"All possible causes are being looked into," Riverside County, Calif., Sheriff Deputy Melissa Nieburger tells PEOPLE.
Police confirmed Thursday that Hug's body was discovered in the San Bernardino National Forest. Other than saying no foul play was suspected, authorities have released little additional information.
A spokeswoman for Hug's parents, Bertrand and Denise Hug, said in a statement Friday: "The family is very distraught and heartbroken and will not comment on any rumors or speculation. They appreciate all of the kind words and condolences they have received and continue to ask for privacy as they mourn privately."
The spokeswoman previously stated that the restaurateur couple "are waiting to find out what happened like everyone else."
In a separate statement, the Hug family fondly recall their only child, who was 35 when he died, as a kind man who had a keen interest in motorcycles, sports cars and skydiving.
The Hugs own two premier San Diego restaurants, Mille Fleurs and Bertrand at Mr. A's, and say Julien helped with the family business "starting at age 14 as a dishwasher and worked his way through the ranks to become restaurant host and manager."
Their son, they add, was a "true gentleman, who treated everyone regardless of status in a kindly and respectful way."
A memorial service has been set for Nov. 12 at the Village Church in Rancho Santa Fe, Calif.