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Tag Archives: wanzie presents
Theater review: '[title of show]'
Posted on October 4, 2010 | 1 comment
There's something about [title of show] that feels very Fringe.
Not Fringe in that this little musical, at the Parliament House's Footlight Theatre, could likely turn up in Orlando's annual spring madness – although, if it were a good bit shorter, that could very well be the case.
But Fringe in that [title of show], which is self-aware to the point of narcissism, feels like innumerable shows we've all seen at the Fringe, the kind in which some blocked young writer says to himself or herself, "Gee, I can't think of anything to write about, so why don't I write about that?"
Posted in Orlando theater, Reviews, Uncategorized
Tagged footlight theatre, Hunter Bell, Jeff Bowen, parliament house, wanzie presents
Opening this week:
Posted on July 12, 2010 | Comments Off on Opening this week:
Here's an early edition, with a whole lot of stuff coming at us all through the week:
Target Family Theatre Festival, Orlando Rep
Splash!, PB&J Theatre Factory
The Little Red Riding Hood Show, Orlando Shakespeare Theater
Jerome Bixby's The Man From Earth, Renegade Theater
Nunsense, Nunsense A-Men!, Tuesdays With Morrie and Greater Tuna, all presented by FantasyLand Theatrical
Hair in Concert, Surfside Players
Hell Freezes Over!, Wanzie Presents
Summer Fantasy V, Orlando Ballet School
Into the Woods, Lakeland Community Theatre
May the Farce Be With You, Henegar Center for the Arts
Flashback!, Ocala Civic Theatre
Bye Bye Birdie, Mount Dora Theatre Company
And here are all the details:
Target Family Theatre Festival, Orlando Rep:
The 4th Annual Target Family Theatre Festival Brings Free Family Entertainment to Central Florida
Orlando Repertory Theatre invites families to attend the 4th Annual Target Family Theatre Festival in Loch Haven Park July 16-25, featuring performances by Milkshake, Airborne Comedians, Hobey Ford (presented along with Ibex Puppetry), Miss Lori, Debi Derryberry and Tony Brent. The festival also features the Rep's Youth Academy production of 42nd Street, which is $12.
The Rep is proud to partner with Target and Orange County Division of Art and Cultural Affairs to present the Target Family Theatre Festival, an event designed to promote cultural awareness in Central Florida, and to develop theatre for family audiences, by producing shows, training young artists, and offering educational experiences in the arts.
Since opening its doors, Target has given 5 percent of its income to organizations that support education, the arts, social services and volunteerism. Today that equals more than $3 million every week. "At Target, our local grants are making a difference in the communities we serve," said Laysha Ward, president, community relations, Target. "We're proud to partner with the Orlando Repertory Theatre as part of our ongoing commitment to give back to the communities where our guests and team members live and work."
This year, the festival kicks off with a free Opening Day Celebration on July 17 from 10 am – 2 pm. The celebration is an exciting family event complete with shows, live entertainment and hands-on activities for kids! This year's Opening Day Celebration features free performances of 42nd Street, Tony Brent, Airborne Comedians, Milkshake, Hobey Ford and more! There will also be free performances on the outdoor stage from the Orlando Science Center, PB&J Theatre Factory, Orlando Shakespeare Theater, Sak Comedy Lab and Pinocchio's Marionette Theater.
The 4th Annual Target Family Theatre Festival begins July 16 with the opening of 42nd Street and continues with a variety of family-friendly and affordable shows through July 25. Tickets for 42nd Street are $12 and tickets for all other performances are free. The festival takes place at the Orlando Repertory Theatre at 1001 E. Princeton Street Orlando, FL 32803. Please visit www.orlandorep.com or call 407-896-7365 for complete festival schedule and information.
Splash! and The Little Red Riding Hood Show, PB&J and Orlando Shakespeare Theater
With Summer Sun Comes Orlando Shakes Fun
The Little Red Riding Hood Show an, Splash!
coming soon to the Orlando Shakespeare Theater
The Orlando Shakespeare Theater in partnership with UCF is proud to announce its summer line-up of performances for theater lovers of all ages. School is out for the summer and Orlando Shakes kicks into high gear for youngsters with its summer performance schedule.
"We are thrilled that Orlando Shakes is able to once again provide summer programming for families with The Little Red Riding Hood Show and PB&J Theatre Factory's production of Splash," says Jim Helsinger, Artistic Director of Orlando Shakespeare Theater.
The summer season opens July 15 with an exciting, new adaptation of a classic folktale with The Little Red Riding Hood Show. This is not your grandmother's Little Red Riding Hood! Follow a crooked path through the forest to grandmother's house with Little Red, her resourceful mother, and Malarkey, the biggest and baddest of the big bad wolves. Directed by Patrick Flick (Schoolhouse Rock Live!, Kiss Me, Kate!, and Into the Woods), the cast includes Orlando audience favorite Robin Olson, as well as several actors new to the Orlando Shakes' stage: Nonalee Davis, Megan Goldman, and Austin Farwell.
This 55-minute play is recommended for audiences in grades pre-K-5 and their families.
What: The Little Red Riding Hood Show by Russell Davis
Where: Orlando Shakespeare Theater in Partnership with UCF
Lowndes Shakespeare Center, 812 East Rollins Street, Orlando, FL 32803
When: July 15 – August 8, 2010
Tuesday – Friday at 10:30 AM
Saturdays at 2 PM and Sundays at 4:30 PM
Tickets: $9-$10 – On Sale Now!
Box Office: (407) 447-1700, or ext. 265 for group sales
online at www.orlandoshakes.org
The summer fun continues with the PB&J Theater Factory's latest production, Splash!, running July 15 –
August 1, 2010 at the Lowndes Shakespeare Center.
Created by the same team behind last summer's hit Snack, Splash! maintains its silent storytelling format, integrating sound effects, music, and props to create hilarious and heartfelt stories for all ages. The sun is out! The surf is up! And summer has arrived! Kick off your vacation with PB&J Theatre Factory's Splash! It's the last day of work, quit daydreaming about summer, pack up your SPF 40 and join the fun-loving gang from PB&J for an all out beach party.
What: Splash!
A PB&J Theatre Factory Production presented by Orlando Shakes
When: July 15-August 1, 2010
Thursday – Friday at 7:00 PM
Saturdays at 4:00 PM and 7:00 PM
Sundays at 2:00 PM
Tickets: $15 – On Sale Now!
Jerome Bixby's The Man From Earth, Renegade Theatre:
Renegade Theater Announces Inaugural Production, Jerome Bixby's The Man From Earth
Renegade Theater, Orlando's newest theater group, is proud to announce its inaugural production, the Southern premiere of the science fiction drama Jerome Bixby's The Man From Earth.
Jerome Bixby's The Man From Earth premieres on Saturday, July 17 and runs for five performances at the Roth JCC of Greater Orlando, Maitland campus (851 N. Maitland Avenue, Maitland, FL 32751).
Performance dates:
Saturday, July 17: 7:30pm
Sunday, July 18: 2:00pm
Thursday, July 22: 7:30pm
Tickets are $15, $12 for students and seniors, and are available cash only at the door before each performance.
PLOT: An impromptu goodbye party for Professor John Oldman becomes a mysterious interrogation after the retiring scholar reveals to his colleagues he is a man who has walked the earth for 14,000 years. Tempers rise and emotion flow as John's fellow professors attempt to poke holes in his story, but it soon becomes clear that his tale is as impossible to disprove as it is to verify. What starts out as a friendly gathering soon builds to an unexpected and shattering climax.
Jerome Bixby's The Man from Earth
Adapted by Richard Schenkman
Directed by Charles Dent
Starring: Alex Carroll, Glen Howard, Amy Pastoor, J.C. Rede, Amy Schwartz, Larry Stallings, David Strauss, Rachael West.
The Man From Earth was the last work from the late, great Jerome Bixby, best known to sci-fi fans as the author of such seminal genre works as "Mirror, Mirror" (one of the most popular episodes of the original Star Trek), "It's a Good Life" (a key Twilight Zone tale), and the film Fantastic Voyage.
WHAT: The Southern premiere of Jerome Bixby's The Man From Earth
WHO: Renegade Theater
WHEN: July 17-25, 2010
WHERE: Roth JCC of Greater Orlando Auditorium, Maitland campus
851 N. Maitland Avenue, Maitland, FL 32751
Near the corner of Maitland Blvd and Maitland Ave.
TICKETS: $15.00 general / $12.00 student/seniors. More information, including group rates, is available by calling 407.385.0204 or emailing artistic director Heather Lam at heatherlam@renegadetheaterorlando.org. All tickets are CASH ONLY at the door.
INTRODUCING RENEGADE THEATER. Renegade Theater, formed in 2010, is a collective group of writers, directors, performers, designers and visual artists who have a desire to take the arts in Central Florida to a new level of excellence, while broadening cultural horizons. Renegade Theater is dedicated to bringing multi-media performance art to all audiences, focusing on our under-served communities. We provide a learning and teaching environment where local artists can experiment, entertain, inspire, and grow. For more information, visit http://www.renegadetheaterorlando.org. Renegade Theater is not affiliated with the Roth JCC or any other group.
Nunsense, Nunsense A-Men!, Tuesdays With Morrie and Greater Tuna, FantasyLand Theatrical:
Summerstock '10 @ The Pointe
FantasyLand Theatrical Productions is proud to announce our Summerstock 2010 season. Our first of many seasons at the Pointe Performing Arts Center at Pointe Orlando (International Drive next to Wonderworks).
This summer we will be presenting 2 plays and a musical. This summer's theme is reflection of life. Observe life through the absurd banter of Greater Tuna or the timeless reflections of Tuesdays with Morrie. Meet the Little Sisters of Hoboken as they share the story of their convent in Nunsense and Nunsense A-Men. Our SummerStock season offers something for everyone: Laughter, Tears, Singing nuns, Famous Characters and Timeless Truths. We hope to see you soon at one or all of these Performances.
Tuesdays With Morrie July 13-Aug. 25
Greater Tuna July 15-Aug. 28
Nunsense A-Men! July 16-Aug. 27
Nunsense July 18-Aug. 29
PointeArts.org.
Hair in Concert, Surfside Players:
Surfside Players is proud to present:
Hair – In Concert!
The American Tribal Love-Rock Musical
By James Rado, Gerome Ragni, and Galt MacDermot
Let the sun shine in! The Star Children, from Surfside Players' May production of Hair, are reuniting for a special concert at our playhouse in Cocoa Beach. We'll be spreading the message of love, peace, and freedom once again with songs from Hair, featuring a live band, and raising money for the theatre with a raffle, T-shirts for sale, and a silent auction. Members of the "tribe" are contributing artwork, clothing, special classes, an autographed show poster, and keepsakes from the heart to this silent auction to benefit Surfside Players. Tickets are just $10, but additional donations are always welcome.
Hair – In Concert! will run for TWO NIGHTS ONLY, Friday July 16 and Saturday July 17, at 8:00 p.m. Tickets are $10 for this special event. Call (321) 783-3127 for reservations. WARNING: The songs contain strong language. For mature audiences only.
Based on the musical Hair, directed by Michael Thompson, with musical director Dan Harrald and choreographer Rachel Cooke.
To get to Surfside, take A1A into Cocoa Beach, and turn west on Fifth Street South. Call (321) 783-3127 or go to www.surfsideplayers.com for more info.
WHO: Surfside Players
WHAT: Hair – In Concert! (Based on the musical)
by James Rado, Jerome Ragni, and Galt MacDermot
Directed by Michael Thompson
WHEN: July 16-17 (two nights only), Friday-Saturday 8:00 p.m.
WHERE: Surfside Playhouse, corner of Brevard Avenue and Fifth Street South, Cocoa Beach, FL
TICKETS: $10 general admission; donations also appreciated
MORE INFO: (321) 783-3127 or info@surfsideplayers.com
Also visit our website, www.surfsideplayers.com
Hell Freezes Over!, Wanzie Presents:
Hell Freezes Over! The Extended Director's Cut * July 17, 24 & 26 @ 8pm
Parliament House Footlight Theatre
410 N Orange Blossom Trail
The gates of Hell are opening once more as Wanzie Presents brings the award-winning Fringe Festival show to the Footlight Theatre at the Parliament House. This production includes scenes and material previously cut from the original script in order to fit in to the Fringe time allotment. Hell Freezes Over! is a follow-up to the 2009 show Welcome to Hell, a show that also enjoyed an extended run at the Parliament House.
In Hell Freezes Over, the audience plays the part of the residents of Hell who have gathered for the annual tenants meeting. Satan, played by Jeff Jones, has stepped out of Hell for a cross-country "soul scouting" mission. Elizabeth Murff, as Satan's assistant, is covering for him in his absence but things have noticeably gone afoul. The residents are running rampant, and an IRS auditor, played by Kevin Bee, is on his way in for an inspection audit. To make matters even worse, Satan's mother is coming in for an unexpected visit. The cast is rounded out by Doug Ba'aser, who portrays a wide variety of the notorious residents of Hell. When the battle of man vs. evil vs. mother begins, all Hell will break loose. The show is written by Jeff Jones and is directed by sixteen time Fringe Festival veteran Michael Marinaccio. Tickets are available at www.wanzie.com/boxoffice
Summer Fantasy V, Orlando Ballet School:
Top Orlando Ballet School Dancers and Alumni Present Special Summer Performance
Even though the Orlando Ballet's official season has come to a close, the award-winning Orlando Ballet School (OBS) will present a special performance to showcase its top current and former students. The Summer Fantasy V performance is the culminating production for the most advanced students participating in the school's five-week Summer Intensive program.
This year's program will feature excerpts from Paquita, Raymonda, Swan Lake and a new work from OBS instructor Olivier Munoz, a former principal dancer with English National Ballet. The performance will also feature former OBS students who are making impressive strides in their ballet careers. Jeffrey Cirio, a soloist with Boston Ballet, who has won nearly every major ballet competition including international ballet competitions in Jackson, New York, Seoul and Helsinki, in addition to the Princess Grace Award, will perform with his sister Lia Cirio, a principal dancer with Boston Ballet. Skyler Martin, who won a full tuition scholarship to study at the Royal Ballet School in England, will dance a solo. Erin Pritchard, who joined the Tulsa Ballet company after previously dancing with Houston Ballet, will also perform.
Three OBS alumni will also return as instructors for the summer intensive, including Cirio who will teach ballet; Eran Bugge, from the Paul Taylor Dance Company in New York and McClaine Timmerman, with Voci Dance in Orlando, will both teach modern dance.
The public performance allows pre-professional dancers to learn classic standards and new choreography from international dance creators. The show gives the Central Florida community a special opportunity to see the next generation of young dancers perform.
WHO: Top students and returning students perform for Orlando Ballet's summer intensive program
WHAT: Summer Fantasy V
WHEN: Sunday, July 18, 2009 at 5PM
WHERE: Garden Theatre, 160 W. Plant St., Winter Garden, FL 34787
TICKETS: $18 – $22 per person
Available at the Garden Theatre box office
Call 407-877-4736, ext. 10
Into the Woods, Lakeland Community Theatre:
Lakeland Community Theatre Presents…
Music by Stephen Sondheim
Lyrics by Stephen Sondheim
Book by James Lapine
An ambivalent Cinderella? A blood-thirsty Little Red Ridinghood? A Prince Charming with a roving eye? A Witch…who raps? They're all among the cockeyed characters in James Lapine and Stephen Sondheim's fractured fairy tale. When a Baker and his Wife learn they've been cursed with childlessness by the Witch next door, they embark on a quest for the special objects required to break the spell, swindling, lying to and stealing from Cinderella, Little Red, Rapunzel and Jack (the one who climbed the beanstalk).
Everyone's wish is granted at the end of Act One, but the consequences of their actions return to haunt them later, with disastrous results. What begins a lively irreverent fantasy in the style of "The Princess Bride" becomes a moving lesson about community responsibility and the stories we tell our children.
Nine Tony nominations: Winner for Best Score and Best Book.
Winner of the New York Drama Critics Circle Award. Grammy for original cast album.
Baker Daniel Ballard
Rapunzel Courtney Cummings
Cinderella's Father Ed Frisbie
Lucinda Stacy Fulford
Cinderella's Prince/Wolf Britt Michael Gordon
Cinderella's Mother Renee Hawk
Jack Benedict Heaps
Sleeping Beauty Jessica Killingsworth
Jack's Mother Anita Kollinger
Steward Marc Lingle
Stepmother Margaret Amanda Marsteller
Little Red Riding Hood Monica Molina
Snow White Ashley Roberts
The Witch Kristi Robinson
Baker's Wife Amber Salaz
Rapunzel's Prince Robert "Dutch" Salaz
Grandmother Gabrielle Thompson
Florinda/Giantess Lauren M. West
Cinderella Maureen Whelan
Narrator/Mysterious Man Roger Williams
Production Staff Director Mary T. Albright, Musical Director Patrick Fleitz, Set Design Alan S. Reynolds, Technical Director/Sound Design Roy Rossell, Light Design Ted Smith, Master Electrician/Scenic Artist Nick Judy, Costume Designs Mary T Albright, Props Mistress Lorraine Nardi, Stage Manager Lindsay Moss, Assistant Stage Manager James Skinner, Special Effects Bob Macey.
Production Dates: July 15-25, 2010
Thursdays, Fridays and Saturdays at 7:30 PM
Lake Mirror Center Theatre
Ticket prices: $20/adults,$15 College Students w/ID, $10/students 18 and under.
Group rates of $2 off the ticket price for groups of 15 or more.
To reserve tickets call the Box Office at 603-7529 or go online at www.LakelandCommunityTheatre.com.
May the Farce Be With You, Henegar Center for the Arts:
Summer Children's Theatre Production Opens at the Henegar Center for the Arts
May the Farce Be With You, featuring a delightful cast of Aliens and Superheroes is presented by Feller Theatre Academy and Stage 1 Stars in historic downtown Melbourne
The Henegar Center for the Arts presents its annual full-scale summer children's theatre production of May the Farce Be With You on Thursday July 15 at 10:00 am &1:00 pm and Saturday July 17 at 2 pm. Tickets are $4. Pam Mankowski and Evanne Floyd direct this full-length production with technical direction by Brock Tucker. May the Farce Be With You is produced by the Henegar Center's Feller Theatre Academy.
David Rogers wrote this hilarious script, where we meet Acrobatman, Albatross, H20 Girl, Astarte, Tapeman, Miss Invisowrap, Electroman, Gadzooks, and that's only part of the cast of this comedy farce. They are members of the Elite Idealistic Earth Individuals Organization, commonly called the E.I.E.I.O. Their annual conference at a health spa called O. MacDonald's Farm (what else?) is attacked by invaders from the evil planet Inferna. Star Raider, the intergalactic villain, has discovered the secret of the E.I.E.I.O.'s strength: they all seem to be some insignificant Earth Person, then they change their clothes and become super heroes. So he has prepared a serum that will rob them of the ability to change clothes! Can the E.I.E.I.O. and the entire Earth be saved from enslavement by nonmembers Superbrain and Wizard Woman? Learn the answer in this takeoff on all the super heroes you've ever heard of. The tale will charm audiences, young and old. Colorful costumes, and a large cast of children performing leave the audience thoroughly entertained with laughter and joy.
The cast of 24 young actors ages, 8 -17 include featured performers Braden Huggins, Ellen Jorgenson, Jennifer Leggett, Nikki Machon, Breanna Earley, Johnny Taulbee, Spencer Reeves Jordan Barnett, Sarah Mays, Scarlette Pizarro, Daniel Griessler, Maggi Marshall, Emma Reeves, Brian Cortez, Bryan McKeone, Karina Elwood, Alyssa Feliciano, Nicole Hall, Chaice Philpott, Kara Marin and Lilli Marshall.
May the Farce Be With You is the ninth annual summer children's theatre production of the Henegar Center's Feller Theatre Academy and Stage 1 Stars. Plans are to include a permanent children's theatre production schedule into the busy programs of the Henegar Center for the Arts. These productions supplement the ongoing schedule of classes, workshops and seminars sponsored by the Feller Theatre Academy's Stage 1 Stars at the center. For information on the performing arts instructional classes call the Henegar Center at (321) 723-8698.
The programs of the Henegar Center for the Arts are sponsored in part by the Margaret Hines Foundation, the Brevard County Board of County Commissioners through the Brevard Cultural Alliance, Inc., and the City of Melbourne.
The Henegar Center for the Arts is fully handicapped accessible and located at 625 East New Haven Avenue in historic downtown Melbourne. Tickets are $4 and available at the Henegar Center box office Monday – Friday during business hours using major credit cards. For information and reservations call (321) 723-8698.
Flashback!, Ocala Civic Theatre:
FLASHBACK! TO CHILDHOOD AT OCALA CIVIC THEATRE
Three performances scheduled for this weekend – Friday and Saturday
Join us for this exuberant musical revue, FLASHBACK!, presented in an intimate setting. You are sitting so close you can see every expression and nuance on the faces of these advanced student actors. Flashback! by Alice Wilson with additional material supplied by Louis Moloney, Johnny Saldana, and Rachael Winfree, uses mime, dance, and song to celebrate the world of the contemporary child. It is a series of sparkling vignettes dramatizing memories common to all children – from receiving report cards to the mixed joys of family vacations. In its entertaining treatment of serious as well as humorous themes, this challenging play explores the years of change known to us all as "childhood."
Tickets are on sale now at $8.00 for adults and $5.00 for students for this dynamic show.
Performances are scheduled for Friday, July 16 at 7:00 p.m. and Saturday, July 17 at 2:00 p.m. and 7:00 p.m. For reservations, call the Ocala Civic Theatre box office at (352) 236-2274 or buy tickets online at http://www.ocalacivictheatre.com. (A service charge applies to online purchases.)
Don't miss this Flashback to childhood!
Bye Bye Birdie, Mount Dora Theatre Company:
The IceHouse Theatre kicks off the 63rd Season
Bye Bye Birdie, the 1960's smash hit musical will light up the IceHouse this summer. Rock and roll idol Conrad Birdie appears on The Ed Sullivan Show to kiss one lucky girl 'goodbye' before he is drafted into the Army. The humorous story revolves around Conrad's business manager/songwriter, Albert Peterson, who is caught between his controlling mother, Mae and his bright secretary/girlfriend, Rosie, as he tries to keep his own career afloat and both of the women in his life happy. This family-friendly show introduced "A Lot of Livin' to Do," "Put On A Happy Face," and that hilarious anthem of parental frustration and worry, "Kids!"
Music: Charles Strouse, Lyrics: Lee Adams, Book: Michael Stewart. Performances are Thursdays-Sundays, July 16 – August 8 at the IceHouse Theatre, 1100 N. Unser St., Mount Dora, 32757.
The talented cast of 34 performers is led by Jeff Lindberg as Albert Peterson, Ashley Willsey as Rosie Alvarez, Michael Kanago as Conrad Birdie and Loraine O'Connell as Mae Peterson. The girl who will be kissed by Birdie, Kim Macafee, is played by Christi Willard and her worried parents are Eddy Coppens and Jessica Hoehn as Mr. & Mrs. Macafee. Kim's younger brother is played by Isaac Hollingsworth as Randolph Macafee and her hapless boyfriend is Britton Hollingsworth in the role of Hugo Peabody.
Box Office Information: Tickets by phone, (352) 383-4616; in person at the IceHouse Box Office at 1100 N. Unser Street, Mount Dora, FL 32757, or online: www.icehousetheatre.com. Box Office Hours are Tuesday – Friday, 1 pm – 5 pm and one hour prior to show time on performance dates.
Individual Tickets: $20 Adults; $18 Seniors (62+ Thurs/Sun); $10 – $15 *Students: $10 (5-12 yrs.); $12 (13-18 yrs.); $15 (with college ID). Groups: $16 tickets for Groups of 15 or more.
Comments Off on Opening this week:
Posted in Children's theater, Community theater, Dance, Orlando theater, Regional theater, Theater festivals, Uncategorized
Tagged bye bye birdie, fantasyland theatrical, flashback!, greater tuna, hell freezes over!, henegar center for the arts, icehouse theatre, into the woods, jerome bixby's the man from earth, lakeland community theatre, little red riding hood show, lowndes shakespeare center, may the farce be with you, mount dora theatre company, nunsense, nunsense a-men!, ocala civic theatre, orlando ballet school, orlando repertory theatre, orlando shakespeare theatre, pb&j theatre factory, pointe arts, renegade theatre, summer fantasy v, target family theatre festival, tuesdays with morrie, wanzie presents
Lots going on this week, including the Tony Awards on CBS Sunday night. Here's a look at what's happening closer to home:
Posted in Cabaret, Children's theater, College and university theater, Community theater, New plays, Orlando theater, Readings, Uncategorized
Tagged aradhana tawari, breakthrough theatre, butterflies are free, footlight theatre, janine klain: gay bar star, kevin kelly, kyona levine, on golden pond, parliament house, play the moment, project F, schoolhouse rock live! jr., son of a preacher man, sweet evalina, ucf summerstage, valencia character company, wanzie presents, winter park playhouse
Openings this week
Posted on March 8, 2010 | Comments Off on Openings this week
Here's a new feature — a list of local theater openings every week.
Comments Off on Openings this week
Posted in Community theater, New plays, Orlando theater, Tours
Tagged 55 west: a high rise church street comedy, come back little sheba, complete works of william shakespeare (abridged), footlight theatre, garden theatre, king center for the performing arts, melbourne civic theatre, shoestring theatre, steel magnolias, stomp, surfside players, the prisoner of second avenue, wanzie presents


