Aviator Casino Game UK 2026: The Crash Game Everyone Pretends to Understand

The Aviator casino game has dominated UK online casino lobbies since Spribe released it in 2019, and by 2026 it remains one of the most-played crash titles across mobile and desktop platforms. This guide breaks down how the game actually works, where UK players can find it, what the maths says about your chances, and which operators on the market offer the smoothest experience for real-money play.

Aviator is not a slot. It is not poker. It is a crash game — a category that punishes greed and rewards discipline, which is exactly why most players lose money at it. Understanding the mechanics, the house edge, and the withdrawal landscape in the UK is the difference between treating it as entertainment and treating it as a deposit-destroying machine.

How Aviator Actually Works: Mechanics Without the Marketing Gloss

Aviator puts a red plane on your screen. It takes off. A multiplier climbs from 1.00x upward — 1.20x, 2.50x, 11.79x, 152.00x — and your job is to cash out before the plane flies away. When it does, the round ends and any unclaimed bets are lost. That is the entire game. No paylines, no bonus rounds, no “special features” dressed up as excitement.

Each round runs on a provably fair system using a combination of server seed and client seed, hashed with SHA-256. The result is generated before the round starts and can be verified independently after it ends. Provably fair does not mean the game is fair to you — it means the operator cannot retroactively change the outcome. The house edge is still baked in, and in Aviator’s case it sits at 3%, meaning the theoretical return to player (RTP) is 97%. For context, most UK online slots run between 94% and 96.5%, so Aviator is technically more generous than the average slot. That is a bit like saying one particular cliff has a softer landing than the others.

The social element is where things get interesting. Every round shows a live feed of other players’ bets and cash-out multipliers. You will see someone cash out at 1.35x for a modest profit, and someone else who held to 847.00x and walked away with a four-figure sum. This feed is not neutral information — it is a carefully designed pressure mechanism. Watching others win big while you cash out early creates a psychological itch that the game’s designers know exactly how to exploit. The “live bets” panel is the casino equivalent of a pub showing lottery winners on a loop.

Rounds are fast. Typically 8 to 30 seconds from takeoff to crash, with a short pause between rounds for betting. This speed is deliberate. Fast rounds mean more bets per hour, and more bets per hour means the 3% house edge grinds harder over time. A slot player might make 400–600 spins in an hour. An Aviator player can place well over 100 bets in the same window, and the auto-cash-out feature makes it trivially easy to keep going without pausing to think.

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Where UK Players Can Find Aviator in 2026

The Aviator casino game UK market in 2026 is not uniform. Some operators carry the original Spribe title, others host knock-off crash games with near-identical mechanics but different names and slightly tweaked multipliers. The difference matters: Spribe’s Aviator has a published 3% house edge and a verifiable provably fair system, while some unlicensed imitators offer no such transparency whatsoever.

Foxy Bingo, William Hill, Ladbrokes, and Monopoly Casino are among the established UK-facing operators that have integrated crash-style games into their lobbies over the past few years. Mystake and AdmiraL represent the newer wave of platforms that built their game libraries around titles like Aviator from the ground up. The point is not which operator is “best” — that depends on your priorities around bonuses, withdrawal speed, and mobile experience — but rather that the game is widely available across both long-standing brands and newer entrants.

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Availability varies by platform. Some operators offer Aviator exclusively through their mobile app, others through the browser-based lobby, and a few through both. If you primarily play on a phone, check whether the operator’s app includes the full crash game library before depositing. Nothing is more frustrating than signing up for a casino app, depositing £20, and discovering the game you came for is desktop-only.

One practical note: the game lobby on most UK-facing platforms is crowded. Aviator often sits alongside similar crash titles like Spaceman, JetX, and various proprietary crash games. They all follow the same basic formula — multiplier climbs, cash out before it crashes — but the house edges, visual presentation, and auto-cash-out options differ. Before committing to any of them, check the published RTP. A crash game running at 95% RTP is giving away 2% more of your money per round than Aviator at 97%, and over hundreds of rounds, that gap adds up to real money.

