Do we know of any hot jupiters that could be orbiting a red dwarf (or, more probably, orbiting a barycenter between the two)? Is this scenario even physically possible?
I found one "hot Jupiter" in the Kepler data (Kepler 45b). The star is a M dwarf with an effective temperature of 3820K. The planet has an estimated mass of 160.5 M(Earth) and radius of 10.76 R(Earth). This gives a density of about 0.8 g/cm2 which is consistent. The planet is located at approximately 0.03 AU from the star with an orbital eccentricity of 0.11. Also, it may be interesting that the star has a very high metallicity of Fe/H = 0.28. (This data is from http://kepler.nasa.gov/Mission/discoveries/ ).
It must be said, however, that M dwarfs only made up about 4.3% of the initial Kepler study which found 82 "Jupiter size" planets (as of my tally in April of 2015). Thus, there is an indication that there are less "hot Jupiters" around M dwarfs, but sample size is way too small. There does seem to be a correlation between stars hosting "hot Jupiters" and their metallicity (more common around higher Fe/H stars).
My gut feeling is that "hot Jupiters" in general are uncommon. They are just the easiest thing to find.