Stochastic Flows and Stochastic Differential Equations
The main purpose of this book is to give a systematic treatment of the theory of stochastic differential equations and stochastic flow of diffeomorphisms, and through the former to study the properties of stochastic flows.The classical theory was initiated by K. Itô and since then has been much developed. Professor Kunita's approach here is to regard the stochastic differential equation as a dynamical system driven by a random vector field, including thereby Itô's theory as a special case. The book can be used with advanced courses on probability theory or for self-study.
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Contents
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Stochastic processes and random fields
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1 |
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Conditional expectations
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4 |
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12 Stochastic processes
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5 |
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Martingales
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6 |
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Markov processes
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9 |
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13 Ergodic properties of Markov processes
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15 |
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Ergodic properties
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21 |
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Ergodic theorems
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25 |
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Global stochastic flows
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180 |
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Random infinitesimal generators
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184 |
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48 Stochastic flows on manifolds
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185 |
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Stochastic differential equations on manifolds
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188 |
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Diffeomorphic property of solutions
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192 |
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Random infinitesimal generators
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195 |
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Itôs formulas for stochastic flows on manifolds
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198 |
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49 Itôs formulas for stochastic flows and their applications
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201 |
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Potential kernels
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28 |
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14 Random fields
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31 |
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Weak convergence of continuous random fields
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36 |
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Kolmogorovs tightness criterion
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38 |
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Continuous semimartingales and stochastic integrals
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43 |
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Local semimartingales
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45 |
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22 Quadratic variational processes
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46 |
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Joint quadratic variations
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52 |
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Continuity of quadratic variations
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53 |
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23 Stochastic integrals and Itôs formula
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56 |
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Orthogonal decomposition of localmartingales
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61 |
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Itôs formula
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64 |
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Applications of Itôs formula
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66 |
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Semimartingales with spatial parameters and stochastic integrals
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71 |
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Regularity of martingales with respect to spatial parameters
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74 |
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32 Stochastic integrals based on semimartingales with spatial parameters
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79 |
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Itô integrals case of semimartingales
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84 |
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Stratonovich integrals
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86 |
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Time change of stochastic integrals
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88 |
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Case of local semimartingales
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90 |
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Generalized Itos formula
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92 |
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Integral and differential rules for stochastic integrals
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94 |
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34 Stochastic differential equations
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100 |
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Existence and uniqueness of the solution
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101 |
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Example
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106 |
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Local solution
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107 |
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Stralonovichs stochastic differential equation
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110 |
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Backward integrals and backward equations
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111 |
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Stochastic flows
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114 |
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42 Brownian flows
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119 |
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Infinitesimal means and covariances
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120 |
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Forward random infinitesimal generators
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128 |
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Backward random infinitesimal generators
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130 |
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43 Asymptotic properties of Brownian flows
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134 |
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Asymptotic behavior of 𝜑₁ˡ11
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140 |
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Asymptotic behavior of 𝜑₁II
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144 |
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44 Semimartingale flows
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147 |
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Itôs formulas for stochastic flows
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151 |
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45 Homeomorphic property of solutions of SDE
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154 |
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Lpestimates of solutions
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156 |
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Homeomorphic property
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159 |
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Differentiability of solutions
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164 |
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Diffeomorphic property
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173 |
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47 Stochastic flows of local diffeomorphisms
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176 |
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Itôs formula for 𝜑 acting on tensor fields
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204 |
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Composition and decomposition of the solution
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208 |
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Supports of stochastic flows of diffeomorphisms
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214 |
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Convergence of stochastic flows
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218 |
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52 Convergence as diffusions I
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221 |
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Uniform Lpestimates
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223 |
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Semimartingale characterization of the limit measure
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225 |
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Proof of Theorem 521
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231 |
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Strong convergence
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232 |
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53 Convergence as diffusions II
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234 |
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Uniform Lpestimates
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238 |
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Characterization of the limit measure
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240 |
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54 Convergence as stochastic flows
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244 |
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Uniform Lpestimate of stochastic flows
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247 |
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Proof of Lemma 544
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250 |
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Convergence as Gˡflows
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255 |
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55 Extensions of convergence theorems
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258 |
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Extensions of convergence theorems as stochastic flows
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261 |
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56 Some limit theorems for stochastic differential equations
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262 |
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Central limit theorem for stochastic flows II Markov case
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270 |
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57 Approximations of stochastic differential equations supports of stochastic flows
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274 |
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A weak approximation theorem
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275 |
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Strong approximation theorems
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281 |
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Supports of stochastic flows
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282 |
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Stochastic partial differential equations
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286 |
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Stochastic characteristic system
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288 |
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Existence and uniqueness of solutions
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291 |
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Quasilinear and semilinear equations
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294 |
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Linear equations
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297 |
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62 Second order stochastic partial differential equations
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301 |
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Existence and uniqueness of solutions
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304 |
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Probabilistic representation of solutions
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307 |
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Adjoint equation
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310 |
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63 Applications to nonlinear filtering theory
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315 |
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Stochastic partial differential equations for nonlinear filter
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318 |
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Robustness of nonlinear filter
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322 |
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Kalman filter
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323 |
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64 Limit theorems for stochastic partial differential equations
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326 |
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Proof of theorems
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329 |
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Frequently used notation and assumptions
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334 |
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Remarks on references
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336 |
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References
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340 |
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Common terms and phrases
Assume B₁ bounded variation Brownian flow Brownian motion called characteristic belonging compact sets Conditions C.1 continuous function continuous localmartingale continuous random field continuous semimartingale converges weakly denote differential equation based exists a positive F(yo Feller process finite flow of diffeomorphisms forward stochastic flow Further homeomorphisms inequality Itô integral Itô's formula Itô's stochastic differential joint quadratic variation Lemma Let f Let F(x linear Markov process martingale maximal solution measure modification o-field parameter partial differential equation positive constant predictable process probability process with values proof is complete proof of Theorem prove quadratic variation random field random infinitesimal random variable respect right hand side satisfies Condition Section semigroup solution of equation space stochastic differential equation stochastic integrals stochastic partial differential stochastic process Stratonovich integral t₁ uniformly unique vector fields X₁