Higher-Order Spectral Densities of Fractional Random Fields

This is the author’s version of a work that was submitted/accepted for publication in the following source: Anh, Vo, Leonenko, Nikolai, & Sakhno, L (2003) Higher-Order Spectral Densities of Fractional Random Fields. Journal of Statistical Physics, 111(314), pp. 789-814. This file was downloaded from: ❤tt♣✿✴✴❡♣r✐♥ts✳q✉t✳❡❞✉✳❛✉✴✷✸✹✾✾✴ Notice: Changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this work, please refer to the published source: 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@ % 0 1 0 1 0 1 - &% 4 ) 7 0 1 %)) * ' 7 ' 7 % " %)) * 0 1! : ' ' % 0 1 4 2 ) 0 1 * ) " * 0 1 4 ) " " "# : ' " "# " # " 2 % "# : ) 2 % ) #" ) % ) %" ' + 2 % # )% + 2 % # % ) %% %" " ) ! " 2 % " # ' &% " " " ' ) %) &% 2 % " % &% ' . 2 % " % =) ' ' % ) % % " # : 7 ' $ ) ( ( " )% 4 . - % 0 1 4 . 0 1 4 7 ( 7 0 1! * +4 $ ! &% = ,! " ) % " ( ( 2 % &% ' " " ) 3 ) ) ( % # ' # " " 0 1! %" % "% " Lp %" " " ( 0 1! "% " "' ( 9 0 1 ? . 0 1 " # 9 %" ) ? . " 0 1 * ' ) % ) % &% " ) ( % # ' &% ' "% ' 2 % $ = " 7 " @ " % $ ( " # $4 % )% " " % " 9 ( "% % % ) % " % % "% %"! # * " 7 CB) "# * ) * ' "% 0 1 " " 4 ( ( '! : " ' 0 1 % &% * + " "% " C " "% %"% % :2 * * * - G * : : 9 9* G : * 9: = @ T : ' "# " % / " &% &% ∂u = ∆u, ∂t @ ,! %)6 >0 " u(0, x) = v(x), x ∈ Rn , @ @! ' 2 % ∆ v(x) "# " v(x) = h(ξ(x)). n "# ξ(x), x ∈ R % h(u), u ∈ Rn , %" 4 G ( ) ' # ξ (x) , x ∈ Rn " % ) " &% % % " % 2 % "# ' Eξ (x) = 0 ( % " B (x) = 1 + x 4 % h h ∞ h (u) = k=1 G ) ( k −1 Hk (u) = (−1) [ϕ (u)] , x ∈ Rn , 0 < κ < n. Eh2 (ξ (0)) < ∞. 4 ) $ % % G 2 −κ/2 Ck Hk (u) , k! Ck = h (u) Hk (u) ϕ (u) du R " " dk ϕ (u) , duk 1 ϕ (u) = √ e−u 2π /2 , u ∈ R1 , k = 0, 1, 2, ... m≥1 % $ C1 = ... = Cm−1 = 0, Cm = 0. m≥1 &&% 0 1! : &% @ ,! ' ! " 7 0 1 " @ @! / " m Xε (t, x) = ' 1 εκm/4 u % )% ( ) h $ " % ) % u(t, x), t > 0, x ∈ R n , ξ(x), x ∈ Rn # κ ∈ (0, n/m)# 7 " ' "% " !" v(x) = h(ξ(x))# h(u), u ∈ R# ( t x ,√ ε ε − C0 , $ %# &# h ) t > 0, x ∈ Rn " C * @ =! ε −→ 0, ( # " Cm [c (n, κ)]m/2 m! Xm (t, x) = ′ ei(x,λ +...