The Math Behind Aviator: What 3% House Edge Actually Means for You

House edge is the single most important number in any casino game, and it is the number most players ignore. In Aviator, a 3% house edge means that for every £100 wagered, the game expects to retain £3 over the long run. This is not a prediction of any individual session — you could win big on your first bet or lose ten in a row — but over thousands of rounds, the maths converges relentlessly.

Here is a concrete example. Suppose you place 200 bets of £1 each, cashing out at a 2.00x multiplier every time. Your average return per bet is £2.00 × (probability the plane reaches 2.00x). With a 3% house edge, that probability works out to roughly 48.5%. So your expected return per bet is £2.00 × 0.485 = £0.97. Over 200 bets, you would expect to lose about £6. That is the house edge doing its quiet, patient work.

Now suppose you get greedy and try to cash out at 10.00x. The probability of the plane reaching 10.00x drops to roughly 9.7%. Your expected return per bet is £10.00 × 0.097 = £0.97. Same result. The house edge does not care about your strategy. Whether you cash out at 1.10x or 100.00x, the expected loss per pound wagered remains the same. This is the fundamental truth that crash game marketing never mentions: there is no “smart” cash-out point that beats the maths.

Where strategy does matter is in variance management. Cashing out early (1.20x–1.50x) gives you frequent small wins and a smoother bankroll curve, but the same long-term expected loss. Cashing out late (10x+) gives you occasional large wins and frequent losses, with the same long-term expected loss. The difference is psychological, not mathematical. Early cash-outs feel like progress. Late cash-outs feel like gambling. Both are.

Operator	Typical Bonus Category	Typical Min. Deposit	Typical Withdrawal Speed	Crash Game Availability
Foxy Bingo	Matched deposit or free bingo tickets	£5–£10	24–72 hours (card), faster via e-wallets	Crash-style games in lobby
Mystake	First-deposit match, sometimes with free spins	£10	E-wallets: 0–24 hours; cards: 1–3 days	Aviator and similar crash titles
William Hill	Deposit match or bet credits depending on product	£5–£10	E-wallets: same day; cards: 1–3 days	Crash games in casino lobby
Monopoly Casino	Free spins or wagering-linked bonus	£10	E-wallets: 0–24 hours; cards: 2–5 days	Crash-style titles available
Sun Bingo	Bingo tickets or slot free spins	£5–£10	24–72 hours typical	Limited crash game selection
AdmiraL	First-deposit match with wagering requirements	£10	E-wallets: 0–24 hours; cards: 1–3 days	Aviator and crash variants
Fabulous Bingo	Bingo ticket bundles or slot spins	£5–£10	24–72 hours typical	Limited crash game selection
Slots Temple	Free-to-play tournaments, real-money options	£10	Varies by method; e-wallets fastest	Crash games in library
Ladbrokes	Deposit match or bet credits	£5–£10	E-wallets: same day; cards: 1–3 days	Crash titles in casino section
Midnite	Free bets or deposit-linked promotions	£10	E-wallets: 0–24 hours; cards: 1–3 days	Crash-style games available

Is Aviator Legal in the UK? The Regulatory Picture in 2026

Aviator is legal to play in the UK provided it is offered by an operator licensed by the Gambling Commission. The UKGC has taken a progressively firmer stance on crash games since 2022, requiring operators to clearly display game rules, RTP figures, and session history. Any crash game available to UK players must meet these transparency requirements, which is one reason the market has consolidated around a smaller number of reputable providers like Spribe.

The Gambling Commission’s approach to online casino licensing in the UK is straightforward: no licence, no legal access to UK players. This applies equally to the casino platform and, in most cases, to the game providers whose titles are hosted on those platforms. A casino holding a UKGC licence is responsible for ensuring the games it offers — including Aviator and other crash titles — meet regulatory standards for fairness, transparency, and responsible gambling tools.

For players, the practical takeaway is simple. If you are playing Aviator at an operator licensed by the UK Gambling Commission, the game is legal, the RTP is published, and you have access to responsible gambling tools like deposit limits, session timers, and self-exclusion through GAMSTOP. If you are playing at an unlicensed offshore site offering Aviator, none of those protections apply, and your deposits are not protected by UK regulatory frameworks.

