Let $(X_t, t \ge 0)$ be a $\mathbb R^d$-valued stochastic process. Let $\lambda>0$. Assume we have $\mathbb E [|X_0|^2] < \infty$ and
$$
\mathbb E [|X_t|^2] - \mathbb E [|X_0|^2] \le -2 \lambda \int_0^t \mathbb E [|X_s|^2] \, \mathrm d s + 2t (\lambda |\mathbb E [X_0]|^2 +1) \quad \forall t \ge 0.
$$

It is mentioned in the proof of *Lemma 5.2* of the paper [Convergence to equilibrium for granular media equations and their Euler schemes](https://projecteuclid.org/journals/annals-of-applied-probability/volume-13/issue-2/Convergence-to-equilibrium-for-granular-media-equations-and-their-Euler/10.1214/aoap/1050689593.full) that
>By Gronwall's lemma,
$$
\mathbb E [|X_t|^2] \le \bigg [ |\mathbb E [X_0]|^2 + \frac{1}{2\lambda} \bigg ] (1- e^{-2\lambda t}) + \mathbb E [|X_0|^2] e^{-2\lambda t}.
$$

Could you explain which form of the Gronwall lemma is used in the paper?

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On the other hand, the form of [Gronwall lemma](https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality?useskin=vector#Integral_form_for_continuous_functions) on Wikipedia seems not applicable in this case, i.e.,

>Let $I$ denote an interval of the real line of the form $[a, \infty)$ or $[a, b]$ or $[a, b)$ with $a<b$. Let $\alpha, \beta$ and $u$ be realvalued functions defined on $I$. Assume that $\beta$ and $u$ are continuous and that the negative part of $\alpha$ is integrable on every closed and bounded subinterval of $I$.
>- (a) If $\beta$ is non-negative and if $u$ satisfies the integral inequality
$$
u(t) \leq \alpha(t)+\int_a^t \beta(s) u(s) \mathrm{d} s, \quad \forall t \in I,
$$
then
$$
u(t) \leq \alpha(t)+\int_a^t \alpha(s) \beta(s) \exp \left(\int_s^t \beta(r) \mathrm{d} r\right) \mathrm{d} s, \quad t \in I.
$$
>- (b) If, in addition, the function $\alpha$ is non-decreasing, then
$$
u(t) \leq \alpha(t) \exp \left(\int_a^t \beta(s) \mathrm{d} s\right), \quad t \in I.
$$

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Below is the screenshot I took from the paper.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/VG6uO.png