The office space where Rebecca Miller has asked me to meet her is on the fourth floor of a building in Manhattan, right above a wig shop.
It does not suggest an endless stream of visitors. In the narrow hallway a small plaque reads: Estate of Inge Morath. Morath, a puckish Magnum photographer, was Miller's mother. I am raising my knuckles to the sign on the metal door that says "please knock loudly" when the elevator doors open behind me. It's the doorman, who is a woman.
"Phobias?" she says. Or I think that's what she says. "You're looking for treatment, right?"
Concerned that I looked lost, she has followed me up. "Er, no - well, not today, anyway," I say. The woman smiles, but she doesn't move. "I'll just knock," I tell her, "Loudly."
I found it hard not to imagine, later in the day, that Rebecca Miller had scripted this introduction herself. It had the flavour of her writing and her films - absurd, wrong-footing, possibly a figment, yet narratively neat.
Miller invites me in, past a room full of archival boxes. She is sleek, amenable and fairly fast-talking, with eyes too pale not to stare at.
"I rented this place for my mother's work to be archived," she explains. Now that she and her husband, Daniel Day-Lewis, have moved with their family from Ireland to New York, she is "subletting the back office for myself, from myself".
Her phone rings as we are about to sit down, and she goes over to check it.
"Daniel," she says, by way of explanation, and calls him back.
Miller's life is so easy to idealise that people overlook her eccentricities. All the pieces of a charmed and glowing existence are there: the culturally privileged upbringing as the daughter of Morath and the playwright Arthur Miller; the exceptional good looks; the Oscar-hoarding husband; the domestic bliss; the professional success. Miller shows no inclination to talk any of it down. But if you take her work into account at all, you'll find that her mind is prone to leaps in the dark. The structure of her life can only be a skeleton for much stranger things.
And sure enough, she'll tell you that she converted to Catholicism - from atheism - at the age of 8; that she was full of superstitions as a child, terrified that the devil lived in the basement. "Catholicism was the perfect religion for me," she suggests, "because of the idea that something truly evil could penetrate the walls of your house."
At art school, where she studied painting, she "had the same dream cycle for years - I dreamed that I had babies that could talk, wise babies. And they weren't always very nice."
What did your parents make of all that? I ask, envisaging proud, indulgent figures perfectly placed to encourage such a vivid imagination.
"Oh, they didn't know," Miller replies. "I had a very light way of expressing myself. And the dark side of my nature I kept secret. My imaginative life was quite private - until later," she adds with a laugh, "when they saw my work and they started to realise what had been going on."
If Angela, her first film, written and directed by Miller in 1995, took those hauntings whole and portrayed them tenderly - a 10-year-old girl persuades her younger sister that Lucifer is staying in the basement - Miller's work has since been more successfully rooted in the everyday.
Her short story collection Personal Velocity (three of which became the film that kick-started her career in 2002) is made up of taut, dark tales: "Kurt used to beat Delia and she took it for eight years until one night he grabbed her by the hair and started banging her head against the kitchen table during dinner. They were having chicken."
The soignee heroine of 2009's The Private Lives of Pippa Lee moves with her older husband to a stifling retirement community and finds that someone is breaking into their house at night, eating their food and trashing the kitchen. After securing the house she begins to fear it might be her husband, that senility has set in. But it turns out to be Pippa herself, somnambulant and feral in her sleep.
Miller's latest novel, Jacob's Folly, is more baroque in parts, swooping between time frames as an 18th-century Jewish valet is reincarnated in the 21st century as a Faustian fly. She has done incredible historical research for the novel, but the daily facts of contemporary life still offer the best material for her sly perspective and laconic style: "Leslie's stepson, Bud, was naturally deadpan, electively Buddhist, possibly a little depressed."
Miller describes herself as a "storyteller" - not when speaking about her profession, but when speaking about her person, the way you might say you were a brunette, or left-handed. "I think that people are so narrative by nature that it's very hard to resist making your own life into a story - and the universe as well," she says. In her hands, each character becomes a fable; most toy with secret selves within claustrophobic lives. Miller is preoccupied, as she puts it, with "the notion of escape".
Her latest heroine, Masha, is an ultra-Orthodox Jew who dreams of being an actress. "She yearned to be one of those people who became other people," Miller writes. "She had to find a story to live in."