UKGC licensing also mandates that operators provide clear information about how to verify your identity and the source of your funds. This is not bureaucratic box-ticking — it is the mechanism that prevents money laundering and protects players from unlicensed operators. Expect to provide photo ID and proof of address when you sign up at any UK-licensed casino. If an operator lets you deposit and play without verification, that is a red flag, not a convenience.

Bonuses and Promotions Around Aviator: Reading the Small Print

Casino bonuses in the UK are marketed with the enthusiasm of a car salesman describing a vehicle with “character.” The reality is that most bonuses are mathematical tools designed to extend your playing time and, by extension, increase the amount the house edge grinds away at your bankroll. Understanding how these bonuses work is essential before you claim one.

The most common bonus types in the UK market in 2026 include matched deposits (where the casino matches a percentage of your deposit), free spins (typically tied to specific slot titles), no-deposit bonuses (small amounts credited without requiring a deposit), and cashback offers (a percentage of losses returned as bonus funds). Each type comes with wagering requirements — the number of times you must bet the bonus amount before you can withdraw any winnings derived from it.

Here is where the maths gets uncomfortable. A “100% match up to £100” bonus with a 35x wagering requirement means you must place £3,500 in bets before withdrawing. If you are playing Aviator with a 3% house edge, your expected loss on £3,500 of wagering is £105. In other words, clearing the wagering requirement on a £100 bonus is expected to cost you more than the bonus itself is worth. This is not a flaw in the system. It is the system.

Bonus Type	Typical Wagering Requirement	Typical Time Limit	Key Restriction	Effective Cost to Player
Matched Deposit (100%)	20x–40x bonus amount	7–30 days	Game weighting: crash games may contribute 10–50% toward wagering	Expected loss from wagering often exceeds bonus value
No-Deposit Bonus (£5–£20)	40x–65x bonus amount	3–7 days	Max cashout cap (often £50–£100)	Low expected value; high variance
Free Spins (No Deposit)	30x–50x winnings from spins	3–7 days	Locked to specific slot titles; crash games excluded	Nominal value; not applicable to Aviator directly
Cashback (10–20%)	1x–5x cashback amount	Weekly or monthly	Only on net losses; bonus funds not cash	Lowest effective cost; closest to “fair” offer
Reload / Deposit Match (50%)	20x–35x bonus amount	7–14 days	Game weighting and max bet limits apply	Similar to matched deposit; slightly better terms

Game weighting deserves special attention for Aviator players. Many UK casinos assign crash games a reduced contribution toward wagering requirements — sometimes as low as 10% or 20%, compared to 100% for slots. This means that £100 wagered on Aviator might only count as £10–£20 toward clearing a bonus. If your plan is to use a welcome bonus specifically to play Aviator, check the terms carefully. You might find that the bonus is effectively unusable for your chosen game.

Some operators offer promotions specifically tied to crash games — “Aviator tournaments,” multiplier challenges, or leaderboards that reward the highest cash-out multiplier of the week. These are structurally different from standard bonuses because they reward performance rather than volume. The catch, as always, is that chasing a leaderboard position requires placing more bets, and more bets means more exposure to the house edge. A tournament that promises £500 to the player with the highest single-round multiplier is asking you to bet aggressively on high-multiplier rounds, which is the strategy with the highest variance and the worst bankroll sustainability.

Mobile Casino Experience: Playing Aviator on the Go

Aviator was designed mobile-first, and it shows. The game’s interface — a single plane, a single multiplier, a single cash-out button — translates perfectly to small screens. Unlike complex video slots with elaborate bonus rounds and dozens of paylines, Aviator’s minimalism is a feature, not a limitation. You can play it one-handed while waiting for a bus, which is either convenient or deeply concerning depending on your perspective.

