Rumer Willis knew she had finally arrived in Hollywood as herself, not just as the oldest daughter of, you know, Bruce Willis and Demi Moore, when she got her first stalker. And a girl stalker at that, who kept sending her e-mails urging her to “just click here” for pictures of what they could be like together. Very Lohanesque, but she resisted. Maybe it’s the short hair. “I do get that. I must be a lesbian, gay, whatever, people judging me by my appearance. But again, why am I surprised?

It is Los Angeles,” she laughs, her voice dropping into a husky range that echoes her mother’s. Were you not curious about this girl? “Nope, not at all. I think she has stopped, she probably got bored,” says Rumer, elegant in a light, strappy dress and cashmere cardigan that would cost a week’s wages for the maître’d at the discreet West Hollywood hotel she has chosen for our rendezvous. We move to the patio so she can blow away the memory of the stalker with a puff from a French cigarette, followed by a Red, then an American Spirit. It’s an impressive effort for a girl whose parents think she has given up.

Not that she is too worried. Rumer Willis left the family nest in north Los Angeles last year and now lives in West Hollywood with her hunky actor-boyfriend, Micah Alberti — a man blessed not only with a difficult name to spell (she writes it down for me, to make sure I get it right), but also guts, given the story that Willis promised to shoot Rumer’s first boyfriend as a warning to all the rest. It’s a threat that she laughs off: “Oh, he’s just a big bear.”

At 20, she is becoming her own woman. Despite pressure from mummy dearest, she has quit the University of Southern California — also known as the University for Spoilt Children — to get on with her life and act for real. These days, she stands financially independent of the fabulous Willis-Moore riches. Not that she doesn’t appreciate the occasional gift, such as the Christian Louboutin boots she lusted after for months before getting them for her last birthday. “I just wanted them so much, but I held off,” she says. “I learnt restraint.”

Rumer was born to drama. Her mother insisted on videoing her first daughter’s entry into this world, a practice Moore repeated with the next two, Tallulah Belle and Scout LaRue, who starred as the Bump on Demi Moore when she appeared naked on the cover of Vanity Fair. She was named after the British writer Rumer Godden — “I don’t know whether my mom had read much of her stuff, I guess she may have just been in a bookshop and liked the sound of it. I used to get teased at school, Rumer Tumour, that kind of thing, but I’ve got used to it. You do.”

One of the reasons why she comes across as polite and interested in life beyond herself, and not as a standard-issue, mad-as-a-Gucci-bag-full- of-snakes LA actress, is that she is, at heart, a country girl, raised in Hailey, Idaho, a tiny mountain town that has just gone a little crazy and put up its fourth set of traffic lights. Her dad owns most of it.

At 10, she had to deal with her parents’ very public divorce. Friends say, despite their determination to stay civilised, it shook her up badly. Today, she is evasive.

“I was 10 years old, so I don’t recall much,” she says. “But our parents handled it pretty well, so we didn’t feel it all too badly.”

And then Ashton Kutcher, pop-culture wunderkind and teenage girl’s dream, landed in Mommy’s life. Didn’t she have his poster on her wall? “Yes, I did, among others. It was a bit strange, but we adapted pretty quickly. Ashton is a wonderful guy, very smart, with lots of ideas coming out of his head all the time. We are very happy to have him in the family.”

Today, the entire Willis clan regularly makes red-carpet appearances together: Demi and Ashton, Bruce and Whoever, Rumer, her boyfriend and sisters. They all spent last Independence Day building a house for the poor in Guatemala. Still, maybe it’s no wonder she chose to spend her teenage years at boarding school in permafrosted Michigan, 1,000 miles from the nearest gossip columnist. By the time she moved to LA to live with Mom in 2004, she was all grown up.

Well, almost. Two years ago there was a flurry of pap-snappery as she stumbled out of LA clubs. There was a panda-eyed goth phase, the remnants of which today are eight tiny little tattoos and nine piercings. Her nights partying with Lindsay Lohan grew so heady that, it was reported, Bruce and Demi gave her a stern parental talking-to. Today she will not even name Lohan, saying: “We were acquaintances, like many in this business. That’s it, full stop.”

It was a brief rebellion. Rumer, unlike many of her peers, has a hinterland. She is a skilled photographer and, according to mutual friends, a funky dancer. Right now, she is into the Ting Tings, and engaged in bitter rivalry with her sisters (and her musically obsessed dad, whose brash rhythm and blues albums she swears she has on her iPod) to spot the next big LA barrio band.

In the past year, after a handful of appearances in her parents’ films — she bullied her way onto the set of Striptease aged seven, and starred with her dad in Hostage — she has started to make her mark as an independent-minded actress with a light touch. She is still at the sidekick stage, playing second banana to Anna Faris in the forthcoming The House Bunny, a chick flick with edge and wit about a Hugh Hefner bunny expelled from the Playboy mansion and reduced to mentoring the most uncool sorority on a college campus.

She gets some good lines as an Ugly Betty student in an ungainly back brace, a prop she did not take home, unlike a taste for pink, which is still surprising her. “I never did pink before. I have never been a girlie girl, but since making The House Bunny, I have found myself wearing a little more of it. I’m not really like my character, though. She is, well, sweet — a word I hate — and I’m more outgoing. But lots of people will recognise her.”

Lots of people will recognise Rumer Willis, too. She is fast becoming public property, and not entirely happy about it. She claims to have the skin of a rhino, but admits she has been genuinely hurt by critics who compare her strong jaw to Jay Leno’s. Maybe it’s because she is flawless Demi’s daughter, but the gloves seem to be off when it comes to criticising her looks. And it’s unfair because, in real life, she has her mother’s extraordinary eyes, and is dainty and pretty, not to mention a million miles away from your standard Hollywood offspring turned wannabe starlet. The gossip columnists tend not to mention that. Now, those are the guys Bruce should be shooting.

The House Bunny opens on October 10

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