+λ )− t λ +...+λ ( λ1 ... λm )(n−κ)/2 ×W (dλ1 ) ...W (dλm ) , t > 0, x ∈ Rn , 0 < κ < n/m, m ≥ 1, R ) @ C! c (n, κ) = Γ W() $ n−κ 2 / 2κ π n/2 Γ (κ/2) , + ′ ) * $ " λi = ±λj , i, j = 1, ..., m, i = j, + * + * , x Xm (t, x) , t > 0, x ∈ Rn , m≥1 κ ∈ (0, n/m) EXm (t, x) = 0 EXm (t, x) Xm (t′ , y) = R (x − y, t + t′ ) . X1 (t, x) , t > 0, x ∈ Rn , - S1,2 (λ) = C12 c (n, κ) @ D! $ (t+t′ ) λ e− λ n−κ x, . λ ∈ Rn , ei(λ,x−y) S1,2 (λ) dλ. EX1 (t, x) X1 (t′ , y) = R ) ∞. / " # 2 EXm (t, x) < ∞. 2 % ) x, # S1,2 (0) = Xm (t, x) , t > 0, x ∈ Rn , ' ' " ) "# @ =! m ≥ 2 2 % Xm (t, x) , t > 0, x ∈ Rn , # + t>0 ' m≥2 ′ ei(λ,x−x ) Sm,2 (λ) dλ, EXm (t, x) Xm (t, x′ ) = R @ B! @ I! Cm 2 e−2 t λ Sm,2 (λ) = (c2 (n, κ))m K (κ, m) , m! λ n−mκ K (κ, m) = π (m−1) "@' ) Γ κ2 Γ n−κ 2 λ ∈ Rn , 0 < κ < n/m, m−1 m−1 Γ iκ 2 i=1 Γ n−iκ 2 C Γ n−(1+i)κ 2 Γ (1+i)κ 2 . :2 * * ) - G * : : 9 9* G : * ) ) m ≥ 2 Xm (t, x) , t > 0, x ∈ Rn , ' 2 % "# % # # % k ≥ 2) % % " k E |Z (x)| < ∞, k ≥ 2. ' 9: D Xm (t, x) , t > 0, x ∈ Rn , m≥2 # Sm,2 (0) = ∞. ) * |λ| → 0 m≥2 "$ 0!!# Sm,2 (λ) / / * $ "# 2 % ) X1 (t, x) &% + ( % Z (x) , x ∈ Rn , "# k ck (x1 , ..., xk ) = ) %"% # 1 ∂k log E exp i uj Z (xj ) ik ∂u1 ...∂uk j=1 % u =...=u =0 (Z (x1 ) , ..., Z (xk )) , k ≥ 2. "( ck (x1 , ..., xk ) = ck (x1 − xk , ..., xk−1 − xk , 0) , " $( % ) % $ Sk (λ1 , ..., λk−1 ) ∈ L1 R(k−1)n % k−1 ck (x1 − xk , ..., xk−1 − xk , 0) = exp i R − j=1 λj , xj − xk Sk (λ1 , ..., λk−1 ) dλ1 ...dλk−1 , Sk (λ1 , ..., λk−1 ) k≥2 # Z (x) , x ∈ Rn , :( ( 7 0 1 ! / % Sk (λ1 , ..., λk−1 ) = Sk (λ1 , ..., λk−1 , λk ) ' λk = − (λ1 + ... + λk−1 ) . 