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text:
\section{Introduction}

Continuous time fractional calculus has been used widely in physics and engineering for memory effect, viscoelasticity, porous media etc \cite{kst06, diethelm10,lll17}. Among them, the Caputo's and Riemann-Liouville's definitions are very popular for power law memory kernels \cite{caputo1967linear, kst06, diethelm10,liliu2018}. Caputo's definition of fractional derivatives was first introduced in \cite{caputo1967linear} to study the memory effect of energy dissipation for some anelastic materials and soon became a useful modeling tool in engineering and physical sciences for nonlocal interactions in time (see \cite{kouxie04,fle2,fle4}). Compared with Riemann-Liouville derivatives, Caputo derivatives remove the singularities at the origin and are suitable for initial value problems \cite{liliu2018}.  There are other models that have power law kernels with certain cutoffs (especially exponential cutoffs) so that they can give transitions between different behaviors \cite{vergara2008,sandev2017generalized,liemert2017generalized,chen2018tempered,molina2018crossover}. In this paper, we are interested in discretizing gradient type time fractional dissipative problems. Moreover, we desire to use the numerical discretization to investigate the properties of the solutions of the time continuous dissipative problems.  The first problem is the time fractional stochastic differential equation (fractional SDE) of dissipative type, which is the overdamped limit of the generalized Langevin equation with fractional noise.  Another problem is the time fractional gradient flows in a separable Hilbert space.