The best mobile casino apps for Aviator in 2026 share a few characteristics. Fast loading times (the game should be live within two seconds of tapping), stable streaming of the social feed (lagging live bets undermine the game’s core appeal), and intuitive cash-out mechanics (a mis-timed tap on a small screen can cost you a round). Operators like William Hill, Ladbrokes, and Mystake have invested heavily in their mobile platforms, and the difference in user experience between a well-built app and a poorly-built one is immediately obvious.

Browser-based play remains a viable alternative for players who do not want to install another app. Most modern casino platforms use HTML5 technology that runs Aviator smoothly in mobile browsers like Safari and Chrome. The trade-off is that browser playmay lack the push-notification alerts for live rounds and the biometric login features that dedicated apps provide. For a game where seconds matter, those small conveniences can influence your overall experience more than you might expect.

Data usage is worth noting too. Aviator’s live social feed streams real-time data from every active player, which means it consumes more mobile data than a standard slot game. On a typical 4G connection, expect roughly 10–15MB per hour of continuous play. Not catastrophic, but enough to notice if you are on a limited data plan. Wi-Fi remains the safer bet for extended sessions, particularly if you are playing at an operator where the live feed is central to the experience.

Withdrawals and Payment Methods: Getting Your Money Out

Withdrawal speed is where casino operators separate themselves from the competition, and it is the area where UK players are most frequently disappointed. The gap between “instant withdrawal” marketing and the reality of 3–5 business days is one of the industry’s oldest traditions. Understanding how payment methods actually work in the UK market saves you from unnecessary frustration.

E-wallets — PayPal, Skrill, Neteller — remain the fastest withdrawal method at most UK-facing casinos. Once an operator processes the withdrawal (which can take anywhere from a few hours to 24 hours depending on the platform), e-wallet transfers typically arrive within minutes. Debit cards are slower: expect 1–3 business days after processing, with some banks adding their own delays. Bank transfers are the slowest route, often taking 3–5 business days end to end, though they are the most reliable for larger sums.

Payment Method	Typical Deposit Speed	Typical Withdrawal Speed	Common Limits (UK Market)	Notes
Debit Card (Visa/Mastercard)	Instant	1–3 business days after processing	Min. £5–£10; max. £5,000–£50,000 per transaction	Most universally accepted; bank delays possible
PayPal	Instant	Within hours after processing	Min. £5–£10; max. varies by operator	Fastest e-wallet option; widely supported in UK
Skrill / Neteller	Instant	0–24 hours after processing	Min. £10; max. varies	Excluded from some welcome bonuses
Bank Transfer	1–3 business days	3–5 business days	Min. £10–£25; max. often higher	Best for large withdrawals; slowest method
Paysafecard	Instant	Not available for withdrawals	Min. £5; max. £250 per voucher	Prepaid; withdrawals require alternative method

One detail that catches many players off guard: most UK casinos require you to withdraw using the same method you deposited with, up to the amount deposited. This is an anti-money-laundering standard, not an operator quirk. If you deposit £50 via PayPal and win £2,000, the casino will typically return the £50 to PayPal and allow you to withdraw the remaining £1,950 via an alternative method like bank transfer. Plan your deposit method accordingly.

Verification delays are the other common bottleneck. UK-licensed operators are required to verify your identity before processing withdrawals, and first-time withdrawals often trigger a document review that adds 24–48 hours to the process. Submitting your ID and proof of address proactively — before you request a withdrawal — can shave a full day off the timeline. Some operators now use automated verification systems that clear most players within minutes, but manual review is still common for larger amounts or flagged accounts.

Choosing an Operator: What Actually Matters When the Options Look Identical

Ten operators, ten “best casino” claims, and roughly the same game library across all of them. Choosing where to play Aviator in the UK market comes down to a handful of practical differentiators that have nothing to do with marketing copy.

Licensing status is the non-negotiable starting point. Every operator on the UK market should hold a licence from the Gambling Commission, and you can verify this through the UKGC’s public register. An unlicensed operator might offer better bonuses or fewer verification requirements, but it also offers zero regulatory protection. In a dispute over a withdrawal or a game outcome, a UKGC licence gives you a formal complaints process and, ultimately, access to an independent adjudicator. Without one, your options are considerably bleaker.