9 % % ' ' % ' "" ( < 1 Sk (λ1 , ..., λk−1 ) = Sk λπ(1) , ..., λπ(k−1) sym k! π∈P {λ ,...,λ :λ +...+λ − +λ =0} Pk ' % k! ' * " 7I ' 2 % / "% ) x + × sym {λ ,λ ,λ :λ +λ +λ =0} ' ) m ≥ 2. {1, ..., k} . ) % ) Xm (t, x) , t > 0, x ∈ Rn , + t > 0 3 E |Xm (t, x)| < ∞. 1 Sm,3 (λ1 , λ2 ) = @ H! p = (π (1) , ..., π (k)) ) % %"" ) "% % % ' Xm (t, x) , t > 0, x ∈ Rn , ' "# 2 Sm,3 (λ1 , λ2 ) + ) C2k k! exp − t λ1 m ≥ 3 (c (n, κ))3k (K (n, κ))3 2 + λ2 2 + λ1 + λ2 2 gm,3 (λ1 , λ2 ) , @ A! dz gm,3 (λ1 , λ2 ) = R ( λ1 + λ2 + z λ2 + z z ) n−kκ , 0 < κ < n/m, λ1 , λ2 ∈ Rn , B m = 2k, k = 1, 2, ... Sm,3 (λ1 , λ2 ) = 0, m = 2k + 1, k = 1, 2, ... ) gm,3 (λ1 , λ2 ) , m = 2k, 3 # gm,3 (tλ1 , tλ2 ) = tH gm,3 (λ1 , λ2 ) , , H = 2 mκ − 2n, * @ ,>! gm,3 (ζ 1 , ζ 2 ) = "=' & / π −kκ ) 3 Γ (kκ/2) Γ ((n − kκ) /2) ( * ( ζ1 ζ 1 − ζ 2 )−kκ . ζ2 C .2 % κ ∈ + %! (0, n/m) mκ 2 mκ gm,3 (0, λ2 ) = k mκ − n, 2 mκ gm,3 (λ, −λ) = k mκ − n, 2 gm,3 (λ1 , 0) = k mκ − n, k (α, β) = π % n/2 Γ Γ α 2 n−α 2 + % gm,3 (λ1 , λ2 ) , λ1 + λ2 = 0. λ2 = 0, λ1 = 0 " ) x + Sm,4 (λ1 , λ2 , λ3 ) Γ β 2 λ1 mκ−2n , λ2 mκ−2n , λ mκ−2n , n−α−β 2 n−β Γ α+β 2 2 Γ Γ ! ) Sm,3 (λ1 , λ2 ) " / + 3!!# ) * " ) Xm (t, x) , t > 0, x ∈ Rn , + t>0 4 E |Xm (t, x)| < ∞ Cm m/2 c Sm,4 (λ1 , λ2 , λ3 ) = (n, κ) m!   3  k  ×3 sym [exp − t  {λ ,λ ,λ ,λ :Σ λ =0} i=1 m≥ 2 4 [m/2] (m!)4 ((m − 2k)!)2 k=1  2 3  λi 2 + λi   i=1 × (Ik,1 (λ1 , λ2 , λ3 ) + Ik,2 (λ1 , λ2 , λ3 ) + Ik,3 (λ1 , λ2 , λ3 ))] ' Ik,1 (λ1 , λ2 , λ3 ) = {K (κ, k)}4 {K (κ, m − 2k)}2 {k (kκ, (m − 2k) κ)}2 @ ,,! × λ1 + λ2 + λ1 + (m−k)κ−n λ1 + λ2 + λ3 + (m−k)κ−n kκ−n d , R Ik,2 (λ1 , λ2 , λ3 ) = {K (κ, k)}4 {K (κ, m − 2k)}2 {k (kκ, (m − 2k) κ)}2 @ ,@! × kκ−n λ1 + R kκ−n λ1 + λ2 + λ3 + kκ−n λ1 + λ2 + (m−k)κ−n (m−k)κ−n d , :2 * * - G * : : 9 9* G : * 9: I Ik,3 (λ1 , λ2 , λ3 ) = {K (κ, k)}4 {K (κ, m − 2k)}2 @ ,=! × * kκ−n −ν λ1 + R kκ−n × & / ν kκ−n −ν −λ λ1 + λ2 + (m−2k)κ−n λ (m−2k)κ−n .2 −λ λ1 + λ2 + λ3 + % d dνdλ. κ ∈ + %!