Do you have any idea how to fix it?
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I've had this issue before, it looks like you have done exactly what I did.
The GeoServer WPS example heat map doesn't support tiles, so what Gery said is correct you need to set tiled=false.
I had the same problem in OpenLayers, so it isn't a Leaflet issue.
Indeed it was a tile problem, as said in the geoserver doc
Rendering transformations may not work correctly in tiled mode, unless they have been specifically written to accomodate it.
but setting tiled to false didn't work.
But I found this custom leaflet class which worked for me : http://bl.ocks.org/ismyrnow/5415445
In the case the code of the WMS class itself would be the problem, you can check this nice heatmap example by Patrick Wied to see if you find a relevant difference with your code: http://www.patrick-wied.at/static/heatmapjs/example-heatmap-leaflet.html