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Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $0$.

Does then $X_1$ have a bounded pdf?

This interesting answer by James Martin shows that, without assuming that $a$ is bounded away from $0$ and replacing $a(s,X_s)$ by $a\big(s,(X_u\colon 0\le u\le s)\big)$, it is possible that $P(X_1=0)>0$. See also comments to that answer.

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  • $\begingroup$ When $a$ is stationary (does not depend on time) and uniformly elliptic, it is a classical result that the density of $X_1$ — the heat kernel at time $1$ for the corresponding second-order differential operator — satisfies the usual Gaussian bounds. The time-inhomogeneous case should also be well-known, but I do not know the references and unfortunately I do not have time now to look them up. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 7, 2021 at 9:33
  • $\begingroup$ One more thought: another keyword might be evolution families. A quick google search leads to, for example, doi.org/10.1002/mma.5978, which seems closely related, but I did not look inside. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 7, 2021 at 9:36
  • $\begingroup$ @MateuszKwaśnicki : Thank you for your comments. $\endgroup$
    – Iosif Pinelis
    Commented Oct 7, 2021 at 13:14

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The "yes" answer follows immediately from Theorem 2.5 in this paper by Kusuoka, which implies that $X_1$ has a normal-like pdf $p$, such that $$c_1 e^{-b_1x^2}\le p(x)\le c_2 e^{-b_2x^2}$$ for all real $x$, where $c_1,b_1,c_2,b_2$ are positive real constants depending only on $\inf_{t,x} a(t,x)>0$, $\sup_{t,x} a(t,x)<\infty$, and $\sup_{t,x,y\ne x}|(a(t,x)-a(t,y)|/|x-y|<\infty$.

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