# (0, n/m) Ik,1 (0, 0, λ3 ) = K1 k ((m + k) κ − 2n, (m − k) κ) λ3 Ik,1 (0, λ2 , 0) = K1 k (mκ − n, mκ − n) λ2 2mκ−2n 2mκ−2n , 2mκ−2n Ik,1 (λ, −λ, λ) = K1 k (2kκ − n, 2 (m − k) κ − n) λ Ik,2 : Ik,2 (0, 0, λ3 ) = K1 k ((2m − k) κ − 2n, kκ) λ3 Ik,2 (0, λ2 , 0) = K1 k (mκ − n, mκ − n) λ2 2mκ−2n 2mκ−2n Ik,2 (λ, −λ, λ) = K1 k (2kκ − n, 2 (m − k) κ − n) λ ) Ik,2 (λ1 , λ2 , λ3 ) m,4 * (λ1 , λ2 , λ3 ) λ1 = 0 , λ2 = 0 0, x ∈ Rn , 2mκ−2n 2mκ−2n , . Ik,1 (λ1 , λ2 , λ3 ) λ1 = 0 , λ3 = 0 λ2 = 0 , λ3 = 0 Sm,p (λ1 , ..., λp−1 ) " " / # ) p " , , Ik,2 (λ1 , 0, 0) = K1 k ((m + k) κ − 2n, (m − k) κ) λ1 λ1 + λ2 = 0 . λ2 + λ3 = 0 p > 4 # Xm (t, x) , t > m=2 m = 3. $ " ) m=2 @ ,C! + X2 (t, x) , t > 0, x ∈ Rn , t>0 # x p, S2,p (λ1 , ..., λp−1 ) , sym {λ ,...,λ :Σ e − t Σ λ + Σ − E |X2 (t, x)|p < 2p−1 (p − 1)! λ g2,p (λ1 , ..., λp−1 ) , λ =0} dλ g2,p (λ1 , ..., λp−1 ) = R & / − p p C2 c (n, κ) 2 S2,p (λ1 , ..., λp−1 ) = × @ ,D! , K1 = {K (κ, k)}4 {K (κ, m − 2k)}2 {k (kκ, (m − 2k) κ)}2 * 4! ∞. 1 , , 2mκ−2n Ik,1 (λ1 , 0, 0) = K1 k ((2m − k) κ − 2n, kκ) λ1 ' kκ−n λ p−1 λ + λ1 ... λ + Σi=1 λi .2 % + %!# 5 S2,2 (λ) = 2k (κ, κ) λ 2κ−n C2 c (n, κ) 2 2 n−κ . H * λ = 0. , p≥3 ! g2,3 (λ1 , λ2 ) const× λi 3κ−2n {λ1 = 0} , {λ2 = 0} , {λ1 = −λ2 } ; const × λi 4κ−3n {λk = λj = 0} , ! g2,4 (λ1 , λ2 , λ3 ) k = j, k, j = 1, 2, 3 {λk = λj = −λi } ; const× λl 5κ−4n {λi = λj = λk = 0} , 4! g2,5 (λ1 , λ2 , λ3 , λ4 ) i = j = k ∈ {1, 2, 3, 4} ; {λi = λj = 0, λk = −λl } , {λi = 0, λj = λk = −λl } , {λi = −λj = λk = −λl } , i = j = k = l ∈ {1, 2, 3, 4} . % * p ≥ 6. ) m=3 + t>0 1 @ ,B! S3,p (λ1 , ..., λp−1 ) = X3 (t, x) , t > 0, x ∈ Rn , x p, S3,p (λ1 , ..., λp−1 ) , C3 (c (n, κ))3/2 3! 