Masha becomes the expression of many of her own ideas about freedom versus happiness - or, as Miller says, "Can I get out of here, and what would it take?" Jacob's Folly is, Miller tells me, "in part a kind of homage to what acting is. And writing really is the same thing in many ways, except it's private."
As the daughter of a playwright, Miller grew up around both disciplines. As a film-maker and novelist married to an actor, she is still continually in the presence of both. Daniel Day-Lewis is the son of a former poet laureate, and his mother was an actress.
So writing and acting - or invention and impersonation - are somehow at the heart of what both of them know.
I can't help wondering what effect that has on their domestic life. Two people pretending to be other people? How do you make time for that, and have a family?
"We do it in different ways," she says, considering this. "I mean, Daniel is somebody who ... It's very hard for him to do more than one thing at a time. He prepares and so on, but then he does the shoot for as long as the shoot lasts. My work takes so long that I couldn't disappear into myself in the same way. I think that he, I, and most people with families, learn to come out of it, and just be with their families - as much as possible. I mean, there are times when you're preoccupied. But he's definitely, even when he's preparing for a part, very connected to the kids and part of life with everybody. And I have to be too."
Miller's career took off more or less at the same time as she became a mother, in 1998.
She had been a painter first (and showed with the prestigious Leo Castelli gallery), then an actress (with small parts in Hollywood films), and after she wrote and directed Angela she found it hard to raise the money for her next film. It occurred to her that writing not for the screen was free.
"I began to realise I might just be waiting for years and years and not get to do the thing I wanted to do," she says. "Because that happens to people. You wake up and a decade's gone by and you haven't told any stories."
So she just went ahead and wrote some. "Having children was very good for me," she explains. "It focused me - the time that I had I had to use really well. I always had such a volume of information coming through my head that I didn't know how to parse it out.
Like jammed signals. And I think I became more productive when I had children because I had to choose what to do."
Now she is strict about her social life and strict about work, which she doesn't do past 3pm.
Miller and Day-Lewis have constructed a formidable machine of a family life. They used to live in Ireland and spend summers in the States, and now they have reversed the order.
For the past two years they have lived in a house in Greenwich Village - Miller describes herself as "a third-generation New Yorker" - and the two older boys can walk to and from school. By two older ones, Miller means Ronan, 14, and Gabriel, Day-Lewis' 17-year-old son with Isabelle Adjani, who now lives with them. (She refers to Ronan, her biological elder boy, as "my middle son".) Ten-year-old Cashel is at primary school nearby.
When they were living in rural Ireland and Day-Lewis had a long shoot, the entire family would accompany him and the children would be home-schooled. (Miller would give the art lessons.)
"We always go places together, up till now," Miller explains. "This last time, with Lincoln, he was in Virginia and we were in New York and it was so close - and also with two kids in high school and one in 10th grade it wasn't really possible to just uproot everybody. I don't know if we'll do another thing where everybody gets home-schooled."
Organised as all of this is, Miller confesses that she is "very, very absent-minded.
I'm almost forbidden to even turn on the stove any more in my house," she says, laughing. "It's actually terrible. Because when I'm taken by a thought I follow that thought, and if I'm making an omelette, the omelette gets left there, burning away ..."
But presumably, I suggest, when Day-Lewis isn't shooting, he's around the house?
"Yes. And he definitely is very much part of the whole running to turn off the stove thing."
There is a wonderful passage in Jacob's Folly in which Jacob, the 18th-century time-travelling fly, notices an array of celebrity magazines on a news stand. "Brangelina's secret wedding!" he reads. "I assumed Brangelina must be the name of the current monarch of this land, or perhaps his courtesan." Is Miller's joke a way of fighting back?
She is, after all, no stranger to the red carpet. Does fame affect her?
"Well," she reflects, "it doesn't come into my real everyday life very much. And it isn't really my own fame, obviously, that I run up against, because I'm not famous.
So the answer to that is 90 per cent no. And then occasionally yes. Occasionally you suddenly remember it."
But then, she says, "I think that once you've been around fame enough, it loses its meaning, and you see the absurdity of it, the arbitrariness of it, the machinery around it.
And in a way that's good, you know? It gives you some freedom. You realise that it has very little to do with the individuals involved. It is its own thing. In my case, I realised that early on."