In some complex systems, the Langevin equations cannot give accurate predictions and the interaction between the system and the surrounding heat bath can no longer be modeled by white noise \cite{kubo66,kouxie04}. The generalized Langevin equation (GLE)
\begin{gather}\label{eq:gle}
\begin{split}
&\dot{X}=v,\\
& m\dot{v}=-\nabla V(X)-\int_{t_0}^t\gamma(t-s)v(s)\,ds+\eta(t)
\end{split}
\end{gather}
was then proposed by Mori and Kubo \cite{mori65,kubo66} to describe such complex systems with memory. In this model, $-\int_{t_0}^t\gamma(t-s)v(s)\,ds$ is the friction acting on the system we consider and $t_0$ is the point where the memory is counted from (usually $0$ or $-\infty$). The friction is the mean effect of the interaction between the system and the heat bath. The last term $\eta(t)$ is the noise which is the fluctuation part of the interaction between the system and the heat bath. Later, the GLE was recovered by dimension reduction from Ford-Kac and Kac-Zwanzig models using Mori-Zwanzig projection (\cite{zwanzig73,coarsegrain1_eric,coarsegrain3_karniadakis,leimkuhler2018}).
In the GLE models, the noise $\eta$ and the kernel for the friction $\gamma(\cdot)$ satisfy the so-called fluctuation-dissipation theorem (FDT)
\begin{gather}\label{eq:fdt}
\mathbb{E}(\eta(t) \eta(t+\tau))= k T \gamma(|\tau|),~ \forall \tau\in \mathbb{R}.
\end{gather}
Intuitively, the random force and the frictional kernel all originate from the interaction between the system and the surrounding environment. When the energy balance is reached, they must be related for the system to achieve the correct temperature. In \cite{kouxie04}, Kou and Xie considered the GLE with fractional Gaussian noise to explain the subdiffusive behaviors for a protein molecule in solution. Later, this model was studied by many authors \cite{kou08,deng2009}. The fractional Gaussian noise is the distributional derivative of the fractional Brownian motion $B_H$ (see \cite{nualart06}  and section \ref{sec:formalfsde} for more details)
\begin{gather}
\eta=\sigma \dot{B}_H(t).
\end{gather}
Using FDT \eqref{eq:fdt} and considering the overdamped limit, we obtain the fractional SDE as the overdamped GLE (see section \ref{sec:formalfsde} for a simple derivation and the rigorous definition):
\begin{gather}
D_c^{2-2H}X=-\nabla V(X)+\sigma dB_H,
\end{gather}
where $D_c^{\alpha}$ is the Caputo derivative (see section \ref{sec:setup} for more explanation).
In \cite{lll17}, the fractional SDE has been studied theoretically. If the force is linear, it was shown that the process converges in law to a unique limiting measure. Moreover, if the FDT is satisfied, the limiting measure is the Gibbs measure. The general potential $V$ cases seem hard to justify. In  \cite{fang2018}, numerical methods have been designed for the overdamped GLE and the numerical results there give positive evidence. One of our goals in this paper is to use the numerical schemes to prove that the limiting measure is unique if the potential $V$ is strongly convex.

Though there might not be strong physical interpretation, the time fractional gradient flow is of its own mathematical interest and can be used for new phase field models (see \cite{chen2002phase,liu2003phase} for the phase field models). In particular, consider a separable Hilbert space $H$ and a functional $\phi: H\to \mathbb{R}$ that is lower semicontinuous. The time fractional gradient flow we consider is
\begin{gather}\label{eq:strongsol}
D_c^{\alpha}u\in -\partial\phi(u),~~u(0)=u_0,
\end{gather}
where the Frechet subdifferential $\partial\phi$ at $u$ is a set defined as
\begin{gather}
\partial \phi(u):= \left\{ \xi\in H:
\liminf_{w\to u}\frac{\phi(w)-\phi(u)-\langle \xi, w-u\rangle }{|w-u|} \ge 0 \right\}.
\end{gather}
If $\partial \phi$ contains a single point $\xi$, then we define $\mathrm{grad}\phi(u):=\xi$. We aim to investigate the discretization using our scheme in this paper and establish error estimates.  For related fractional gradient flow, one can see \cite{vergara2008} where the memory kernel takes the form $t^{-\gamma}e^{-\mu t}$ with exponential decay. If $\phi$ is convex, then $\partial\phi$ is accretive and some related Volterra equations have been discussed in \cite{cn78,cn81}, where the existence of generalized solutions have been established using the Yosida approximations. The equation we will consider is not included in these papers.

Numerical discretizations of time fractional differential equations and related equations have already been investigated by many authors \cite{lubich1986,linxu2007,lixu2009space,nochetto2016,jin2017correction,jin2018discrete,atangana2018new,shens2019a,shens2019b}.  In particular, the authors of \cite{lixu2009space,nochetto2016} applied the $L^1$ schemes, which approximate the Caputo derivative directly, for several dissipative problems. In \cite{jin2017correction}, some corrections are made for the first $k-1$ steps so that the non-smoothness at $t=0$ does not pollute the desired accuracy of the schemes. In \cite{shens2019a,shens2019b}, some spectral methods have been developed for fractional differential equations. Moreover, in \cite{fllx18}, some comparison principles for the discrete fractional equations have been established.
Unfortunately, using these discretizations to study the fractional SDE and time fractional gradient flows is not appropriate because the time continuous problems are not well understood yet.
Our approach is to consider the discretization of the integral formulation first and apply the deconvolution (see \cite{liliu2018note}) to obtain the discretization of the Caputo derivatives in differential form. Since the integral formulation is more suitable for passing the limit, we are then able to conclude the important results regarding the time-continuous problems and establish some error estimates. Note that the new scheme is not just discretization of Volterra integrals since some important properties will be proved based on the de-convolved sequence, which seems very hard using the discretization of the integral form. Besides the problems considered in this paper, the scheme proposed here or schemes derived
using the same philosophy may be applied for other problems \cite{molina2018crossover,owolabi2017mathematical}.

The rest of the paper is organized as follows. In section \ref{sec:setup}, we give the basic notations and propose the discretization of the Caputo derivatives using deconvolution. In section \ref{sec:property}, we prove some important properties of the new discretization. Section \ref{sec:fsde} and section \ref{sec:fgradientflow} are devoted to fractional SDE and time fractional gradient flows.  In particular, we show that the overdamped GLE with fractional noise has a unique limiting measure for strongly convex potentials; we also establish some error estimates for the strong solutions of time fractional gradient flows.

\section{Notation and setup for the discretization}\label{sec:setup}

Let $B$ be a Banach space. Consider the following equation for a mapping: $X: [0, T]\to B$:
\begin{gather}\label{eq:caputoexp}
D_c^{\alpha}X(t)=f(t),
\end{gather}
where $f: [0, T]\to B$ is some mapping. $D_c^{\alpha}$ represents the Caputo derivative
of order $\alpha\in (0, 1)$ (\cite{kst06, diethelm10}). If $X(\cdot)$ is regular enough, for example, absolutely continuous, the Caputo derivative traditionally is defined as
\begin{gather}\label{eq:caputotradition}
D_c^{\alpha}X(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{\dot{X}(s)}{(t-s)^{\alpha}}\,ds.
\end{gather}
In \cite{liliu2018,liliu2018compact}, a generalized definition of Caputo derivative based on convolution groups was proposed.  To explain this generalized defintion, we first recall the distributions $\{g_\beta\}$ in \cite{liliu2018}:
\begin{gather}\label{eq:g}
g_{\beta}(t)=
\begin{cases}
\frac{1}{\Gamma(1+\beta)}D\left(\theta(t)t^{\beta}\right),& \beta\in (-1, 0)\\
\frac{1}{\Gamma(\beta)}t_+^{\beta-1}, & \beta>0.
\end{cases}
\end{gather}
Here $\theta(t)$ is the standard Heaviside step function, $\Gamma(\cdot)$ is the gamma function, $t_+=\theta(t)t=\max(t, 0)$, and $D$ means the distributional derivative on $\mathbb{R}$. Indeed, $g_{\beta}$ can be defined for $\beta\in \mathbb{R}$ (see \cite{liliu2018}) so that $\{g_{\beta}: \beta\in\mathbb{R}\}$ forms a convolution group. In particular, we have
\begin{gather}\label{eq:grouprelation}
g_{\beta_1}*g_{\beta_2}=g_{\beta_1+\beta_2}.
\end{gather}
 Note that the support of $g_{\beta_i}$ ($i=1,2$) is bounded from left, so the convolution is well-defined.
\begin{definition}[\cite{liliu2018,liliu2018compact}] \label{def:caputo}
Let $0<\alpha<1$. Consider $X\in L_{\loc}^1([0, T),B)$. Given $X_0\in B$, we define the $\alpha$th order generalized Caputo derivative of $X$, associated with initial value $X_0$, to be a distribution as $D_c^{\alpha}X: C_c^{\infty}(-\infty, T; \mathbb{R})\to B$ with support in $[0, T)$, given by
\begin{gather}\label{eq:generalizeddef}
D_c^{\alpha}X=g_{-\alpha}*\Big((X-X_0)\theta(t)\Big).
\end{gather}
If$\ \lim_{t\to 0+}\frac{1}{t}\int_0^t\|X(s)-X_0\|_Bds=0$, we call $D_c^{\alpha}X$ the Caputo derivative of $X$.
\end{definition}
The weak Caputo derivatives  in \cite{liliu2018compact} for mappings in general Banach spaces was defined through a dual equality using right derivatives. One can verify easily that
the one in \cite{liliu2018compact}  agrees with Definition \ref{def:caputo}.
This generalized definition appears complicated. However, it is theoretically more convenient, since it allows us to take advantage of the underlying group structure. In fact, making use of the convolutional group structure \eqref{eq:grouprelation} (see \cite{liliu2018} for more details), it is straightforward to convert \eqref{eq:caputoexp} with \eqref{eq:generalizeddef} into the Volterra type equation
\begin{gather}\label{eq:Voltter}
X(t)=X_0+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}f(s)\,ds.
\end{gather}
Indeed \eqref{eq:Voltter} is well-known for regular enough $f$; see \cite[Lemma 2.3]{df02}. The theory in  \cite{liliu2018,liliu2018compact} tells us that it still holds for $f$ to be distributions.

For absolutely continuous functions, Definition \ref{def:caputo} reduces to \eqref{eq:caputotradition}.
In this paper, we sometimes need the generalized definition, Definition \ref{def:caputo}, and its equivalence to \eqref{eq:Voltter} since we need to consider the Caputo derivative of a continuous function later.

For numerical setup, we fix the terminal time $T$ and consider time step
\begin{gather}
k=T/N.
\end{gather}
Define $t_n=nk$. We will use $X_n$ to represent the numerical solution at $t_n$.

\subsection{Discretization of the fractional derivatives: two options}

Depending on whether we discretize \eqref{eq:caputoexp} or \eqref{eq:Voltter}, we can possibly have different schemes (see \cite[Section 6]{fllx18} for some relevant discussions).
Discretization of \eqref{eq:Voltter} and deconvolution yields a discretization of the Caputo derivative, whose implicit scheme turns out very useful for studying two important time fractional dissipative problems. In particular, we can conclude the asymptotic behavior of the fractional SDEs and study the time fractional gradient flows in separable Hilbert spaces ( see sections \ref{sec:fsde} and \ref{sec:fgradientflow} respectively).

\subsubsection{Discretization of the differential form}

Discretizing \eqref{eq:caputoexp} directly is well studied in literature (see \cite{diethelm1997,linxu2007}). The $L^1$ scheme in \cite{diethelm1997,linxu2007} is widely used in applications due to the good sign of the coefficients (see \cite{nochetto2016,liumazhou2017}). The scheme is given by
\begin{gather}
(\bar{\mathcal{D}}^{\alpha}X)_n
=k^{-\alpha}(\bar{c}_0X_n-\bar{c}_1X_{n-1}-\ldots-\bar{c}_n^n X_0).
\end{gather}
Here, the coefficients are given by
\begin{gather}
\begin{split}
& \Gamma(2-\alpha)\bar{c}_0=1,\\
& \Gamma(2-\alpha)\bar{c}_j=-((j+1)^{1-\alpha}-2j^{1-\alpha}+(j-1)^{1-\alpha}),~~1\le j\le n-1, \\
& \Gamma(2-\alpha)\bar{c}_n^n=(n^{1-\alpha}-(n-1)^{1-\alpha})
\end{split}
\end{gather}
We have the following observations:
(i) $\bar{c}_j>0$,  $\bar{c}_n^n>0$; (ii) $\bar{c}_0-\sum_{j=1}^{n-1}\bar{c}_j-\bar{c}_n^n=0$;
(iii)
\begin{gather}
\bar{c}_j=\frac{-1}{\Gamma(-\alpha)}j^{-1-\alpha}\left(1+O\left(\frac{1}{j}\right)\right),~
j\to\infty,\  \ \
\bar{c}_n^n=\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\left(1+O\left(\frac{1}{n}\right)\right).
\end{gather}

\subsubsection{Discretization of the integral form and deconvolution}
Alternatively, we can consider the discretization of the integral form \eqref{eq:Voltter} and then take deconvolution to get the approximation for
the Caputo derivative. In fact, discretizing the integral form has been well-studied in literature (see for example \cite{lubich1986,galeone2009}). Slightly different from the discretizations in these works, what we choose to do is to approximate $f$ with piecewise constant functions. Then, we take deconvolution and get the approximation to the differential form.

To start, we approximate $f(t)$ by
\begin{gather}
\tilde{f}(t)=F_n,~~t\in (t_{n-1}, t_n].
\end{gather}
Then, \eqref{eq:Voltter} gives the following scheme
\begin{gather}\label{eq:discretefracint}
X_n-X_0=k^{\alpha}\sum_{m=1}^{n}a_{n-m}F_m =: (J_kF)_n ,
\end{gather}
where the right hand side is a discrete integral and the sequence $a$ is given by
\begin{gather}\label{eq:seqa}
a=(a_0, a_1, \ldots, a_n,\ldots)= \frac{1}{\Gamma(1+\alpha)}(1, 2^{\alpha}-1, 3^{\alpha}-2^{\alpha}, \ldots).
\end{gather}

For convenience, we define $F_0=0$ and introduce the sequence $F\in B^{\mathbb{N}}$ by
\begin{gather}
F=(F_0, F_1, \ldots, F_n, \ldots),
\end{gather}
so that for $n\ge 0$, $X_n-X_0=k^{\alpha}(a*F)_n$. The convolution between $u$ and $v$ is given by
\begin{gather}
(u*v)_n=\sum_{m=0}^n u_m v_{n-m}=(v*u)_n.
\end{gather}

Let $a^{(-1)}$ be the convolution inverse of $a$ such that
\begin{gather}
a*a^{(-1)}=a^{(-1)}*a=\delta_d :=(1, 0, 0,\ldots).
\end{gather}
Then, we obtain for $n\ge 0$ that
$k^{-\alpha}(a^{(-1)}*(X-X_0))_n=F_n$. We therefore obtain a new scheme for discretizing the Caputo derivative
\begin{gather}\label{eq:capnewdis}
(\mathcal{D}^{\alpha}X)_n=k^{-\alpha}(a^{(-1)}*(X-X_0))_n.
\end{gather}
Though equivalent to the discretization of Volterra integral, we regard this as a new scheme because some important properties (e.g. Theorem \ref{thm:implicitcomp} (2)-(3) and \eqref{eq:fractionalgradientkeyestimate}) will be proved based on this differential form \eqref{eq:capnewdis}, which will be hard using the integral form \eqref{eq:discretefracint}.

