CONTENTS 19

68.4.2 The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . .234268.4.3 The Ito And Skorokhod Integrals . . . . . . . . . . . . . . . . .2347

69 Gelfand Triples 235369.1 An Unnatural Example . . . . . . . . . . . . . . . . . . . . . . . . . . .235569.2 Standard Techniques In Evolution Equations . . . . . . . . . . . . . . . .235969.3 An Important Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .236869.4 The Implicit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237469.5 Some Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . .2384

70 Measurability Without Uniqueness 239170.1 Multifunctions And Their Measurability . . . . . . . . . . . . . . . . . .239170.2 A Measurable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . .239370.3 Measurability In Finite Dimensional Problems . . . . . . . . . . . . . . .239970.4 The Navier−Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . .240470.5 A Friction contact problem . . . . . . . . . . . . . . . . . . . . . . . . .2412

70.5.1 The Abstract Problem . . . . . . . . . . . . . . . . . . . . . . .241470.5.2 An Approximate Problem . . . . . . . . . . . . . . . . . . . . .241670.5.3 Discontinuous coefficient of friction . . . . . . . . . . . . . . .2421

71 Stochastic O.D.E. One Space 242571.1 Adapted Solutions With Uniqueness . . . . . . . . . . . . . . . . . . . .242571.2 Including Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . .242671.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . .2430

71.3.1 The Lipschitz Case . . . . . . . . . . . . . . . . . . . . . . . . .243171.3.2 The Locally Lipschitz Case . . . . . . . . . . . . . . . . . . . .2434

72 The Hard Ito Formula 243972.1 Predictable And Stochastic Continuity . . . . . . . . . . . . . . . . . . .243972.2 Approximating With Step Functions . . . . . . . . . . . . . . . . . . . .244172.3 The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244272.4 The Main Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244572.5 Converging In Probability . . . . . . . . . . . . . . . . . . . . . . . . . .245272.6 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2452

73 The Hard Ito Formula, Implicit Case 246173.1 Approximating With Step Functions . . . . . . . . . . . . . . . . . . . .246173.2 The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246373.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246473.4 The Main Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246973.5 A Simplification Of The Formula . . . . . . . . . . . . . . . . . . . . . .247973.6 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248073.7 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2486

74 A More Attractive Version 249774.1 The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2498