Tuyến tính hóa phương trình vi phân ngẫu nhiên phi tuyến

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» » » ! "   # σ% & ' ) %Ω* - + δ% + % + + } , 01 ', & / ∈ $ ! / ={ ( !! $ ! 29 = 29 34 » : 2% ! ; < !+ ! 12 > ! ? $ ! @ 0A2 !0 ! " ! # '! $ " %&& ( ) * # * !+ + , - ((( ) / ! ∈ 78 + 0Ω+ 1→02+ 0Ω+ $ σ3 ∈ ( ; 1+ , # 5&+ × : ω1 ∈ 2( < = + ! ! ! 2( -   2( > ! * * σ% < − '   = ? @ ( ! = *   / 55 ! * ' ( ' ! A B −∫ = / = " #$ %& ' ) * σ () − ) /*1 ' ( ≥ ! ) * ! + )& , ( " * /*1 C ⇒σ ( 56 ( D* σ ( '  *  / - /0 " 12#$ %/ / )⊆ ≤ (& " (& " $ 8* * ≤ σ )≡ ( / ' ( ( ; D * /*, ' ( ( ! / *1 / -0 ∫ ! $9 A - (& ≥4 ≤ * ! ! ! σ )$ G /0 $ " 1F#$ 45 (& " B&7 (7 ! 7* φ " B ! + (& " *(&'! ' " H = + ∫ (φ )$ " * = )   $ *; =H I  A B β = * - −∫  /L ⇒9 ) & ( 0⇒ @+ < *) )& , ( " IK- < −∫ = ( ) J9 + )+ 6∫ +∫ J9    =β K = % - 2( ) * - ! & / *, /*1 / *, * D A B α = β −6  α = * - ∫ 4 * /L ⇒ M ) < *) & 0⇒ @+ ( ) -    − ∫ ! % 2( ( ! - =α )% ! & * @3 = + ∫ (φ ) J 78 + 9# $ 78 + ! !' ( + 9# * * % * < = - +∫D +% ( ) % % 2( ! & A B = ' ! ( ! ≡ σ% − ' −∫ D 5N / D* σ% % =σ ( ' - ( : ≤ ) /L @O P Q ) 5N ⇒ = +∫D + ∫ (φ /L IK- < Q = ∫ (φ / ( , R - ' ( , ) * ' 64 % 64 /L @ +, D* φ = − = +L +∫ D ( +∫ ( 64 / − *, < *) +L ) !< 66 ( / L = L6 L / ! ∫L /L ) L ' ( % !/ ( ) = % : 78 !ξ + ! * * ' / / *1 ξ = * -0 - / ( ( !0 * -< -< + γ I ?I - D ξ ' ( +∑∫ - 2L B D ξ ?I D - + - I !0 * D - ∂ D ∂- ' =D ξ ( ( +∫ D ξ ! I I =6 I 56 *1 ? $ 0* D ) * D ξ / L ξ *  8 D - = ∑  I =6   D - =∑  I =6 D ξ = D ξ4 + ∫ ( @ D ξ = D ξ4 + ∫ I I =6 ! ) ' ( ξ )+ ∑ - F 0 *, ! - IK- < * −D ξ +L D ) ?I : I = ∫γ I S / B * ∑∫ I =6 * /L @ D ξ I 6⇒ L = ∑ γ I L L D = L6 D = ∑γ I I =6 γI$ ' / ( ( I /L D L D D - =6 γI ) 56  →  → ξ @* ! ' D ξ ω = L ! γ I ω =γ I D ξ I D ξ I !0 - D I I =6 I =6 A B L = ∫ ∑γ ?I : * ( * /L - ( @ / L / /L / - * / ) - * ( @ +, , π " ξ )& π =ω + + 3π (M M + (& " ) + ! * C =ω = " " A ! (ξ ∈ = )ω $9 * N π D = π D + ∫π (8 T ∫ (π D D) ) −π ( D : D+ + (& " ' !) -# ) M! ' "  π D = π4 D + ∫π   T ∫ (π ; 78 5H ! - π D π - ( < 6; ) 6; '! + ( )) D - ) * - @ 65 ' ≥4 > ' A B ψ D ≡∫D =ψ Y ( 65 @* 6; - ( ( @ 65 @* & X < ( ) * = / / ψ D = ψ D + ∫ψ ( / D+ * * @ L / * ∀D ∈ ,B D = L5 ! W $ 78 ψ 35 - @* ; / < ! ! ) + π (L π ) = $ 7)    D < UV ∂ D ∂  + ) −π ( D (D ) + π (L π "* π D =∫D = π A D ∈ ,B %& + D / D = ( 36 - 2L D3 B + ∫ (ψ D) D ( D X  → D D) ) S4 ; = $ 78 < +, $ : π D = ψ & ψ D ψ B$ : " ! & " ! + (& " π ψ D =π D α & " !  α = * -  ∫ (π 4 *; ( + (& " ! ψ = ψ + ∫ (ψ $ N ! - ( ) = ψ + ∫ (π 4  ⇒ ψ = ψ * -  ∫ (π 4 "*    ) ψ −6 6 F + π  X / )+ ∫ π ) ψ    )− ∫π  ⇒ ψ −6 = ψ − 6 * -  − ∫ ( π  ψ −6 = − π = E$F$E ψ IK- < $ )−∫ π ( ) & " ! + (& " , ( " + (& "    ψ −6 @ ψ D ψ −6 = ψ D ψ −6 + ψ −6 ψ D + ψ −6 ψ D D S6 ψ D ψ −6 E − (π D π ψ −6 ψ D = π 8D ψ −6 ψ D , ") ) *(&'! π D =π (8 A @ D) & - ( π * IK- < F = − (π π D + (π D −π D π ! - ψ D ≡π D α O 6! * α π D π D )α π 3D − π ! > D) ) −π ( + (ψ ) D X = D* DED - ( X *> ψ ( D ) =ψ ( D ψ ) −π + (π ( 3D )α − π D π ψ D =ψ ( ( ) α = (π α π D - ( α +α π D + α = (π D π > A @ ) ) * α π D = π 8D α G + (π 3D π D @ π D α =π D π D )+ π !  ) + ∫ψ   ψ D = !0 +    D ξ * - ∫  4 * * ∂ D ∂  + ∫ (ψ D − ∫    / - )    S > 78 + ! < ! 56 ∫ψ ( 35 D) !0 = ∫ψ / = "# $ * * ( 35 % ? / D) + ! ψ ( 35 D) ∑ ∫ψ ( 35 D) (3 35 =6 # ψ ( 35 D) / ψ ( 35 D) = ∑ ∫ψ =6 ( E ∫ψ 35 < @3 Y ψ ( X  D = ψ D + ∫ψ   − ) 6 = δ5 =  4 ≠ = δ5 D D) / 0* ∑ 35 =6   D   + ∑ ∫ ( 35 =6 D) S; ! " !# $ %# & ! ' ( ! σ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