\section{Properties of the discretization}\label{sec:property}

In this section, we discuss in detail the properties of discretization \eqref{eq:capnewdis}.
One can refer to \cite{liliu2018note} for some discussion of using deconvolution to define discrete fractional calculus.

We first introduce some definitions for the discussion. We say a sequence $v=(v_0, v_1, \ldots)$ is completely monotone if $((I-S)^jv)_k\ge 0$ for any $j\ge 0, k\ge 0$ where $(Sv)_j=v_{j+1}$.  A sequence is completely monotone if and only if it is the moment sequence of a Hausdorff measure (a finite nonnegative measure on $[0,1]$)  (\cite{widder41}). Another description is given below in Lemma \ref{lmm:generating}.  The generating function of a sequence $v=(v_0, v_1, \ldots)$ is defined by
\begin{gather}
F_v(z)=\sum_{n=0}^{\infty} v_n z^n.
\end{gather}
Another concept we introduce is the Pick function. A function $f:\mathbb{C}_+\to\mathbb{C}$ (where $\mathbb{C}_+$ denotes the upper half plane, not including the real line) is Pick if it is analytic such that $\mathrm{Im}(z)>0\Rightarrow \mathrm{Im}(f(z))\ge 0$.  Now, we state some properties of sequences in terms of the generating functions, for which we omit the proofs.
\begin{lemma}\label{lmm:generating}
\begin{enumerate}[(1)]
\item For convolution,  $F_{u*v}(z)=F_u(z)F_v(z)$, and
$F_{v^{(-1)}}(z)=(F_v(z))^{-1}$.

\item (\cite[Corollary VI.1]{fs09}) Assume $F_v(z)$ is analytic on $\Delta :=\{z: |z|<R, z\neq 1, |\mathrm{arg}(z-1)|>\theta\}$, for some $R>1, \theta\in (0, \frac{\pi}{2})$. If $F_v(z)\sim (1-z)^{-\beta}$ as $z\to 1, z\in \Delta$ for $\beta\neq 0, -1, -2, -3, \ldots$, then $v_n \sim \frac{1}{\Gamma(\beta)}n^{\beta-1},~n\to \infty$.

\item $\lim_{n\to\infty}v_n=\lim_{z\to 1^-}(1-z)F_v(z)$.

\item (\cite{lp16}) A sequence $v$ is completely monotone if and only if the generating function $F_v(z)=\sum_{j=0}^{\infty}v_jz^j$ is a Pick function that is analytic and nonnegative on $(-\infty, 1)$.
\end{enumerate}
\end{lemma}

For convenience, define a sequence $c=(c_0, c_1, \ldots, c_n,\ldots)$ as (see \eqref{eq:seqa} for $a$)
\begin{gather}
c_0=a_0^{(-1)},~~c_i=-a_i^{(-1)},~\forall i\ge 1.
\end{gather}
Moreover, it is convenient to introduce
\begin{gather}
c_n^n=c_0-\sum_{i=1}^{n-1}c_i.
\end{gather}
Then,  \eqref{eq:capnewdis} can be reformulated as
\begin{gather}\label{eq:capnewdis2}
\begin{split}
(\mathcal{D}^{\alpha}X)_n & =k^{-\alpha}\Big(c_0(X_{n}-X_0)-\sum_{i=1}^{n-1}c_i (X_{n-i}-X_0)\Big)\\
&=k^{-\alpha}\Big(c_0X_{n}-\sum_{i=1}^{n-1}c_i X_{n-i}
-c_n^n X_0\Big).
\end{split}
\end{gather}
Using the result in \cite{liliu2018note}, we have the following claims.
\begin{proposition}\label{pro:coefficients}
Consider scheme \eqref{eq:capnewdis2}. The following claims hold:
\begin{enumerate}[(1)]
\item $c_i>0$ for $i\ge 0$ and $c_0=\sum_{i=1}^{\infty}c_i=\Gamma(1+\alpha)$.
Consequently, $c_n^n=\sum_{i=n}^{\infty}c_i>0$ and $c_0=\sum_{i=1}^{n-1}c_i+c_n^n$.

\item We have the following asymptotics for the coefficients.
\begin{gather}
c_j=\frac{-1}{\Gamma(-\alpha)}j^{-1-\alpha}\left(1+o(1) \right),~
j\to\infty,\  \ \
c_n^n=\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\left(1+o(1) \right).
\end{gather}

\end{enumerate}
\end{proposition}

\begin{proof}
(1). First of all, we recall that (see \eqref{eq:g} for the definition of $g_{\alpha}$)
\[
a_n=\int_{n}^{n+1}g_{\alpha}(t)\,dt.
\]
Since $g_{\alpha}(\cdot)$ is completely monotone (which means $(-1)^m \frac{d^m}{dt^m}g_{\alpha}(t)\ge 0$ for any $t\in (0,\infty)$), then the sequence $a=\{a_n\}$ is completely monotone.
Since $a^{(-1)}=(c_0, -c_1, -c_2, \ldots)$,  by \cite[Theorem 2.3]{liliu2018note}, one has that
$c_0>0$ while $(c_1, c_2, \ldots)$ is a completely monotone sequence.
The sign of $c_i$ is thus proved.

Using the explicit formula for $a_n$,  $F_{a}(z)\to \infty$ as $z\to 1^-$. Hence, we find that
$F_{a^{(-1)}}(z)\to 0,~~z\to 1^-$.
Noting the sign of elements for $a^{(-1)}$, the monotone convergence theorem holds and thus  $\sum_{i=0}^{\infty}a_i^{(-1)}=0$. The other claims then follow accordingly.

(2). We consider the function
\[
H(z)=F_a(z)-(1-z)^{-\alpha}=: \sum_{n=0}^{\infty} d_n z^n.
\]
By \cite[Theorem VI.1]{fs09}, we have
$[z^n](1-z)^{-\alpha}=\frac{1}{\Gamma(\alpha)}n^{\alpha-1}(1+O(\frac{1}{n}))$, $n\to\infty$,
where $[z^n]F(z)$ means the coefficient of $z^n$ in the series expansion of $F(z)$ about $0$.
Hence, $|d_n|\le C\frac{1}{n^{2-\alpha}}$, and $H(z)$ is a locally bounded function (bounded on any compact set). Hence,
\[
F_{a^{(-1)}}(z)=\frac{(1-z)^{\alpha}}{1+(1-z)^{\alpha} H(z)}.
\]
Clearly, this function is analytic in $\{z: |z|<1+\varepsilon, z\neq 1, |\mathrm{arg}(z-1)|>\frac{\pi}{4}\}$ for some $\varepsilon>0$.
Applying the second claim in Lemma \ref{lmm:generating} gives the asymptotics for $c_j=-a_j^{(-1)}$ ($j\ge 1$).
Using the fact $c_n^n=\sum_{i=n}^{\infty}c_i$, the asymptotics for $c_n^n$ then follows.

\end{proof}

The discrete comparison principles are important for stability of numerical schemes. Below, we prove several important comparison criteria that are helpful for the stability of the implicit schemes. For the stability of some explicit schemes, one may refer to \cite{fllx18}.

\begin{theorem}\label{thm:implicitcomp}
Consider discretization \eqref{eq:capnewdis2}. Let $u=\{u_n\}$, $v=\{v_n\}$ and $w=\{w_n\}$ be three sequences in $\mathbb{R}^{\mathbb{N}}$, with $u_0\le v_0\le w_0$.
\begin{enumerate}[(1)]

\item (Convex functional) Suppose $E(\cdot): \mathbb{R}^d\to \mathbb{R}, X\mapsto E(X)$ is convex. Then,
\[
(\mathcal{D}^{\alpha}E(X))_n \le (\mathcal{D}^{\alpha}X)_n\cdot\nabla E(X_n).
\]

\item (Comparison principle for nonincreasing $f$) Suppose $f(s, \cdot)$ is non-increasing. Assume $u,v,w$ satisfy the discrete implicit relations
\[
(\mathcal{D}^{\alpha}u)_n\le f(t_n, u_n),~~(\mathcal{D}^{\alpha}v)_n = f(t_n, v_n),~~
(\mathcal{D}^{\alpha}w)_n \ge f(t_n, w_n).
\]
Then,  $u_n\le v_n \le w_n$.

\item (Comparison principle for Lipschitz $f$) Assume $f$ is Lipschitz continuous in the second variable with Lipschitz constant $L$. If
\[
(\mathcal{D}^{\alpha}u)_n\le f(t_n, u_n),~~(\mathcal{D}^{\alpha}v)_n = f(t_n, v_n),~~
(\mathcal{D}^{\alpha}w)_n \ge f(t_n, w_n),
\]
then for step size $k$ with $c_0>k^{\alpha}L$, $u_n\le v_n \le w_n$.

\item (Comparison principle for integral form)
Assume $f(s,\cdot)$ is non-decreasing and Lipschitz continuous in the second variable with Lipschitz constant $L$. Introduce $f_u, f_v, f_w$ by, for example, $f_u=(0, f(t_1,u_1), f(t_2,u_2), \ldots)$. If
\begin{gather*}
u_n \le u_0+(J_k f_u)_n,~~
 v_n =v_0+(J_k f_v)_n,~~ w_n \ge w_0+(J_k f_w)_n,
\end{gather*}
 then for step size $k$ with $k^{\alpha}a_0L=k^{\alpha}L/c_0<1$, $u_n\le v_n \le w_n$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1).  By \eqref{eq:capnewdis2}, Proposition \ref{pro:coefficients} and the convexity of $E(\cdot)$, we have
\begin{gather*}
\begin{split}
(\mathcal{D}^{\alpha}X)_n\cdot\nabla E(X_n) & =\nabla E(X_n)\cdot
k^{-\alpha}\left( \sum_{i=1}^{n-1}(X_n-X_{n-1})+c_n^n(X_n-X_0)\right)\\
& \ge k^{-\alpha}\left(\sum_{i=1}^{n-1}(E(X_n)-E(X_{n-1}))+c_n^n(E(X_n)-E(X_0))\right)
=(\mathcal{D}^{\alpha} E(X))_n.
\end{split}
\end{gather*}

(2). Let $\xi_n=u_n-v_n$. Then,
\[
\mathcal{D}^{\alpha}\xi \le f(t_n, u_n)-f(t_n, v_n).
\]
Multiplying $1(\xi_n\ge 0)$ on both sides, and defining $\eta_n=\xi_n \vee 0=\max(\xi_n, 0)$, we have
\[
1(\xi_n \ge 0)k^{-\alpha} \Big(c_0\xi_n-\sum_{i=1}^{n-1}c_i \xi_{n-i}
-c_n^n\xi_0 \Big) \le 1(\xi_n \ge 0)(f(t_n, u_n)-f(t_n, v_n))\le 0.
\]
Since $\xi_n 1(\xi_n\ge 0)=\xi_n\vee 0=\eta_n$, $\xi_i 1(\xi_n\ge 0)\le \xi_i\vee 0=\eta_i$, one easily finds that
\[
(\mathcal{D}^{\alpha}\eta)_n \le 1(\xi_n \ge 0)(\mathcal{D}^{\alpha}\xi)_n \le 0.
\]
Since $\eta_0= 0$, one easily finds $\eta_n\le 0$, and hence $u_n\le v_n$.  It is similar to compare $v_n$ and $w_n$.

(3). We compare $u_n$ with $v_n$. We know already $u_0\le v_0$. Now, suppose $n\ge 1$ and assume for all $m\le n-1$, we have proved $u_m\le v_m$ already. We now consider $m=n$.
\[
k^{-\alpha}c_0(u_n-v_n)\le (\mathcal{D}^{\alpha}(u-v))_n
\le f(t_n, u_n)-f(t_n, v_n)\le L |u_n-v_n|.
\]
If $u_n> v_n$, we then have $(k^{-\alpha}c_0-L)(u_n-v_n)\le 0$ which is clearly not true. Hence, induction shows that the claim is true for all $n$.
Comparing the sequence $v$ with $w$ is similar and we omit.

(4).  Direct computation shows that
\[
u_n-v_n\le u_0-v_0+k^{\alpha}a_0(f(t_n, u_n)-f(t_n, v_n))+k^{\alpha}\sum_{m=1}^{n-1}a_{n-m}(f(t_m, u_m)-f(t_m, v_m)).
\]
If we have proved that $u_m\le v_m$ for $m\le n-1$, then
$u_n-v_n\le k^{\alpha}a_0 L|u_n-v_n|$.
The proof then follows by induction similarly as in 3. Comparing the sequence $v$ with $w$ is similar and we omit.
\end{proof}

We now consider the stability of the implicit scheme applied to the simple FODEs
\begin{gather}\label{eq:simpleode}
D_c^{\alpha}X=\lambda X,~~X(0)=x_0>0,
\end{gather}
whose solution is given by $X(t)=x_0 E_{\alpha}(\lambda t^{\alpha})$, where
\[
E_{\alpha}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(n\alpha+1)}
\]
is the Mittag-Leffler function \cite{HMS}.
\begin{theorem}\label{thm:schemeforlinear}
Consider the implicit scheme applied on the fractional ODE \eqref{eq:simpleode}:
\begin{gather}\label{eq:simplefodescheme}
(\mathcal{D}^{\alpha}X)_n=\lambda X_n \Leftrightarrow X_n=X_0+\lambda k^{\alpha}\sum_{m=1}^{n}a_{n-m}X_m.
\end{gather}
\begin{enumerate}[(1)]
\item If $\lambda>0$ and $k^{\alpha}\lambda<c_0$, then $X(t_n) \le X_n \le X_{n+1}$.
If otherwise $\lambda<0$,
$\lim_{n\to\infty}X_n=0$.