Withdrawal speed is the second differentiator that matters in practice. If you value getting your money quickly — and most players eventually do — prioritise operators that support e-wallet withdrawals with same-day processing. The difference between a 6-hour withdrawal and a 72-hour withdrawal is not just convenience; it affects how you manage your bankroll across sessions. Money stuck in a 3-day processing queue is money you cannot re-deploy or, more importantly, money you cannot walk away with.

Third: the mobile experience. If you primarily play on a phone — and statistics consistently show that the majority of UK online casino activity now happens on mobile devices — the quality of the operator’s app or mobile site is not a secondary concern. It is the primary one. Test the platform before committing significant funds. Most operators offer demo modes or low-minimum-deposit options that let you evaluate the experience without risking much.

Fourth: bonus terms, specifically game weighting for crash games. An operator that offers a generous welcome bonus but assigns Aviator a 10% wagering contribution is functionally offering you no bonus at all for your chosen game. Compare the effective value of bonuses across operators, not the headline figures.

Responsible Gambling: The Part Everyone Skips

Aviator’s design — fast rounds, instant feedback, near-miss psychology — makes it one of the more habit-forming casino games on the market. The combination of speed and variable reinforcement (sometimes you win, sometimes you do not, and the unpredictability is the point) is precisely the pattern that behavioural psychologists identify as most conducive to compulsive behaviour. This is not a moral judgement; it is a description of how the game works on the human brain.

UK-licensed operators are required to offer a suite of responsible gambling tools, and using them is not an admission of weakness. Deposit limits — daily, weekly, or monthly — are the most effective tool for most players because they remove the decision-making from the moment of temptation. Session time limits and reality checks (pop-up reminders showing how long you have been playing and how much you have won or lost) serve a similar function. Self-exclusion through GAMSTOP blocks access to all UK-licensed gambling sites for a chosen period, ranging from six months to indefinitely.

The 3% house edge in Aviator means that sustained play will, with mathematical certainty, erode your bankroll. There is no betting pattern, no cash-out strategy, and no “system” that changes this. The only variable you control is how much you wager and how long you play. Set a budget before you start, treat it as the cost of entertainment, and stop when it is gone — not when you have “won it back,” because the maths says you will not.

Support is available. GamCare operates a national helpline (0808 8020 133) and provides live chat for anyone affected by gambling-related harm. Gambling Therapy offers free online support groups. These services are confidential, free, and staffed by people who understand the specific pull of games like Aviator. Reaching out early — before a bad session turns into a bad month — is the single most effective thing a player can do.

Is Aviator Worth Playing? A Sober Assessment

Aviator is a well-designed game with a published house edge, transparent mechanics, and a format that is genuinely entertaining in short bursts. The 3% house edge is competitive within the UK market, and the provably fair system means you can verify that the game is not rigged against you beyond its stated mathematical advantage. For players who treat it as paid entertainment with a known cost, it delivers exactly what it promises.

The problem is not the game itself. The problem is the environment around it — the social feed designed to trigger FOMO, the fast rounds designed to maximise bets per hour, the marketing that frames a 3% house edge as an opportunity rather than a cost. Aviator is a knife. It is useful in the right hands and dangerous in the wrong ones, and the casino lobby does not distinguish between the two.

Play it at a UKGC-licensed operator. Check the RTP before you deposit. Set a deposit limit before your first bet. Cash out early and often, not because it improves your expected value — it does not — but because it keeps your bankroll alive longer, which means more entertainment per pound. And when the session is over, close the app. The plane will still be there tomorrow. It always is.

Browser-based play remains a viable alternative for players who do not want to install another app. Most modern casino platforms use HTML5 technology that runs Aviator smoothly in mobile browsers like Safari and Chrome. The trade-off is that browser play may lack the push-notification alerts for live rounds and the biometric login features that dedicated apps provide. For a game where seconds matter, those small conveniences can influence your overall experience more than you might expect.