4! E |X3 (t, x)| < ∞. p p p (S1 (λ1 , ..., λp−1 ) + S2 (λ1 , ..., λp−1 )) p = 2k, k = 1, 2, ... S3,p (λ1 , ..., λp−1 ) = 0 p = 2k + 1, k = 1, 2, ... # S1 (λ1 , ..., λp−1 ) = (2k − 1)! {λ i 2k−1 × γ ∈Γ (1,...,1) R i=1 × j=1 δ @ ,I! λ λ κ−n κ−n j j=1 j=1 dλdλ 1 ...dλ k − 2k j k ]; R i=1 ,...,λ :Σ i j=1 δ λj − k + − − t Σ [e sym λ =0} 2k+1 − + Σ − λ κ−n 2k+2 i κ−n j j=1 d 1 ...d k λ j j=1 k+1 ]. (k ,k )∈K(γ ) ) ( "% %"" " ' 2k ( ' ( $ " " ' 2k ( ' ( @( E K (γ) "! : ! ) x Sm,p (λ1 , ..., λp−1 ) , + Σ (k ,k )∈K(γ ) × : + k 2k−1 γ ∈Γ (2,1,...,1,2) λj + λ − λ κ−n i S2 (λ1 , ..., λp−1 ) = (2k − 1)! {λ × [e λ =0} ,...,λ :Σ − − t Σ sym p Sm,p (λ1 , ..., λp−1 ) = 7 ( Γc (1, ..., 1) , ( ( Γc (2, 1, ..., 1, 2) , (2, 1, ..., 1, 2) ) ' " γ ' "% ' Xm (t, x) , t > 0, x ∈ Rn , E |Xm (t, x)|p < ∞. ) Cm (c (n, κ))m/2 m! " ( $4 % t>0 + # p (S1 (λ1 , ..., λp−1 ) + S2 (λ1 , ..., λp−1 )) , p, :2 * * * - G * : : S1 (λ1 , ..., λp−1 ) = (p − 1)! {λ @ ,H! p−1 × − R γ ∈Γ (m−2,...,m−2) 9 9* G : ,...,λ :Σ j=1 λj + λp − @ ,A! × κ−n k k=1 (k ,k )∈K(γ ) S2 (λ1 , ..., λp−1 ) = (p − 1)! {λ γ ∈Γ (m−1,m−1,m−2,...,m−2) − R ! δ p−1 i i=1 j=1 κ−n k + k d k − k " λ p(m−2)+1 − dλp ], λ + Σ κ−n j(m−2) i λj − λ + Σ j=1 k=(j−1)(m−2)+1 [e λ =0} ,...,λ :Σ λ A j(m−2) − t Σ sym 9: − i p(m−2) × λp − t Σ [e λ =0} sym i i=1 κ−n * k j=1 k=(j−1)(m−2)+1 p(m−2)+2 κ−n × : δ k k=1 ) ( "% %"" " " ' p ( Γcp (m − 1, m − 1, m − 2, ..., m − 2) , ' ( m−1 ( ( m−2 ( ( ! , 6 *2 k + d k k ]. (k ,k )∈K(γ ) 7 ( (m − 2) ( " ) ' $4 Γc (m − 2, ..., m − 2) , ( ( " ' p ( @ ( " p−2 ( "! 7 ) " * * L1 " * # *8 = T : ' " # 5 .% &% " ∂β u = − (I − ∆)γ/2 (−∆)α/2 u, β ∂t = ,! %)6 = @! 7 >0 " u (0, x) = v (x) , α > 0, β ∈ (0, 1], γ ≥ 0 " v (x) = h (ξ (x)) , x ∈ Rn , ξ (x) , x ∈ Rn , A−C ∆ n " ( (−∆)α/2 , α > 0, ' % x ∈ Rn , " " % % 4 v (x) h( ) " # "# ( % − (I − ∆)γ/2 , γ ≥ 0, * + ( ,> ' 7 " ( 7 0 1! 