$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\mathscr{P}} \newcommand{\KKK}{\mathscr{K}} \newcommand{\eps}{\varepsilon} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $T \in (0, \infty)$ and let $\TT$ be the interval $[0, T]$. Let $\sigma : \TT \times \RR^d \to \RR^d \otimes \RR^m$ be measurable. Let $a := \sigma \sigma^\top$. For simplicity, we denote $\sigma_t(x) := \sigma (t, x)$ and $a_t(x) := a (t, x)$. We assume there exist constants $C>0, 0 < \beta < 1$ such that for $t \in \TT$ and $x, y \in \RR^d$: \begin{align} \frac{1}{C} |y|^2 \le \langle a_t (x) y, y \rangle & \le C |y|^2, \label{main_assmpt1:ineq1} \\ |\sigma_t (x) - \sigma_t (y)| &\le C |x-y|^\beta. \label{main_assmpt1:ineq2} \end{align}
Let $(B_t, t \ge 0)$ be a $m$-dimensional Brownian motion and $\FF := (\mathcal F_t, t \ge 0)$ an admissible filtration on a probability space $(\Omega, \mathcal A, \PP)$. Let $X_0 : \Omega \to \RR^d$ be a $\mathcal F_0$-measurable random variable that admits a probability density function (p.d.f.) denoted by $\ell_0$. Let $$ X_t := X_0 + \int_0^t \sigma(s, X_0) \diff B_s, \quad t \in \TT. $$
From this answer, each $X_t$ admits a p.d.f. denoted by $\ell_t$. We assume that $\ell_0$ belongs to the Hölder space $C^\alpha_b (\RR^d)$ for some $\alpha \in (0, \beta)$.
Is it true that $$ \sup_{t \in \TT} \sup_{\substack{x,y \in \RR^d \\0 <|x-y| \le 1}} \frac{|\ell_t (x) - \ell_t (y)|}{|x-y|^\alpha} < \infty \, ? $$
Thank you so much for your elaboration.