\item Consider $\lambda>0$. Suppose $k_i,i=1,2$ satisfy $k_i^{\alpha}\lambda<c_0$ and $k_1=2^{m_1} k_2$ for some $m_1\in\mathbb{N}$. Let $X_n^{(i)}$ be the numerical solutions. Define the piecewise constant functions $\bar{X}_i(t)$ by $\bar{X}_i(t)=X_n^{(i)},~t\in (t_{n-1}^{(i)}, t_n^{(i)}]$ for $i=1,2$. Then,
$\bar{X}_1(t)\ge \bar{X}_2(t)$.
Consequently, there exists a constant $C(\alpha, T)>0$ such that  for any $k$ with $k^{\alpha}\lambda\le \frac{1}{2}c_0$,
\begin{gather}\label{eq:stabilityineqa}
\sup_{n: nk\le T}X_n\le C(\alpha, T)X_0.
\end{gather}

\item When $k$ is sufficiently small, $\sup_{n: nk\le T}|X_n - X(t_n)|\le C_1(\alpha, T) k^{\alpha}$.
\end{enumerate}
\end{theorem}

\begin{proof}

(1). Consider $\lambda>0$. The induction formula from the differential form  reads
\[
(c_0-k^{\alpha}\lambda)X_n=\sum_{i=1}^{n-1}c_i X_{n-i}+c_n^n X_0.
\]
If $n=1$, $c_1^1=c_0$. Then, we clearly have $X_1=\frac{c_0}{c_0-k^{\alpha}\lambda}X_0>X_0$.
Suppose we have proved $X_m\ge X_{m-1}$ for $m\le n$ with $n\ge 1$. For $n+1$,
\[
(c_0-k^{\alpha}\lambda)X_{n+1}= \sum_{i=1}^{n-1}c_i X_{n+1-i}+(c_n X_1+c_{n+1}^{n+1}X_0)
\ge \sum_{i=1}^{n-1}c_i X_{n-i}+(c_n X_0+c_{n+1}^{n+1}X_0).
\]
Since $c_n+c_{n+1}^{n+1}=c_n^n$, the claim then follows.

Now, consider the equivalent integral form (second in \eqref{eq:simplefodescheme}).
\[
X_n=X_0+\frac{\lambda}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}X_{j}\,ds.
\]
The accurate solution satisfies
\[
X(t_n)=X_0+\frac{\lambda}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}X(s)\,ds
\le X_0+\frac{\lambda}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}X(t_j)\,ds.
\]
By the third claim in Theorem \ref{thm:implicitcomp},  $X(t_n)\le X_n$.

Now, we consider $\lambda<0$. Recall that $F_0=0$ and
\[
X_n-X_0=k^{\alpha}\lambda a*(X-X_0\delta_{n0}).
\]
The generating function of $X^n$ is thus given by
\[
F_X(z)=X_0\frac{(1-z)^{-1}-k^{\alpha}\lambda F_a(z)}{1-k^{\alpha}\lambda F_a(z)}.
\]
As $z\to 1^-$, $F_a(z)\to\infty$, $(1-z)F_a(z)\to 0$, and hence
\[
\lim_{n\to\infty}X_n=\lim_{z\to 1^-}(1-z)F_X(z)=0.
\]

(2). We only need to consider $m_1=1$ (or $k_1=2k_2$).  By (1), the piecewise constant functions $\bar{X}_i(t)$'s are nondecreasing. Suppose that for $n\ge 1$,  one has
$\bar{X}_1(t)\ge \bar{X}_2(t)$, $t\in [0, (n-1)k_1]$. Then, for $t\in ((n-1)k_1, nk_1]$, one only needs $\bar{X}_1(nk_1)\ge \bar{X}_2(nk_1)=\bar{X}_2(2nk_2)$ since $\bar{X}_2$ is nondecreasing. By the integral formulation (second in \eqref{eq:simplefodescheme}),
\begin{align*}
\bar{X}_1(nk_1)& =X_0+\frac{\lambda}{\Gamma(\alpha)}\int_0^{t_{n-1}^{(1)}}(nk_1-s)^{\alpha-1}\bar{X}_1(s)\,ds
+\frac{\lambda}{\Gamma(\alpha)}\int_{t_{n-1}^{(1)}}^{t_n^{(1)}}(nk_1-s)^{\alpha-1}\bar{X}_1(nk_1)\,ds\\
&\ge X_0+\frac{\lambda}{\Gamma(\alpha)}\int_0^{(n-1)k_1}(nk_1-s)^{\alpha-1}\bar{X}_2(s)\,ds+\frac{\lambda k_1^{\alpha}}{\Gamma(1+\alpha)}\bar{X}_1(nk_1)
\end{align*}

On the other hand,
\begin{align*}
\bar{X}_2(nk_1)=X_0+\frac{\lambda}{\Gamma(\alpha)}\int_0^{t_{n-1}^{(1)}}(nk_1-s)^{\alpha-1}\bar{X}_2(s)\,ds+\frac{\lambda}{\Gamma(\alpha)}\int_{t_{n-1}^{(1)}}^{t_n^{(1)}}(nk_1-s)^{\alpha-1}\bar{X}_2(s)\,ds.
\end{align*}
The last term is simply controlled by $\frac{\lambda k_1^{\alpha}}{\Gamma(1+\alpha)}\bar{X}_2(2nk_2)$ due to monotonicity of $\bar{X}_2$.
Since $c_0=\Gamma(1+\alpha)$ and $\bar{X}_2(2nk_2)=\bar{X}_2(nk_1)$, we then find
\[
\bar{X}_1(nk_1)\ge \frac{1}{1-k_1^{\alpha}\lambda/c_0}\left(X^0+\frac{\lambda}{\Gamma(\alpha)}\int_0^{(n-1)k_1}(nk_1-s)^{\alpha-1}\bar{X}_2(s)\,ds \right)\ge \bar{X}_2(2nk_2).
\]

We now prove the stability. For any step size $k$, we choose $k_0=2^m k$ such that $k_0^{\alpha}\lambda
\in (\frac{c_0}{2^{1+\alpha}}, \frac{c_0}{2}]$.
Then, with time step $k_0$, there are
\[
N_0=\frac{T}{k_0} \le T (2^{1+\alpha}\lambda/c_0 )^{1/\alpha}
\]
steps. Then consider the induction using the differential form:
\[
(c_0-k_0^{\alpha}\lambda)X_n \le \sum_{i=1}^{n-1}c_i X_{n-m}+c_n^n X_0
\le c_0 X_{n-1}.
\]
Hence, $X_n\le 2X_{n-1}\le 2^{N_0}X_0$.
The claim then follows.

(3). By the explicit formula of the solution for \eqref{eq:simpleode}, we know $X$ is $\alpha$-H\"older continuous and for $t>0$, it is smooth. Inserting $X(\cdot)$ into the integral form, we have
\begin{gather*}
X(t_n)=X_0+\frac{\lambda}{\Gamma(\alpha)}\sum_{m=1}^n \int_{t_{m-1}}^{t_m}(t_n-s)^{\alpha-1}X(s)\,ds
=X_0+\lambda k^{\alpha}\sum_{m=1}^n a_{n-m}X(t_m)+R_n,
\end{gather*}
where
\[
R_n=\frac{\lambda}{\Gamma(\alpha)}\sum_{m=1}^n \int_{t_{m-1}}^{t_m}(t_n-s)^{\alpha-1}(X(s)-X(t_m))\,ds,
\]
and thus
\[
|R_n|\le C(T)k^{\alpha}\frac{\lambda}{\Gamma(\alpha)}\sum_{m=1}^n \int_{t_{m-1}}^{t_m}(t_n-s)^{\alpha-1}\,ds
=C_1(T,\alpha)k^{\alpha}.
\]
Hence, the error $E^n:=|X_n-X(t_n)|$ satisfies
\[
E^n\le k^{\alpha}|\lambda|\sum_{m=1}^n a_{n-m}E_m+Ck^{\alpha}.
\]
Using the comparison principle for integral formulation in Theorem \ref{thm:implicitcomp} and the stability result \eqref{eq:stabilityineqa}, we have for $k$ sufficiently small that $E^n\le C(\alpha, T) k^{\alpha}$.
\end{proof}

By Theorem \ref{thm:schemeforlinear}, the following claims hold when we compare the numerical solution with the exact solutions of some FODEs.
\begin{corollary}
Suppose $f(\cdot)\in C^2[A,\infty)$ is nondecreasing and globally Lipschitz for some $A\in\mathbb{R}$. Let $u(\cdot)$ be the solution to the FODE $D_c^{\alpha}u=f(u)$ with $u(0)=U>A$ and $f(U)>0$.  Let $\{u_n\}$ be the numerical solution of the implicit scheme $(\mathcal{D}^{\alpha}u)_n=f(u_n),~~u_0=u(0)$.
Then, for $k$ sufficiently small, $u(t_n)\le u_n \le u_{n+1}$. Moreover, for any $T$ such that $u$ exists on $[0, T]$, we have for some $C(T)>0$ that
\begin{gather}
\sup_{n: nk\le T}|u(t_n)-u_n|\le C(T)k^{\alpha}.
\end{gather}
\end{corollary}
\begin{proof}
The solution to the FODE satisfies the following (\cite{fllx18}):
\begin{gather}\label{eq:utnineq}
u(t_n) =u_0+\frac{1}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}f(u(t_j))\,ds+R_n,
\end{gather}
where
\[
R_n=\frac{1}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}\Big(f(u(s))-f(u(t_j))\Big)\,ds.
\]
By the theory in \cite{fllx18}, $u(\cdot)$ is nondecreasing and thus $R_n\le 0$. Consequently, applying Theorem \ref{thm:implicitcomp} (4), we have $u(t_n)\le u_n$.

To prove that $\{u_n\}$ is nondecreasing, we use induction. It is clear that $u_0\le u_1$. Now, assume we have proved $u_0\le u_1\le \cdots \le u_{n}$ for $n\ge 1$. We now prove $u_n\le u_{n+1}$. Using the equivalent integral form, we find
\begin{gather}\label{eq:unineq}
\begin{split}
u_{n}-k^{\alpha}a_0f(u_n) &=u_0+\frac{1}{\Gamma(\alpha)}\sum_{j=1}^{n-1}\int_{t_{j-1}}^{t_j}(t_{n+1}-s)^{\alpha-1}f(u_j)\,ds \\
&\le u_0+\frac{1}{\Gamma(\alpha)}\sum_{j=2}^{n}\int_{t_{j-1}}^{t_j}(t_{n+1}-s)^{\alpha-1}f(u_j)\,ds \\
& \le u_{n+1}-k^{\alpha}a_0 f(u_{n+1}).
\end{split}
\end{gather}
This implies that $u_n\le u_{n+1}$ when $k$ is sufficiently small.

Finally, by \eqref{eq:utnineq} and \eqref{eq:unineq}, we have
\[
|u(t_n)-u_n|\le \frac{1}{\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}|f(u(t_j))-f(u_j)|\,ds
+|R^n|
\]
It is well-known that $u$ is $\alpha$-H\"older continuous on $[0, T]$ (one can for example combine \cite[Lemma 3.1]{fllx18} and \cite[Theorem 3.1]{skm93}). Consequently, $\sup_{n: nk\le T}|R_n|\le C(T)k^{\alpha}$. Hence,
\[
|u(t_n)-u_n|\le \frac{L}{\Gamma(\alpha)}\sum_{j=1}^{n}|u(t_j)-u_j|\int_{t_{j-1}}^{t_j}(t_n-s)^{\alpha-1}\,ds
+C(T)k^{\alpha}.
\]
Applying Theorem \ref{thm:schemeforlinear} (2), we thus find $\sup_{n: nk\le T}|u(t_n)-u_n|\le C(T)k^{\alpha}$.
\end{proof}

\section{Limiting behavior of fractional SDE}\label{sec:fsde}

In this section, we use the implicit scheme corresponding to the discretization \eqref{eq:capnewdis2} to study the fractional SDE as advertised in the introduction.
In particular, we first of all provide some details for the derivation of the FSDE, and then prove that when the potential is strongly convex, there is a unique limiting measure for the FSDE.

