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?




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This [interesting answer by James Martin](https://mathoverflow.net/a/84289/36721) 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.



Bounded density for diffusions with diffusion coefficients bounded away from $0$
The "yes" answer follows immediately from [Theorem 2.5 in this paper by Kusuoka](https://reader.elsevier.com/reader/sd/pii/S0304414916300850?token=83AF411073D65047F42CC4F378ABD1D841ADA751E9CEE4DCE66CD933F7D9311AC9D96047079D3456564E461FAA7C60B7&originRegion=us-east-1&originCreation=20211007170037), 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 $\sup\_{t,x,y\ne x}|(a(t,x)-a(t,y)|/|x-y|<\infty$.


