In Illustrator, Select & Copy the object with the gradient fill.
In PS, create a new document with 300ppi. resolution, or you could use one that’s already open. (The resolution will affect the overall quality.)*
Paste, then in the dialog, choose “Paste As: Smart Object,” then Accept the transform box. Apply the Noise (or whichever) Filters. Select All, then Copy.
Return to Illustrator & Paste. Should be a silo–no white BG.
*The pasted graphic will be a rasterized, embedded “.tif” that has the same resolution as the PS document where you applied the Filter.