\subsection{A formal derivation of the fractional SDE}\label{sec:formalfsde}

We first of all derive the auto-correlation function for the fractional noise.  Recall that the fractional Brownian motion has the following:
\begin{gather}\label{fb:cov}
\mathbb{E}(B_t^HB_s^H)=R_H(s, t) := \frac{1}{2}\left(s^{2H}+t^{2H}-|t-s|^{2H}\right).
\end{gather}
Fix $\tau\neq 0$. Formally, for $t>0$ with $t+\tau>0$, it holds that
\begin{multline}
\mathbb{E}(\dot{B}_H(t)\dot{B}_H(\tau+t))=\lim_{h\to 0, h_1\to 0}\mathbb{E}\left(\frac{B_H(t+h_1)-B_H(t)}{h_1}\frac{B_H(t+\tau+h)-B_h(t+\tau)}{h}\right)\\
  =\lim_{h\to 0, h_1\to 0}
  \frac{1}{2hh_1}\bigl(|\tau+h|^{2H}-|\tau+h-h_1|^{2H}-|\tau|^{2H}+|\tau-h_1|^{2H}\bigr)=H(2H-1) |\tau|^{2H-2}.
\end{multline}
Assume there is no extra singularity for $\tau=0$, we check formally for $s<t$:
\begin{gather}\label{fb:cov2}
\begin{split}
\mathbb{E}(B_t^HB_s^H) &=\int_0^t\int_0^s \mathbb{E}\left(\dot{B}_H(z)\dot{B}_H(w) \right) \,dz dw \\
&=\int_0^s\int_0^tH(2H-1)|z-w|^{2H-2}dzdw =\frac{1}{2}(s^{2H}+t^{2H}-(t-s)^{2H}).
\end{split}
\end{gather}
Now that \eqref{fb:cov2} agrees with \eqref{fb:cov}. Hence, the assumption for no extra singularity at $\tau=0$ is reasonable.
According to FDT \eqref{eq:fdt}, the GLE \eqref{eq:gle}
 is then reduced to the following dimensionless equation
\[
\varepsilon\dot{v}=-\nabla V-\frac{1}{\Gamma(2H-1)}\int_0^t(t-s)^{2H-2}v(s)\,ds
+\frac{\sqrt{2}}{\sqrt{\Gamma(2H+1)}}\dot{B}_H.
\]
Here, $\varepsilon=\frac{mT^{2H}}{\gamma_0}$ with $T$ being the scale for time, $\gamma_0$ being typical scale for the friction (see \cite{lll17} for detials). In the overdamped regime, $\varepsilon\ll 1$, the  GLE with fractional Gaussian noise formally corresponds to the fractional SDE
\begin{gather}
D_c^{\alpha}X=-\nabla V(X)+\sigma \dot{B}_H,
\end{gather}
with $\alpha=2-2H, ~\sigma=\frac{\sqrt{2}}{\sqrt{\Gamma(2H+1)}}$.
This overdamped generalized Langevin equation (overdamped GLE) is rigorously defined through the following integral formulation
\begin{gather}\label{eq:fsde2}
X(t)=X_0+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}b(X(s))\,ds+G(t)
\end{gather}
where
\begin{gather}
G(t) :=\frac{\sigma}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1} dB_H(s).
\end{gather}
We can consider generally $\alpha\in (0, 1)$ and $\sigma>0$ for fractional SDEs. Of course, only the one with $\alpha=2-2H$ has physical significance, which is the overdamped GLE. It has  been shown in \cite{lll17} that when $\alpha=2-2H$ and $\sigma=\frac{\sqrt{2}}{\sqrt{\Gamma(2H+1)}}$, $G(t)$ is another fractional Brownian motion with Hurst parameter $1-H$ up to some multiplicative constant:  $G(t)\sim \beta_H B_{1-H}$ with $\beta_H=\frac{\sqrt{2}}{\sqrt{\Gamma(3-2H)}}$.

When the force $-\nabla V(x)$ is linear, the distribution of $X$ converges algebraically to the Gibbs measure (\cite{lll17}). For general cases, whether it converges to the Gibbs measure is unknown.   Recently, in the case of overdamped GLE, some numerical experiments indicate that the law of $X$ still converges algebraically to the corresponding Gibbs measure for general potential (\cite{fang2018}).

\subsection{Convergence to equilibrium for strongly convex potentials}
In this subsection, we will try to use our discretization to study the limit behaviors of the FSDE for the strongly convex potential $V$. In particular, we show that there is a unique limiting measure as $t\to\infty$. Letting
\begin{gather}
b(x)=-\nabla V(x),
\end{gather}
we will assume the following.
\begin{assumption}
There exists some $\mu>0$ such that
\begin{gather}
(x-y)\cdot (b(x)-b(y))\le -\mu |x-y|^2,~\forall x, y\in\mathbb{R}^d.
\end{gather}
Moreover, $b(\cdot)$ is Lipschitz continuous so that for some $L>0$,
\begin{gather}\label{eq:lip}
|b(x)-b(y)|\le L|x-y|.
\end{gather}
\end{assumption}

The Lipschitz condition of $b$ may be relaxed by proving that the probability density of the process decays fast at infinity. Since this is not our focus, we assume the Lipschitz condition for simplicity.
 As proved in  \cite{lll17} and \cite{fang2018}, with assumption \eqref{eq:lip}, the fractional SDE \eqref{eq:fsde2} has a unique strong solution and for any $T>0$ there exists $C(T)>0$ such that the following hold.
\begin{gather}\label{eq:momentholder}
\sup_{t\ge T} \mathbb{E}|X(t)|^2\le C(T),\quad \sup_{t\ge T}\sqrt{\mathbb{E}|X(t+\delta)-X(t)|^2}\le C(T)\delta^{H+\alpha-1}.
\end{gather}

Now, we consider the limiting behavior of the law for $X(t)$.
Given two different initial data $X^{(i)}(0)$, $i=1,2$, we consider the strong solutions of \eqref{eq:fsde2}.
We will use the synchronization coupling to compare the distributions of the two processes.
Taking the difference between two solutions, we have
\[
X^{(1)}(t)-X^{(2)}(t)=X^{(1)}(0)-X^{(2)}(0)+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}(b(X^{(1)}(s))-b(X^{(2)}(s)))\,ds.
\]
To get the idea of a proof, we apply the theory in \cite{liliu2018} so that the Caputo derivatives can be defined pathwise for $X^{(1)}-X^{(2)}$ (see also Section \ref{sec:setup} for the brief introduction). In the distributional sense, it holds that
\begin{gather}
D_c^{\alpha}(X^{(1)}-X^{(2)})(t)=b(X^{(1)}(t))-b(X^{(2)}(t)),~\text{almost surely}.
\end{gather}
If $X^{(1)}-X^{(2)}$ is regular enough, applying \cite[Proposition 3.11]{liliu2018}, we have
\begin{gather}\label{eq:continuousineq}
\frac{1}{2}D_c^{\alpha}|X^{(1)}-X^{(2)}|^2(t)\le (X^{(1)}-X^{(2)})\cdot(b(X^{(1)})-b(X^{(2)}))\le -\mu |X^{(1)}-X^{(2)}|^2.
\end{gather}
If we define $u(t):=\mathbb{E}\Big(|X^{(1)}(t)-X^{(2)}(t)|^2\Big)$, it then holds that
\[
D_c^{\alpha}u(t)\le -2\mu u(t).
\]
Applying the comparison principle for $D_c^{\alpha}u=f(t, u)$ with nonincreasing $f(t,\cdot)$ (see, for example, \cite[Theorem 2.1]{fllx18note}) yields
\begin{gather}\label{eq:desiredcontrol}
u(t)\le u(0)E_{\alpha}(-2\mu t^{\alpha}).
\end{gather}
If this is true, we then are able to compare the laws of the two strong solutions of \eqref{eq:fsde2}  under Wasserstein-$2$ distance. Recall that the Wasserstein-$2$ distance is given by \cite{santambrogio2015}
\begin{gather}\label{eq:W2}
W_2(\mu, \nu)=\left(\inf_{\gamma \in \Pi(\mu,\nu)}\int_{\mathbb{R}^d\times\mathbb{R}^d}|x-y|^2 d\gamma\right)^{1/2},
\end{gather}
where $\Pi(\mu,\nu)$ is the set of joint distributions whose marginal distributions are $\mu$ and $\nu$ respectively. Equation \eqref{eq:desiredcontrol} will imply the convergence of the law of the process to the unique limiting measure.

The issue in the above argument is that \eqref{eq:continuousineq} is not justified rigorously. In the following, we shall utilize the implicit scheme based on discretization \eqref{eq:capnewdis2} to prove the convergence of the law.  In fact, we have the following claims.
\begin{theorem}\label{thm:limitsde}
Suppose assumption \eqref{eq:lip} holds and $X^{(i)}(t)$ are the two strong solution to the FSDE \eqref{eq:fsde2} with initial data $X^{i}(0)=X_0^{(i)}\sim \mu_0^{(i)}$ ($i=1,2$), where $\mu_0^{(i)}$ are some given probability measures.
Then, the laws of $X^{(i)}(t)$ satisfy in Wasserstein-$2$ distance that
\begin{gather}
W_2(\mu^{(1)}(t), \mu^{(2)}(t))\le W_2(\mu_0^{(1)}, \mu_0^{(2)})\sqrt{E_{\alpha}(-2\mu t^{\alpha})}.
\end{gather}
Consequently, the FSDE model has a unique limiting measure $\pi$.
\end{theorem}

We will apply the following backward Euler scheme based on \eqref{eq:capnewdis2} to FSDE \eqref{eq:fsde2}.
\begin{gather}\label{eq:fsdescheme}
X_n=X(0)+k^{\alpha}\sum_{m=1}^n a_{n-m} b(X_m)+G(t^n).
\end{gather}
We need some preparation for the complete proof. The first is the following convergence result of the scheme \eqref{eq:fsdescheme}.
\begin{lemma}\label{lmm:convresult}
Suppose assumption \eqref{eq:lip} holds and $X(t)$ is the unique strong solution to \eqref{eq:fsde2}.
Let $X_n$ be the numerical solution to \eqref{eq:fsdescheme}.
Then, for $k$ with $k^{\alpha}L<c_0/2$,
\begin{gather}
\sup_{n: nk\le T}\sqrt{\mathbb{E}(|X_n-X(nk)|^2)} \le C(\alpha, T) k^{\alpha+H-1}.
\end{gather}
\end{lemma}
\begin{proof}
The proof is very similar to that for the third claim in Theorem \ref{thm:schemeforlinear}.
In fact, the strong solution of \eqref{eq:fsde2} satisfies
\begin{gather}\label{eq:localtruncation}
X(t_n)=X_0+k^{\alpha}\sum_{m=1}^{n}a_{n-m} b(X(t_m))+R_n,
\end{gather}
where
\[
R_n:=\frac{1}{\Gamma(\alpha)}\sum_{m=1}^n \int_{t_{m-1}}^{t_m}(b(X(s))-b(X(t_m))).
\]
Using \eqref{eq:lip} and \eqref{eq:momentholder}, one finds
\[
(\mathbb{E}|R_n|^2)^{1/2}\le C k^{\alpha+H-1}.
\]
Taking the difference between \eqref{eq:fsdescheme} and \eqref{eq:localtruncation} and defining
$E_n:=(\mathbb{E}|X^n-X(t_n)|^2)^{1/2}$,  one then has
\begin{gather}
E_n \le k^{\alpha}L\sum_{m=1}^n a_{n-m} E_m +Ck^{\alpha+H-1}.
\end{gather}
Finally, using the comparison principle for integral formulation in Theorem \ref{thm:implicitcomp} and the stability result \eqref{eq:stabilityineqa}, the claim follows.
\end{proof}

Consider two numerical solutions $\{X^{(1)}_n\}$ and $\{X^{(2)}_n\}$ with initial data $X_0^{(i)}$ ($i=1,2$), with the synchronization coupling. The variable $Z_n:=X^{(1)}_n - X^{(2)}_n$ satisfies the following relation:
\begin{gather}
Z_n=Z_0+k^{\alpha}\sum_{m=1}^n a_{n-m} (b(X^{(1)}_m)-b(X^{(2)}_m)).
\end{gather}
Equivalently, one has almost surely that
\[
k^{-\alpha} \left(\sum_{j=1}^{n-1}c_j  (Z_n-Z_{n-j})+c_n^n(Z_n-Z_0) \right)=b(X^{(1)}_n)-
b(X^{(2)}_n).
\]
Applying the first claim in Theorem \ref{thm:implicitcomp} for $E(u)=\frac{1}{2}u^2$, one has almost surely that
\begin{gather}\label{eq:directineqZn}
(\mathcal{D}^{\alpha}|Z|^2)_n
\le - 2 Z_n\cdot (b(X^{(1)}_n)-
b(X^{(2)}_n))\le -2\mu |Z_n|^2.
\end{gather}
The point is that one may pass this inequality somehow to the strong solutions of \eqref{eq:fsde2} by taking $k\to 0$.

\begin{proof}[Proof of Theorem \ref{thm:limitsde}]
Define
\[
u_n :=\mathbb{E}(|Z_n|^2),
\]
and correspondingly
\[
Z(t):=X^{(1)}(t)-X^{(2)}(t),~~u(t):=\mathbb{E}(|Z(t)|^2).
\]

A direct consequence of inequality \eqref{eq:directineqZn} is
\[
\sup_{n\ge 0}u_n\le u_0.
\]
Applying Theorem \ref{thm:implicitcomp} (2), $u_n \le v_n$, where $v_n$ solves the following induction formula:
\[
(\mathcal{D}^{\alpha}v)_n=-2\mu v_n,~~v_0=u_0=\mathbb{E}(|Z_0|^2).
\]
Let $v(t)$ solve the FODE, $D_c^{\alpha}v=-2\mu v$, ~$v(0)=u_0$. By Theorem \ref{thm:schemeforlinear},
$|v_n-v(t_n)|\le C k^{\alpha}$.

Applying \eqref{eq:momentholder} and Lemma \ref{lmm:convresult}, one has
\[
|u(t_n)-u_n|\le (\sqrt{u}_0+C_1(T))\sqrt{\mathbb{E}(|Z_n-Z(t_n)|^2)} \le C k^{\alpha+H-1}.
\]
Hence, for all $n, nk\le T$, it holds that
\[
u(t_n)\le v(t_n)+C(k^{\alpha+H-1}+k^{\alpha}).
\]
Taking $k\to 0$ then gives
\begin{gather}
u(t)\le v(t)\le u_0 E_{\alpha}(-2\mu t^{\alpha}).
\end{gather}
This inequality clearly implies the claim about the Wasserstein distance using \eqref{eq:W2}.
\end{proof}

As in section \ref{sec:formalfsde}, the overdamped GLE with fractional noise corresponds to
\begin{gather}
\alpha=2-2H,~\sigma=\frac{\sqrt{2}}{\sqrt{\Gamma(2H+1)}}.
\end{gather}
 We guess that the limiting measure is the Gibbs measure $\pi(x)\propto \exp(-V(x))$.
Rigorously justifying this seems challenging, and we leave it for the future.

