"Yesterday, I lost my father and a best friend and the world got a little grayer," says the actor's oldest son, Zak. "I will carry his heart with me every day. I would ask those that loved him to remember him by being as gentle, kind and generous as he would be. Seek to bring joy to the world as he sought."
Daughter Zelda offers a heartfelt message to Williams's countless fans and to any critics as well. "My family has always been private about our time spent together. It was our way of keeping one thing that was ours, with a man we shared with an entire world," she says. "But now that's gone, and I feel stripped bare."
"My last day with him was his birthday, and I will be forever grateful that my brothers and I got to spend that time alone with him, sharing gifts and laughter. He was always warm, even in his darkest moments," she says.
"While I'll never, ever understand how he could be loved so deeply and not find it in his heart to stay, there's minor comfort in knowing our grief and loss, in some small way, is shared with millions. It doesn't help the pain, but at least it's a burden countless others now know we carry, and so many have offered to help lighten the load. Thank you for that."
"To those he touched who are sending kind words, know that one of his favorite things in the world was to make you all laugh. As for those who are sending negativity, know that some small, giggling part of him is sending a flock of pigeons to your house to poop on your car. Right after you've had it washed. After all, he loved to laugh too."
"Dad was, is and always will be, one of the kindest, most generous, gentlest souls I've ever known, and while there are few things I know for certain right now, one of them is that not just my world, but the entire world is forever a little darker, less colorful and less full of laughter in his absence," she adds. "We'll just have to work twice as hard to fill it back up again."
Son Cody is finding strength in his belief that he will see Williams again in the afterlife. "There are no words strong enough to describe the love and respect I have for my father," he says. "The world will never be the same without him. I will miss him and take him with me everywhere I go for the rest of my life, and will look forward, forever, to the moment when I get to see him again."
Marsha Garces, who was married to Williams until 2008 and is Zelda and Cody's mother, is grateful for all that the actor gave to their family and the entire world.
"My heart is split wide open and scattered over the planet with all of you," she says. "Please remember the gentle, loving, generous – and yes, brilliant and funny – man that was Robin Williams. My arms are wrapped around our children as we attempt to grapple with celebrating the man we love, while dealing with this immeasurable loss."
If you or someone you know is considering suicide, please contact the National Suicide Prevention Lifeline at 1-800-273-TALK (8255).
The Williams family is asking well-wishers to send contributions to charities close to the actor's heart in lieu of flowers. Suggested organizations include St. Jude Children's Research Hospital, Challenged Athletes, USO, the Muhammad Ali Parkinson Center, the Christopher and Dana Reeve Foundation and Glide Memorial Church in San Francisco.
For more on Robin Williams's tragic death and his legacy of comic genius, pick up the latest issue of PEOPLE, on newsstands Friday