Who's taming whom?

Deya Pichardo does not care for Rebecca Miller, the 32-year-old wife of Daniel Day-Lewis. Until last weekend, the fitness trainer from the Dominican Republic believed that she had first call on Day-Lewis, the actor who had a child with the French actress Isabelle Adjani and has been linked with a string of other beautiful women, including Julia Roberts. "He said he moved to New York to be with me," says Deya, with more than a hint of anger in her voice. "Now I know it must have been for Rebecca."

It is not clear if Miller knew about Pichardo. If she didn't, that seems naive; if she did, that appears calculating. But it is doubtful that either she or her new husband will be talking about that aspect of their relationship in the near future, if at all.

Ms Pichardo is entitled to feel confused. She says she learnt that Day- Lewis was married over the telephone, from a friend who read her a story about the wedding. "He must have been cheating on me with Rebecca," she says, from a Manhattan apartment she claims to have been sharing with Day-Lewis. "I can't understand what Daniel sees in her. For somebody who likes exotic women, she's pretty plain."

On that, Pichardo is wrong. Rebecca Day-Lewis, nee Miller, can look stunning when she tries. It's just that most of the time the daughter of Arthur Miller affects a grunge look, one more in keeping with the bold, creative temperament that acquaintances have grown used to seeing on her sleeve. This is a woman who wants to be an artist in the grand tradition of her father, a one-time actress who said she disliked the job because she was "uncomfortable with having to be so concerned with what you look like".

Miller, now a novice movie director, is not one for swanky parties or expensive dress stores. Preferring to entertain at home, her dinner parties are a secret pleasure for arty New Yorkers, who liken them to the literary salons of the Twenties. She is a good cook, and has what one friend calls a "wickedly complex sense of humour".

Before now, Miller has rarely surfaced in the New York gossip columns and style pages. She prefers to wear jeans, Doc Martens and baggy sweaters. Friends say she feels more comfortable when she sports this casual look and apparently feels guilty if she spends too much money on clothes.

She has edited parts of her childhood, maybe to match a too carefully manufactured self-image. She used to tell people that she grew up with her parents in the notorious Chelsea Hotel, an artists' villa in downtown New York that has been home to Bob Dylan, Lou Reed, Milos Forman, Norman Mailer and Sid Vicious. In fact, she spent her childhood in Roxbury, Connecticut, one of the most privileged neighbourhoods in America. Property prices there make even Manhattan seem cheap, and Rebecca's father owned - still owns - a 350-acre slice, complete with a 10-bedroom house.

Rebecca Miller was born one year after the death of Marilyn Monroe, her father's second wife. He already had two children, Robert and Jane, by his first wife, Mary Slattery, who he divorced in 1956. Rebecca's mother is Inge Morath, a photographer Miller met on the set of The Misfits, the movie he wrote for Monroe. Many people believe that Miller began his relationship with Morath well before his divorce from Monroe. Morath and Arthur Miller remain married and attended Rebecca's wedding in Vermont last weekend.

"I think she had an idyllic upbringing," says a friend from Yale, where Rebecca studied painting and literature in the early Eighties. "She could run around that huge house doing pretty much what she pleased and getting most of what she wanted. She's a true member of America's aristocracy, as upper-class as they come in this country."

Rebecca remembers it differently. She says she had a happy childhood, but spent much of her time in a state of terror. "I believed Lucifer lived in the basement," she said earlier this year. "I couldn't go down there until I was 11, and soon after I did for the first time I developed a fervent need to get baptised as a Catholic." (Her father is Jewish, her mother is a Protestant.)

Rebecca's teachers say that she was "contemplative and introspective", "with a powerful need for self-expression". According to a tutor at Yale, she was always especially concerned to establish an artistic identity separate from that of her parents. She fervently did not want to write novels, plays or take photographs, but they say she did have a driving ambition to succeed.

For 10 years after Yale, she laboured as an artist, producing dream-like canvases that were never much in demand. "They were very opaque. I'm an obsessive dreamer," she told the Chicago Tribune last year. "I was never able to break the dreams into true narrative, partly because of the nature of painting. I stayed with it for almost a decade because I had a sense that painting was something totally my own."

The family connection helped when Rebecca tired of painting. "At a certain point, I realised I wanted to make films, but only tiny ones," she told People magazine last year. "I wanted the images to move in time, the way my dreams did."

It was convenient, then, that Arthur Miller's agent, Sam Cohn, was at hand. The man who also represents Woody Allen was happy to help realise Rebecca's dreams of movie-making. He suggested that she begin to learn her craft by working as an actress and arranged a number of auditions. Between 1988 and 1994, Miller appeared in several TV films and movies. She won praise from some quarters, but the majority of critics were harsh: The New Republic called her 1992 performance in Consenting Adults with Kevin Kline "affectedly inept".

Miller's defence is that she never intended to be an actress. She was "acting" the role of starlet as a staging post to the director's chair. Although for most aspiring actresses that's an attitude that would have won them unemployment, in Miller's case it is a claim that has some merit. Her first feature film, Angela, won two prizes at this year's Sundance Film Festival.

The main female character in Miller's Angela, a film about the childhood dread of isolation and abandonment, is an obvious ringer for Marilyn Monroe, the silent ghost in Rebecca's life. Some of Miller's friends say the shadow of Marilyn has hung heavily over her head since she was a child, but she has always been reluctant to discuss the screen idol. In interviews she has become icy whenever Monroe's name is mentioned. Denying that there was a Marilyn connection in her film, she told one magazine: "I can't answer for my unconscious, but I wasn't making a personal statement."

She has high ambitions for her future films. "I think my place is to tell women's stories of all kinds," she said in February. "To investigate women's psychology and sexuality and spirituality. That to me is fascinating. I think there's a need for that."

I'm sure Deya Pichardo thinks so. Rebecca might want to start by investigating her new husband's ex-lover's psychology, before moving on to the other women he has dropped - most notably Isabelle Adjani, to whom he faxed the news that their relationship was over.

Miller and Day-Lewis themselves may be an inflammable mixture. She seems to be content in her relationship with her father, although she has taken pains to try to establish a separate identity to hold up for him to admire. That has involved her distancing herself from his kind of writing and his most famous relationship, with Monroe.

By contrast, Day-Lewis seems to have been looking for a father since his own died when he was 15. Earlier this year he implied he would like to be adopted by Arthur Miller. He spoke of turning up on the writer's doorstep with adoption papers and said: "There's something about Arthur Miller that makes you wish he was your father."

In 1989, Daniel Day-Lewis ran from the stage at the National Theatre during his performance of Hamlet. The highly strung actor later suggested that he had seen the ghost of his own father. In this new drama, Rebecca Miller had better take care that she has not been cast as Ophelian