\subsection{A numerical simulation}

\begin{figure}
\begin{center}
	\includegraphics[width=0.8\textwidth]{meansquare.pdf}
\end{center}
\caption{Means square distance of the FSDE with potential \eqref{eq:V} for $H=0.6$ and $H=0.8$.}
\label{fig:meansquare}
\end{figure}

In this section, we apply the implicit numerical scheme \eqref{eq:fsdescheme} to a 1D FSDE example with
\begin{gather}\label{eq:V}
V(x)=\frac{1}{4}x^4.
\end{gather}
We choose $\alpha=2-2H$ so that $G\sim \beta_H B_{1-H}$ as we have mentioned.
This potential is convex but not strongly convex, and the corresponding force $-\nabla V(x)=-x^3$ is nonlinear. Justification of convergence to a limiting measure is by no means easy, not to mention whether the limiting measure is the Gibbs distribution
\[
\pi(x)\propto \exp\left(-\frac{1}{4}x^4\right).
\]

Fig. \ref{fig:meansquare} shows the trend of mean square distance $\mathbb{E}X^2=:\langle X^2\rangle$ with $X(0)=X_0=1$. Fig.  \ref{fig:meansquare} (b) enlarges the portion $t\in [0, 5]$ of Fig.  \ref{fig:meansquare} (a). If the distribution of the FSDE converges to the Gibbs measure, then $\mathbb{E}X^2\to 0.675$.
In the figures, the green curve is the numerical simulation for $H=0.6$ with $k=5/2^7\approx 0.391$ and
$N_s:=10^4$ samples, while the red solid curve is the fitting curve $0.675+0.02(0.12+t)^{4H-4}$.
The blue curve is the numerical simulation for $H=0.8$ with the same $k$ and number of samples, while the black curve is the fitting curve $0.675+0.015(0.05+t)^{4H-4}+0.04(3+t)^{4H-4}$. We use $4H-4$ power to fit because the variance of $X$ in the linear forcing case has been shown to converge with rate $t^{4H-4}$ in \cite{lll17}. In this sense, the rate in Theorem \ref{thm:limitsde} might not be optimal.

\begin{figure}
\begin{center}
	\includegraphics[width=0.8\textwidth]{density.pdf}
\end{center}
\caption{Empirical density. When $\alpha=2-2H=0.8$, the convergence is obtained after a reasonable time; when $\alpha=2-2H=0.4$, an intermediate distribution lingers for long time before converging to the final equilibrium.}
\label{fig:density}
\end{figure}

When $H=0.6$ and $\alpha=0.8$, the mean square distance already converges after $t=5$ or so. However, for $H=0.8$ or $\alpha=0.4$, the mean square distance has a rapid drop at the early stage, but then the memory lingers for long time so that the convergence is very slow. This is also the case for normal fractional ODE \cite{fllx18}. In fact, in Fig. \ref{fig:density}, we plot the empirical density versus the Gibbs measure (in black line). When $\alpha=2-2H=0.8$, the distribution is already close to the desired Gibbs distribution at $t=7.5$. However, when $\alpha=2-2H=0.4$, the distribution is roughly like the Gibbs distribution but still has some difference even at $t=20$. In fact, this kind of distribution stays for long time ($t\sim 10^2$). We expect that when $t$ is very large, it can be close to the Gibbs distribution as in \cite{fang2018}. We choose not to do the simulation for $H=0.8$ and $t\sim 10^3$ since the complexity for obtaining a sample path is $O(N^2)$ and the simulation is expensive (we need $10^4$ samples).  For long time simulation when $H$ is close to $1$ (like $H=0.8$ for $T\gtrsim 100$), it is good to adopt the fast scheme in \cite{fang2018}. However, our scheme here is appropriate for dissipative problems due to its good stability properties, and can be used to analyze the time continuous problems compared with the one in \cite{fang2018}.

\section{Time fractional gradient flows}\label{sec:fgradientflow}

In this section, we investigate the time fractional gradient flows using the implicit scheme based on our discretization \eqref{eq:capnewdis2} and establish the error estimates of the numerical scheme. We will use $\langle \cdot, \cdot\rangle$ to denote the inner product in $H$ and $\|\cdot\|$ to denote the norm on $H$.
We will focus on convex functionals $\phi$:
\begin{assumption}\label{ass:conv}
Suppose the functional $\phi$ is lower semi-continuous, convex and
$\inf_{u\in \mathbb{R}^d} \phi(u)>-\infty$.
\end{assumption}
\begin{remark}
All the claims in section \ref{sec:fgradientflow} regarding convex functionals have analogies for $\lambda$-convex functionals (i.e. $\exists \lambda\ge 0$, $u\mapsto \phi(u)+\frac{\lambda}{2}|u|^2$ is convex; of course, the proof is more involved). Considering clarity of presentation, we only focus on convex functionals.
\end{remark}

The Frech\'et subdifferential of convex $\phi$ satisfies
\begin{gather}
\xi\in \partial\phi(u) \Leftrightarrow \partial\phi(v)\neq \emptyset, \forall w\in H, \phi(w)-\phi(v)
-\langle \xi, w-v\rangle \ge 0.
\end{gather}
The following strong-weak closure property is a straightforward consequence of this characterization.
\begin{lemma}
Suppose Assumption \ref{ass:conv} holds. Assume sequences $\{\xi_n\}$ and $\{u_n\}$ satisfy $\xi_n \in \partial\phi(u_n)$ for all $n$, $u_n \to u$ strongly, and that $\xi_n \rightharpoonup \xi$ weakly.
Then $\xi\in \partial\phi(u)$.
\end{lemma}

Fix time $T>0$. Similarly as in \cite[Definition 2.2]{crandall1971}, we define
the following.
\begin{definition}\label{def:strongsol}
$u\in L_{\loc}^1([0, T), H)$ is called a strong solution to \eqref{eq:strongsol}, if  (i) $D_c^{\alpha}u$  is locally integrable on $[0, T)$; (ii)  $\lim_{t\to 0^+}\frac{1}{t}\int_0^t \|u(s)-u_0\|\,ds=0$  (iii) For almost every $t\in [0, T)$, we have $D_c^{\alpha}u\in -\partial\phi(u)$.
\end{definition}

\begin{remark}
If $\alpha=1$, the local integrability of the distributional derivative $Du$ clearly implies that $u$ is absolutely continuous on $[0, T_1]$ for any $T_1\in (0, T)$. The conditions (i)-(iii) in \cite[Definition 2.2]{crandall1971} are automatically satisfied. For $\alpha\in (0, 1)$, imposing $D_c^{\alpha}u\in L_{\loc}^1[0, T)$ does not ensure the uniqueness of $u_0$ (see \cite{liliu2018compact}). To kill this ambiguity, we impose $\lim_{t\to 0^+}\frac{1}{t}\int_0^t \|u(s)-u_0\|\,ds=0$.
\end{remark}

We aim to approximate the solutions of \eqref{eq:strongsol} (though the existence is unclear at this point), following  the method of De Giorgi \cite{de1993new,rossi2006}.
\begin{gather}\label{eq:discreteU}
U_n=\argmin\Big(\frac{1}{2k^{\alpha}}(\sum_{j=1}^{n-1}c_j\|u-U_{n-j}\|^2+c_n^n\|u-U_0\|^2)+\phi(u)\Big).
\end{gather}
Note that the functional on the right hand side of \eqref{eq:discreteU} is the sum of a convex function and some quadratic functionas. Then
\begin{gather}\label{eq:subdiffsum}
\partial \Big(\frac{1}{2k^{\alpha}}(\sum_{j=1}^{n-1}c_j\|u-U_{n-j}\|^2+c_n^n\|u-U_0\|^2)+\phi(u)\Big)
=k^{-\alpha}\Big(c_0u-(\sum_{j=1}^{n-1}c_jU_{n-j}+c_n^nU_0)\Big)+\partial\phi(u),
\end{gather}
and the numerical solution satisfies
\begin{gather}\label{eq:discretescheme}
 -\xi_n :=(\mathcal{D}^{\alpha}U)_n \in  -\partial\phi(U_n).
\end{gather}

Motivated by the proof of Theorem \ref{thm:schemeforlinear}, we consider the set of time steps
\begin{gather}
E_T=\{k>0: k=2^{-m}T,~m\in\mathbb{N} \}.
\end{gather}
The following results from \eqref{eq:subdiffsum} and the strong convexity of the functional in \eqref{eq:discreteU} (proof omitted).
\begin{lemma}\label{lmm:equivfortwominimalsol}
Suppose Assumption \ref{ass:conv} holds. Then, for sufficiently small $k\in E_T$, the discrete schemes \eqref{eq:discreteU} and \eqref{eq:discretescheme} are equivalent and they have a unique solution $\{U^n\}$.
\end{lemma}

\subsection{Properties of the discrete solutions}

Consider the solution given by \eqref{eq:discretescheme}. Define the function $V(t)$ such that
\begin{gather}
V(t) := -\xi_n,~t\in (t_{n-1}, t_n].
\end{gather}
Using the function $V$, define a natural continuous version interpolation of $U_n$ by
\begin{gather}\label{eq:continuousextenU}
U(t)=U_0+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}V(s)\,ds
\end{gather}
with $U(t_m)=U_m$. This continuous interplocation justifies why the discretization \eqref{eq:capnewdis2} is suitable for \eqref{eq:strongsol}.  By \eqref{eq:grouprelation} and the generalized definition (Definition \ref{def:caputo}), one has
\begin{gather}
D_c^{\alpha}U(t)=V(t).
\end{gather}

\begin{lemma}\label{lmm:controlfracint}
Assume Assumption \ref{ass:conv}. For $k \in E_T$ small enough, $\sup_{n: nk\le T}|\phi(U_n)|\le C(U_0, T)$ and
\begin{gather}
\sup_{t\le T}\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}\|V(s)\|^2\,ds\le C(U_0, T).
\end{gather}
\end{lemma}

\begin{proof}
Paring with $\xi_n=-(\mathcal{D}^{\alpha}U)_n\in \partial\phi(U_n)$, and noting $\langle \xi_n, U_n-U_{j}\rangle \ge \phi(U_n)-\phi(U_j)$, one has
\[
-\|\xi_n\|^2=k^{-\alpha}\left(\sum_{j=1}^{n-1}c_j\langle \xi_n, U_n-U_{n-j}\rangle
+c_n^n\langle \xi_n, U_n-U_0\rangle \right)
\ge (\mathcal{D}^{\alpha}\phi(U))_n.
\]
Using the equivalence between \eqref{eq:capnewdis} and \eqref{eq:discretefracint}, and nonnegativity of $\{a_m\}$, one has
\[
\phi(U_n)-\phi(U_0)\le -k^{\alpha}\sum_{m=1}^n a_{n-m}\|\xi_m\|^2
= -\frac{1}{\Gamma(\alpha)}\int_0^{t_n}(t_n-s)^{\alpha-1}\|V(s)\|^2\,ds.
\]
The first claim and the second claim with $t=t_n$ holds.
For general $t\in (t_{n-1}, t_{n})$, the following trivial observation with the result just proved yields the claim in the statement of the lemma.
\begin{gather*}
\begin{split}
\int_0^{t}(t-s)^{\alpha-1}\|V(s)\|^2\,ds
& =\int_0^{t_{n-1}}(t-s)^{\alpha-1}\|V(s)\|^2\,ds
+\|\xi_{n}\|^2\int_{t_{n-1}}^{t}(t-s)^{\alpha-1}\,ds\\
& \le \int_0^{t_{n-1}}(t_{n-1}-s)^{\alpha-1}\|V(s)\|^2\,ds
+\|\xi_{n}\|^2\int_{t_{n-1}}^{t_{n}}(t_{n}-s)^{\alpha-1}\,ds \\
& \le \int_0^{t_{n-1}}(t_{n-1}-s)^{\alpha-1}\|V(s)\|^2\,ds+\int_0^{t_n}(t_n-s)^{\alpha-1}\|V(s)\|^2\,ds.
\end{split}
\end{gather*}

\end{proof}

Now, we compare the numerical solutions with different time steps.
\begin{lemma}\label{lmm:functionseq}
There exists $C(T, U_0)$ independent of $k$ such that when $k$ is small enough,
\begin{gather}
\|U(t)-U(t+\delta)\|\le C(T, U_0)|\delta|^{\alpha/2},~\text{if}~\max(t, t+\delta)\le T.
\end{gather}
Let $U_{i}(t)$ be two functions given by \eqref{eq:continuousextenU} for step sizes $k_i\in E_T$ ($i=1,2$).  Then,
\begin{gather}
\sup_{0\le t\le T}\|U_1(t)-U_2(t)\|^2
\le C(U_0,\alpha)(k_1^{\alpha/2}+k_2^{\alpha/2}).
\end{gather}
\end{lemma}

\begin{proof}

Without loss of generality, we assume $\delta>0$. Then, by \eqref{eq:continuousextenU},
\[
\begin{split}
\|U(t)-U(t+\delta)\| \le &~\frac{1}{\Gamma(\alpha)}\Big(\int_0^t [(t-s)^{\alpha-1}-(t+\delta-s)^{\alpha-1}]\|V(s)\|\,ds \\
&+\int_{t}^{t+\delta}(t+\delta-s)^{\alpha-1}\|V(s)\|\,ds\Big)=:I_1+I_2.
\end{split}
\]
The second term is estimated easily by
\begin{gather*}
I_2\le \frac{1}{\Gamma(\alpha)}\left(\int_t^{t+\delta}\|V(s)\|^2(t+\delta-s)^{\alpha-1}\,ds \right)^{1/2}
\left(\int_t^{t+\delta}(t+\delta-s)^{\alpha-1}\,ds \right)^{1/2}\le C\delta^{\alpha/2}.
\end{gather*}
For the first term $I_1$, we have by H\"older inequality:
\begin{gather}\label{eq:discreteaux1}
I_1\le \frac{1}{\Gamma(\alpha)}\left(\int_0^t(t-s)^{\alpha-1}\Big(1-(\frac{t+\delta-s}{t-s})^{\alpha-1}\Big)^2\,ds\right)^{1/2}
\left(\int_0^t (t-s)^{\alpha-1}\|V(s)\|^2\,ds \right)^{1/2}.
\end{gather}
Clearly,
\begin{gather}\label{eq:discreteaux2}
\int_0^t(t-s)^{\alpha-1}\Big(1-(\frac{t+\delta-s}{t-s})^{\alpha-1}\Big)^2\,ds
\le \int_0^t((t-s)^{\alpha-1}-(t+\delta-s)^{\alpha-1})\,ds
\le C\delta^{\alpha}.
\end{gather}
The claim follows.