Withdrawals and Payment Methods: Getting Your Money Out

Payment Method	Typical Deposit Speed	Typical Withdrawal Speed	Common Limits (UK Market)	Notes
Debit Card (Visa/Mastercard)	Instant	1–3 business days after processing	Min. £5–£10; max. £5,000–£50,000 per transaction	Most universally accepted; bank delays possible
PayPal	Instant	Within hours after processing	Min. £5–£10; max. varies by operator	Fastest e-wallet option; widely supported in UK
Skrill / Neteller	Instant	0–24 hours after processing	Min. £10; max. varies	Excluded from some welcome bonuses

The first table above covers operator-level comparison across bonuses and crash game availability; this one maps payment mechanics so you can plan deposits around withdrawal reality rather than marketing claims.

Beyond speed limits and fees — which range from zero on most e-wallets to occasional percentage charges on certain card transactions — there is one detail that catches many players off guard: most UK casinos require you to withdraw using the same method you deposited with, up to the amount deposited. This is an anti-money-laundering standard baked into UKGC licence conditions since 2024 tightened source-of-funds checks across all licensees.

If you deposit £50 via PayPal and win £2,000 on Aviator at 47x multiplier cash-out during a late-night session fueled by cold coffee and questionable life choices, expect this sequence: first £50 returns to PayPal within hours because that matches your deposit rail; remaining £1,950 routes through bank transfer or whichever alternative method supports larger sums at your chosen operator.

