would like to obtain. But of course, $W\_t \neq t Y$ for a random variable $Y$, so I cannot directly apply this.



Bound moments wrt. known initial and final moments
With $Y:=W\_1$ and $t\in[0,1]$, the expectation in question is
$$
\begin{aligned}
E|X+\sqrt t\,Y|^p&=E|(1-\sqrt t)X+\sqrt t\,(X+Y)|^p \\ 
&\le(1-\sqrt t)^p E|X|^p+(\sqrt t)^p E|X+Y|^p \\
&\le E|X|^p+E|X+Y|^p \\
&=E|X|^p+E|X+W\_1|^p. \\
\end{aligned}
$$