Miller was first in touch with Day-Lewis in the mid-1990s when she sent him the script for the film that would eventually become The Ballad of Jack and Rose. They had never met. "I remember saying to my producer at the time: why don't we just send it to the best actor in the world? You know, why don't we just start with that?"
She also saw Day-Lewis' very particular gifts. Jack is a widower who lives on a remote island in possibly over-intimate proximity to his teenage daughter, and recedes, literally, as he succumbs to a terminal illness. Miller was taken with "how deeply [Daniel] seemed to feel things. And his intelligence. Jack is completely crippled by both guilt and desire.
And I thought: I bet he can find that. But at the time, he was really too young. If I'd been thinking about it, I wouldn't have cast him. It would have been a very different story. He saw that. But his basic qualities as a human being is why I wanted to cast him."
When they did make the film together, a decade or so later and in a place that "felt like the end of the world", much had changed.
Miller had originally written the script partly inspired by fear of her father's death; Arthur Miller died in the year of its release, 2005.
By then, its star - whom she had met through her father when Day-Lewis was due to appear in a production of The Crucible - was the father of her own children.
"When I started it I was a daughter, and when I finished it I was a parent," Miller says. "It started as Rose's story, and it ended up as very much both of theirs, as I saw it from his point of view, too."
Miller tells me that it's hard to know how her father's death has affected her. As a child, she identified with him perhaps more than she did with her mother (who died in 2002). "I wanted to be like my father - I didn't want to be female, feminine. Of course, I was very feminine."
And she became a painter in part because "at that time being a writer was overwhelming to me, because of my father. I needed to find something that was really my own". Now she describes herself as having gone into "the family circus".
"I think when people die," she suggests, "they become part of you, even more than they were. You absorb them a little bit more."
Miller says that she and Day-Lewis are hopeful that they will work together again, but that, rather inconveniently, "I tend to tell stories about women. And in order to be worthy of him, it would have to be a pretty great part. So it would have to be me becoming obsessed by the same sort of thing that he could be obsessed by.
And when that's going to happen I'm just not sure."
For someone who is so driven by narrative, and whose fictions are so controlled, it's curious - and in another person might have been crippling - that Miller should be shadowed by so many stories not her own.
For instance, though she was born into a happy marriage, she was also born in the wake of one of the most publicised tragedies of the 20th century. Five weeks before her birth, in 1962, her father's ex-wife, whom he had divorced a year earlier, was found dead. His ex-wife was Marilyn Monroe. The making of The Misfits - the last film in which Monroe appeared, which Arthur Miller had written, and which Morath had photographed as one of many Magnum photographers allowed on set - was famously ill-starred.
Neither of Miller's parents, who met on the set of The Misfits and remained married for the rest of their lives, were especially keen to go over that ground.
Second, Rebecca Miller has a younger brother - named Daniel, and now part of her life - who was born with Down syndrome and brought up in an institution. This was not really a secret - her mother visited him every week - but when Vanity Fair magazine reported it after her father's death it was seen as a scandal. The scandal? That Arthur Miller, who had left Daniel $3 million in his will, reportedly thought it would be unfair on 4-year-old Rebecca to grow up with a brother who needed that level of care.
Third, she seems condemned to be seen as part of a triangle - the daughter and wife of famous men. I have done it, too: because clearly, she loves them, and it would be meaningless to defend her against them. But I found myself wondering about the pattern: is it just plain chauvinism? Is it that their work is louder and more commercial, hers intricate and independent? Or could it be that Arthur Miller and Daniel Day-Lewis are, genuinely, stronger personalities? In other words, can it actually tell us anything about Rebecca Miller?
"Stronger personalities?" she repeats, when I ask her. "Stronger than mine?"
She laughs and says, with rueful emphasis, "Oh, no ..." A corner of the fragile veil has been lifted to reveal a character of steel.
"I'm a very secure person in some ways," she elaborates, "in the sense that I know what I'm interested in, and even though I get hurt if people don't like my work, I follow my own curiosity, and I'm going to do that whatever happens. I've noticed that quite a lot of people aren't interested in my work - I don't need the whole world to be.
"I've come to think of my audience as a group of individuals. What I do feels so personal to me that the miracle of reaching anybody ... that's amazing."
Jacob's Folly (published by Canongate) is out now.