To compare the numerical solutions with steps $k_i,i=1,2$  ($V_i$, $\xi_n^{(i)}$, $i=1,2$ similarly defined), we fix $t\in [0, T]$. Then, there exist $n_1$ and $n_2$ such that
$t\in ((n_1-1)k_1, n_1k_1]\cap ((n_2-2)k_2, n_2k_2]$ and such that $V_i(t)=-\xi^{(i)}_{n_i}$.
Denote
\[
\Delta_i(t) :=U^{(i)}_{n_i}-U_i(t),~~i=1,2.
\]

By the definition of $U_i(t)$ and convexity of $\phi$, one has
\begin{gather}\label{eq:fractionalgradientkeyestimate}
\begin{split}
\langle D_c^{\gamma}U_1-D_c^{\gamma}U_2, U_1-U_2\rangle
 &=\langle V_1(t)-V_2(t), U_1(t)-U_2(t)\rangle \\
 & =-\langle \xi^{(1)}_{n_1}-\xi^{(2)}_{n_2}, U^{(1)}_{n_1}-U^{(2)}_{n_2}\rangle+R(t) \le R(t),
\end{split}
\end{gather}
where $\langle \xi^{(1)}_{n_1}-\xi^{(2)}_{n_2}, U^{(1)}_{n_1}-U^{(2)}_{n_2}\rangle \ge 0$ by convexity of $\phi$ and
\begin{gather*}
R(t)=-\langle V_1(t)-V_2(t), \Delta_1(t)\rangle
+\langle V_1(t)-V_2(t), \Delta_2(t)\rangle .
\end{gather*}
It suffices to estimate $\Delta_i(t)$. We take $i=1$ as the example.
By the definition of $U_1(t)$,
\begin{gather*}
\Gamma(\alpha)\Delta_1(t)
=-\int_{t}^{n_1k_1}(n_1k_1-s)^{\alpha-1}\xi^{(1)}_{n_1}\,ds
+\int_{0}^{t}[(n_1k_1-s)^{\alpha-1}-(t-s)^{\alpha-1}]V_1(s)\,ds=:I_1^1+I_1^2.
\end{gather*}
The terms corresponding to $I_1^1$ are controlled by (noting $0< n_1k_1-t\le k_1$)
\begin{gather*}
-\langle V_1(t)-V_2(t), I_1^1\rangle=
\langle \xi^{(1)}_{n_1}-\xi^{(2)}_{n_2}, I_1^1\rangle
=\frac{(n_1k_1-t)^{\alpha}}{\Gamma(1+\alpha)}\langle \xi^{(2)}_{n_2}-\xi^{(1)}_{n_1}, \xi^{(1)}_{n_1}\rangle
\le Ck_1^{\alpha}(\|V_1(t)\|^2+\|V_2(t)\|^2).
\end{gather*}
The terms corresponding to $I_1^2$ can be estimated similarly as in \eqref{eq:discreteaux1}-\eqref{eq:discreteaux2}.
\begin{gather*}
\begin{split}
\langle \xi^{(1)}_{n_1}-\xi^{(2)}_{n_2}, I_1^2\rangle
& \le \frac{1}{\Gamma(\alpha)}\|V_1(t)-V_2(t)\|
\int_0^t[(t-s)^{\alpha-1}-(n_1k_1-s)^{\alpha-1}]\|V_1(s)\|\,ds\\
& \le C(\alpha, U_0)\|V_1(t)-V_2(t)\| k_1^{\alpha/2}.
\end{split}
\end{gather*}

By the explicit formula \eqref{eq:continuousextenU},   $U_i(t)$ is absolutely continuous. Proposition 3.11 in \cite{liliu2018} can be easily generalized to show that
\[
\frac{1}{2}D_c^{\alpha}\|U_1(t)-U_2(t)\|^2
\le \langle D_c^{\alpha}(U_1-U_2)(t), U_1(t)-U_2(t)\rangle.
\]

Overall,
\[
D_c^{\alpha}(\|U_1(t)-U_2(t)\|^2)
\le C(k_1^{\alpha/2}+k_2^{\alpha/2})(\|V_1(t)\|^2
+\|V_2(t)\|^2).
\]
Lemma \ref{lmm:controlfracint} then yields the result.
\end{proof}

\subsection{Well-posedness and numerical error estimates}

In this subsection, we establish the existence and uniqueness of the time fractional gradient flow under some assumptions and give the error estimate of the numerical scheme. Besides Assumption \ref{ass:conv}, we also need a certain regularity property of the subdifferential mapping $\partial\phi$.
\begin{assumption}\label{ass:lambdaconv}
Let Assumption \ref{ass:conv} hold. Moreover, $v_n\to v$ strongly implies any sequence $\{\xi_n\}$ with $\xi_n\in \partial\phi(v_n)$ converges weakly to some $\xi\in H$.
\end{assumption}

\begin{theorem}\label{thm:fracgradient}
Suppose Assumption \ref{ass:lambdaconv} holds. For any $T>0$, the fractional gradient flow \eqref{eq:strongsol} has a unique strong solution $u$ on $[0, T)$ in the sense of Definition \eqref{def:strongsol}. The strong solution is H\"older continuous on $[0, T)$:
\begin{gather}
\|u(t+\delta)-u(t)\|\le C|\delta|^{\alpha/2}.
\end{gather}
Besides, we have the following error estimates for the numerical solution \eqref{eq:discreteU}:
\begin{gather}
\sup_{n: nk\le T}\|U_n-u(t_n)\|\le C(\alpha, T)k^{\alpha/4}.
\end{gather}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:fracgradient}]
By Lemma \ref{lmm:functionseq}, the family $\{ U_i(\cdot): i\in E_T\}$ is a Cauchy sequence in $C([0, T], H)$.
 Therefore, there exists $u(\cdot)\in C([0, T]; H)$, such that $U_i(\cdot)$ converges to $u$ in $C([0, T]; H)$.

Consider the piecewise constant interpolation of the numerical fractional derivative, $\{V_i(t)\}$.
Lemma \ref{lmm:controlfracint} implies that  $V_i(t)\in L^2(0, T)$ with the $L^2(0, T)$ norm uniformly bounded. Then, there is a further subsequence so that we have the weak convergent sequence $V_i(t) \rightharpoonup v(t)$ in $L^2(0, T; H)$, with the estimate
\[
\sup_{t\le T}\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}\|v(s)\|^2\,ds\le C_2(U_0, T).
\]
With the convergence in hand, $u(t)=u_0+\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}v(s)\,ds$.
Hence, in the distributional sense,
\begin{gather}\label{eq:caputoutov}
D_c^{\alpha}u=v(t).
\end{gather}
Similarly as in the proof of Lemma \ref{lmm:functionseq}, $u(\cdot)$ is H\"older continuous
with order at least $\alpha/2$.

By Lemma \ref{lmm:controlfracint}, we have found that $\phi(U_n)$ is uniformly bounded.
Now we consider piecewise linear interpolation of $\{U_n\}$, denoted by $\bar{U}_i(\cdot)$,
\[
\bar{U}_i(t)=U_n^{(i)},~~t\in (t_{n-1}, t_n].
\]
Then, $V_i(t)\in -\partial\phi(\bar{U}_i(t)),~\forall t\in [0, T),~\forall i$.  By the uniform H\"older continuity of $U_i(t)$ in Lemma \ref{lmm:functionseq}  and $\bar{U}_i(t)=U_i(t_n)$
for $t\in (t_{n-1}, t_n]$, we find that for all $t$, $\bar{U}_i(t)$ converges
strongly to $u(t)$.  With Assumption \ref{ass:lambdaconv}, we find that for all $t$,
\begin{gather}\label{eq:pointwiseweak}
V_i(t) \rightharpoonup \bar{v}(t),~H,
\end{gather}
so that
\begin{gather}\label{eq:limitsubgrad}
\bar{v}(t)\in -\partial\phi(u(t)),
\end{gather}
by the weak--strong closure property.

Hence, for any $w\in H$ so that $w1_{[0, T]} \in L^2(0, T; H)$, we find that
$\langle V_i(t), w\rangle$ converges in $L^2(0, T; \mathbb{R})$ to $\langle v(t), w\rangle$.
By \eqref{eq:pointwiseweak}, we also have $\langle V_i(t), w\rangle \to \langle \bar{v}(t), w\rangle$
for all $t\in [0, T]$. Consequently, for a.e. $t\in [0, T]$, we have
\[
\langle v(t), w\rangle = \langle \bar{v}(t), w\rangle.
\]
Since $H$ is separable, we can find a basis consisting of countable elements $\{w_n\}$. Consequently, we have
\[
\langle v(t)-\bar{v}(t), w_n\rangle =0,~\forall n, a.e. t\in [0, T]
\]
Moreover, $v-\bar{v}\in L^2(0, T; H')$ (where $H'$ means $H$ equipped with the weak star topology), we have then
\[
\int_0^T \langle v(t)-\bar{v}(t), w(t)\rangle\,dt=0
\]
for $w(t)$ to be simple functions, and then $L^2(0, T; H)$ functions. Hence, $v(t)=\bar{v}(t)$ for a.e. $t\in [0, T]$.
By \eqref{eq:limitsubgrad} and \eqref{eq:caputoutov}, we find that $u(t)$ is a strong solution under Definition \ref{def:strongsol}. Taking limit $k_2\to 0$ in $\|U_1(t)-U_2(t)\|^2\le C(U_0, \alpha)(k_1^{\alpha/2}+k_2^{\alpha/2})$ yields $\|U_n(t) -u(t)\|^2 \le C(U_0, \alpha)k^{\alpha/2}$.

For the uniqueness, suppose we have two strong solutions $u_i(t),i=1,2$ such that $-\xi_i(t):=D_c^{\alpha}u_i(t)\in -\partial\phi(u_i(t))$. Then, $\langle u_1-u_2, D_c^{\alpha}(u_1-u_2)\rangle\le 0$. Some regularization procedure can yield $\frac{1}{2}D_c^{\alpha}\|u_1-u_2\|^2\le \langle u_1-u_2, D_c^{\alpha}(u_1-u_2)\rangle\le 0$. This then implies uniqueness.
\end{proof}

As a concluding remark, the orders of estimates in Theorem \ref{thm:fracgradient} are not optimal. If one can show that $\xi_n$ is bounded, then one can improve the orders.  Lastly, we give a quick glimpse of the case $H=\mathbb{R}^d$, $\phi\in C^1(\mathbb{R}^d)$ (instead of requiring $\nabla\phi$ to be Lipschitz as in \cite{liliu2018,fllx18}) so that \eqref{eq:strongsol} becomes the FODE:
\begin{gather}
D_c^{\alpha}u=-\nabla \phi(u).
\end{gather}
The following asymptotic behavior holds when $\phi$ is strongly convex.
\begin{proposition}\label{pro:fracgradientsmooth}
Assume that $\phi\in C^1(\mathbb{R}^d)$ and $\phi-\frac{\mu}{2}|u|^2$ is convex for some $\mu>0$.
Let $u^*$ be the global minimizer of $\phi$. Then, for some $C$ depending on $u_0,\mu$,
\[
\phi(u(t))-\phi(u^*)\le (\phi(u_0)-\phi(u^*))E_{\alpha}\left(-Ct^{\alpha} \right).
\]
Moreover, $|u(t)-u^*|\le C(1+t)^{-\alpha/2}$.
\end{proposition}
\begin{proof}

Consider the implicit scheme
\[
(\mathcal{D}^{\alpha}U)_n=-\nabla\phi(U_n).
\]
Using the first claim in Theorem \ref{thm:implicitcomp}, one has
\[
(\mathcal{D}^{\alpha}\phi(U))_n\le -|\nabla\phi(U_n)|^2\le 0.
\]
Hence $\phi(U_n)$ is bounded. Since $\phi$ is strongly convex and thus $\lim_{R\to\infty}\inf_{|u|\ge R}\phi(u)=+\infty$,  $\{U_n\}$ is in a compact domain $K$ that only depends on $u_0$.
By Theorem \ref{thm:implicitcomp} and the Polyak-Lojasiewicz inequality ($|\nabla \phi(x)|^2\ge 2\mu(\phi(x)-\phi(u^*))$), one has
\[
(\mathcal{D}^{\alpha}(\phi(U)-\phi(u^*)))_n=(\mathcal{D}^{\alpha}\phi(U))_n\le -|\nabla\phi(U_n)|^2\le -2\mu(\phi(U_n)-\phi(u^*)).
\]
The Polyak-Lojasiewicz inequality is obtained by
\[
\phi(y)\ge \phi(u)+\nabla\phi(u)\cdot(y-u)+\frac{\mu}{2}|y-u|^2\ge \phi(u)-\frac{1}{2\mu}|\nabla \phi(u)|^2.
\]

By the second claim in Theorem \ref{thm:implicitcomp} and the third claim in Theorem \ref{thm:schemeforlinear}, it holds for any $nk\le T$ that
\[
\phi(U_n)-\phi(u^*)\le (\phi(u_0)-\phi(u^*))E_{\alpha}(-2\mu(nk)^{\alpha})+o(k).
\]
Taking $k\to 0$ and by Theorem \ref{thm:fracgradient} (convergence and continuity of $u(t)$), one thus has for any $t\le T$:
\[
\phi(u(t))-\phi(u^*)\le (\phi(u_0)-\phi(u^*))E_{\alpha}(-2\mu t^{\alpha})
\]
Since $T$ is arbitrary, the first claim is true for all $t$.

Similarly,  $(\mathcal{D}^{\alpha}(U-u^*)^2)_n
\le 2\langle U_n-u^*, -\nabla\phi(U_n)\rangle \le -2\mu |U_n-u^*|^2$.
Theorem \ref{thm:fracgradient} allows us to take $k\to 0$ to obtain $|u(t)-u^*|\le |u_0-u^*|\sqrt{E_{\alpha}(-2\mu t^{\alpha})}$.
Since $E_{\alpha}(-s)\sim C_1s^{-1}$ as $s\to\infty$, the second claim follows.
\end{proof}

\section*{Acknowledgement}
The work of L. Li was partially sponsored by Shanghai Sailing Program 19YF1421300. The work of J.-G. Liu was partially supported by KI-Net NSF RNMS11-07444 and NSF DMS-1812573.

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