Deposit-to-Withdrawal Round Trip: What Actually Happens Across Methods (UK Market Averages)
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shifts divide-and-conquer MRRR multiple relatively robust representations ARPACK implicitly restarted Lanczos LOBPCG locally optimal block preconditioned conjugate gradient Jacobi-Davidson JD thick-restart JD harmonic Ritz Rayleigh quotient RQI inverse iteration shift-invert mode folding symmetry exploitation group representation theory irreducible characters character tables Young tableaux symmetric group alternating group dihedral groups cyclic groups permutation representations tensor products Clebsch-Gordan coefficients Wigner-Eckart theorem selection rules angular momentum coupling Clebsch-Gordan series Kronecker products decompositions multiplicity-free decompositions branching rules restriction induction Clifford algebras exterior algebra wedge products Grassmannians Plücker relations projective geometries incidence structures matroids greedy algorithm augmentation circuits independent sets bases rank function Tutte polynomial deletion-contraction recurrence chromatic polynomial Whitney rank-two deletion theorem graph minors Robertson-Seymour theorem well-quasi-orderings obstruction sets forbidden minors planar graphs Kuratowski Wagner characterization five-color six-color four-color theorem Appel Haken Hales-Jewett van der Waerden Ramsey numbers Schur numbers happy ending problem Erdős-Szekeres monotone subsequences Dilworth chain decomposition Mirsky antichains Sperner families Lubell-Yamamoto-Meshalkin LYM inequality normalized matching property LYM chains symmetric chain decompositions Gray codes combinatorial designs Steiner systems projective planes affine planes Hadamard matrices conference matrices difference sets Singer difference sets Paley construction quadratic residues Weil bound Kloosterman sums exponential sums character sums Gauss sums Stickelberger p-adic valuations lifting solutions Hensel lemma Newton polygon Eisenstein criterion irreducibility criteria rational root test Descartes rule signs Sturm sequence Sturm-Habicht polynomial gcd subresultant PRS polynomial remainder sequence Euclidean algorithm half-gcd fast algorithms modular composition multipoint evaluation Horner scheme Paterson-Stockmeyer baby-step giant-step Cantor-Kaltofen division Karatsuba multiplication Toom-Cook Schönhage-Strassen FFT-based multiplication NTT number theoretic transform complex FFT Cooley-Tukey radix-2 radix-mixed Bluestein chirp-z Goertzel prime factor FFT split-radix recursive FFT Stockham autosort out-of-place bit-reversal permutation digit-reversal permutation Gray-code ordering interleaved butterflies twiddle factors precomputed tables cache-oblivious layouts recursive blocking Strassen matrix multiplication Coppersmith-Winograd new approaches laser method tensor decomposition CP Tucker HOSVD HOVD TT-cross QTT quantized tensor train hierarchical Tucker format structured tensors low-rank approximations Eckart-Young-Mirsky SVD truncated principal component analysis PCA whitening decorrelation ZCA Mahalanobis distance covariance estimation shrinkage Ledoit-Wolf OAS oracle approximating shrinkage graphical models Markov random fields Gibbs distributions Boltzmann machines RBM restricted Boltzmann machines energy-based models score matching denoising score matching Langevin dynamics annealed importance sampling AIS thermodynamic integration path sampling Jarzynski equality Crooks fluctuation theorem non-equilibrium statistical mechanics entropy production dissipation function fluctuation-response relations Green-Kubo Onsager reciprocal relations linear response susceptibility correlations relaxation times correlation length critical exponents universality classes renormalization group fixed points relevant irrelevant marginal operators scaling laws finite-size scaling data collapse Binder cumulant Lee-Yang zeros Fisher zeros partition function free energy internal energy heat capacity magnetic susceptibility Curie temperature critical op temperature Ising model exact solution Onsager solution Bethe ansatz Yang-Baxter equation quantum integrability transfer matrix technique Pfaffian techniques Sherman-Morrison-Woodbury determinant lemma matrix determinant lemma adjugate cofactor expansion minors Laplace expansion Gaussian elimination pivoting partial complete rook pivoting threshold pivoting iterative refinement condition number backward stability forward stability perturbation analysis Bauer-Fike theorem Kreiss matrix theorem pseudospectra resolvent norm pseudospectral abscissa numerical range numerical radius compression Hermitian dilation Cayley transform fractional powers sectorial operators m-sectorial accretive dissipative monotone operators maximal monotone Yosida approximation Moreau envelope proximal mapping firmly nonexpansive averaged nonexpansive firmly quasi-nonexpansive fixed point iterations Picard Mann Halpern viscosity solutions Crandall-Liggett formula nonlinear semigroups evolution equations maximal monotone subdifferentials convex conjugate Fenchel duality Moreau-Yosida regularization Legendre transforms Bregman divergences f-divergences Jensen-Shannon Hellinger total variation Wasserstein distances optimal transport Monge-Kantorovich problem Brenier map McCann displacement interpolation Benamou-Brenier dynamic formulation JKO scheme Jordan-Kinderlehrer-Otto minimizing movements minimizing movementschemes gradient flows entropy dissipation mobility coefficients porous medium equation fast diffusion equation nonlinear diffusion degenerate parabolic equations viscosity solutions comparison principle maximum principle Aronson estimates Gaussian bounds heat kernel estimates Davies-Gaffney ultracontractivity Sobolev inequalities Nash inequalities log-Sobolev inequalities Poincaré inequalities concentration of measure Talagrand inequality transportation cost inequality bounded differences McDiarmid inequality Azuma-Hoeffding martingale inequalities Freedman inequality Bernstein inequality Bennett inequality sub-Gaussian tails Orlicz norms ψ_α norms Orlicz spaces Luxemburg norm modular function Young functions complementary pairs Hölder conjugate exponents Young inequality convolution inequality Minkowski integral inequality Brascamp-Lieb inequalities rearrangement inequalities Hardy-Littlewood-Pólya symmetrization Schwarz symmetrization Steiner symmetrization bathtub principle decreasing rearrangement increasing rearrangement equimeasurable distribution functions Lorentz spaces weak L^p Marcinkiewicz interpolation Riesz-Thorin Stein interpolation vector-valued extensions Calderón-Zygmund singular integrals Littlewood-Paley decomposition atomic decomposition Hardy space H^1 Carleson measure tent space BMO John-Nirenberg oscillation maximal function Cotlar inequality commutator estimates Coifman-Rochberg paraproducts bilinear estimates multilinear Calderón-Zygmund theory pseudodifferential operators symbol classes Shubin classes exotic symbols quantization rules Weyl calculus Kohn-Nirenberg functional calculus Fourier integral operators Lagrangian distributions Mas