4 4 &% " ( β ∈ (0, 1] * + ) " # ) ( '< ∂u ∂t (t, x) , Dtβ u (t, x) , ∂β u = ∂tβ # β = 1, β ∈ (0, 1), ' % / ' t ∂ 1 (t, x) = Γ (1 − β) ∂t Dtβ u + 0 ( 7 0 1 % ( ( ' j=0 zj , Γ (βj + 1) F Eβ (−x) = j=0 # . ) " dk Eβ (−x) ≥ 0, dxk β < 1, 9 x≥0 Eβ (−x) = : % $ < x → ∞, ' " = @! ( ) ' " 9 β<1( )% ( " 7 %# &# 6) k=1 ∞ 0 " ! exp − (xt)1/β dt t2 + 2t cos (πβ) + 1 % . Eβ , ' " (−1)k x−k + O |x|−N−1 Γ (1 − βk) D! % 7 ( % " 0 1! u (t, x) , 0 < t ≤ T, x ∈ Rn , $ n L1 (R ) = D! % 0 < β < 1, 0 1 4 ! 0 1! : (−1)j xj Γ (βj + 1) x ≥ 0, E1 (−x) = e−x . 9 Eβ (−x) = − = C! 6) x ≥ 0, 0 < β < 1, k = 0, 1, 2, ... sin (πβ) πβ N 6) z ∈ C1 , β > 0. " (−1)k % ,< % ∞ = =! G 1/β 7 ' x ≥ 0, β > 0, 0 < t ≤ T, ! % Eβ (z) = u (0, x) , tβ ( " ∞ % (t − τ )−β u (τ , x) dτ − κ < min (2α, n) /m, % ) " = ,! "% 4 ! 7 :2 * * * - G * : : 9 9* G : h. ) " Uε (t, x) = * 1 εmκβ/(2α) ε→0 ( = B! t x , ε εβ/α u x∈R Cm m/2 (n, κ) c m! ′ ei(x,λ , m≥1 +...+λ ) α "1 = * * Um (t, x) ′ ei(x−y,λ +...+λ ) ( λ1 ... λm )n−κ R α Eβ − sβ λ1 + ... + λm 2 (t, x) < ∞. κ < min (2α, n) /m, EUm n U1 (t, x) , t > 0, x ∈ R , 4 >! m=1 dλ1 ...dλm . x) $ α S1,2 (λ) = C12 c (n, κ) Eβ2 − tβ λ ) 0 < t ≤ T, x ∈ Rn , + $ ′ < W() κ < min (2α, n) /m, ×Eβ − tβ λ1 + ... + λm = H! ( ( λ1 ... λm )(n−κ)/2 R 2 Cm EUm (t, x) Um (s, y) = cm (n, κ) m! = I! ' 0 < t ≤ T, x ∈ Rn , , ×Eβ − tβ λ1 + ... + λm α W (dλ1 ) ...W (dλm ) , *" : 4 4!# W Eβ c (n, κ) ;! ' # * +$ λi = ±λj , i, j = 1, ..., m, i = j, * + " , ,, " Um (t, x) = n 9: m≥1 % κ * λ κ−n λ ∈ Rn . , 4 3! C12 c (n, κ) λ λ → 0. ' * ? , x ∈ (0, min (2α, n)) U1 (t, x) , t > 0, x ∈ Rn , ( 7 $ + # λ →∞ S1,2 (λ) = % κ−n / 1 C12 c (n, κ) n+2α−κ + O β t λ ( 0 1! ) 1/ λ 0 1! % # ) 3 1 λ n+2α−κ+1 n+2α−α " n ≥ 1, β = 1, γ = 0, α = 2. ) 2 % # Um (t, x) , t > 0, x ∈ Rn , ' # m ≥ 2. ) m≥2 ( $ ' Um (t, x) , t > 0, x ∈ Rn , x, # ′ ei(λ,x−x ) SU EUm (t, x) Um (t, x′ ) = R (λ) dλ " " + t>0 ,@ * = A! SU 2 2 Cm 2 cm (n, κ) K (κ, m) Eβ − tβ λ α λ mκ−n . m! "$ Um (t, x) , t > 0, x ∈ Rn , 1 # 4 @! (λ) = $ ) / λ → 0. , 4 ;! SU ,2 2 Cm m! (λ) = : Rn . mκ−n const × λ $ κ ∈ (0, min (2α, n)) , cm (n, κ) × 2K (κ, m) tβ λ "' 1 λ n+2α−mκ+1 . Um (t, x) , t > 0, x ∈ # Um (t, x) , t > 0, x ∈ Rn , E |Um (t, x)|p < ∞, 5 p * 1 +O n+2α−mκ ) $ ) x m ≥ 2 t>0 + !) SU × ,3 sym {λ ,λ ,λ :Σ × λ =0} 3 C2k k! (λ1 , λ2 ) = [Eβ − tβ λ1 ( λ1 + λ2 + λ (c (n, κ))3k (K (κ, m))3 α α Eβ − tβ λ2 Eβ − tβ λ1 + λ2 λ )kκ−n dλ] λ2 + λ α m = 2k R SU ,3 (λ1 , λ2 ) = 0 m = 2k + 1. !) SU × {λ ,4 (λ1 , λ2 , λ3 ) = sym λ =0} ,λ ,λ ,λ :Σ # Ik,1 , Ik,2 , Ik,3 4! ) SU Cm m/2 (n, κ) c m! $ 3 β i=1 Eβ − t λi α * 4 [m/2] k=1 3 β − t Eβ !" (m!)4 3k+1 ((m − 2k)!)2 α% λi i=1 & {Ik,1 + Ik,2 + Ik,3 } , 4! p ,p (λ1 , ..., λp−1 ) = Cm (c (n, κ))m/2 m! p (1) (2) × SU (λ1 , ..., λp−1 ) + SU (λ1 , ..., λp−1 ) , × {λ sym ,...,λ :Σ λ =0} #p−1 i=1 (1) SU (λ1 , ..., λp−1 ) = (p − 1)! $ p−1 Eβ − tβ λi α Eβ − tβ i=1 λi α% Σ1 (λ1 , ..., λp−1 ) & :2 * * * - G * : : 9 9* G : * 9: ,= (2) ' × {λ sym λ =0} ,...,λ :Σ #p−1 i=1 # Σ1 (λ1 , ..., λp−1 ) SU (λ1 , ..., λp−1 ) = (p − 1)! $ p−1 Eβ − tβ λi Eβ Σ2 (λ1 , ..., λp−1 ) 3! @! ) − tβ λi i=1 @ Σ2 (λ1 , ..., λp−1 ) . * * A # * " " * 5 ' * @ # = % ) * & 7 * ) 0! ' 7 9 α α% / ( " A " " C P / ( % % * % ( ( ( "% 4) % 7 ) B ) + " / ' " 7 ei(y,λ +...+λ ) ei(y,λ ) R 2 Cm (c (n, κ))m m! ei(y,λ +...+λ )−2 t λ +...+λ × ( λ1 ... λm )n−κ R ( λ1 − λ2 ei(y,λ) e−2 e−2 t λ ... λm−1 − λm t λ R − dλ1 ...dλm λm )n−κ dλ2 ...dλm ( λ − λ2 λ2 − λ3 ... λm−1 − λm Xm (t, x) ' Sm,2 (λ) = e−2 t λ R × dλ1 ...dλm e−2 t λ +...+λ dλ1 ...dλm ( λ1 ... λm )n−κ R : % &% ( % ( " λ1 = λ′1 − λ′2 , λ2 = λ′2 − λ′3 , ..., λm−1 = λ′m−1 − λ′m , λm = λ′m ) ′ Σm i=1 λi = λ1 = " "% "% G ( R " 0 1 Cov (Xm (t, x) , Xm (t, x + y)) = = " @ "% :; " G " − ( λ − λ2 2 Cm (c (n, κ))m . m! ( λm )n−κ ) dλ2 ...dλm λ2 − λ3 ... λm−1 − λm λm )n−κ dλ. ,C f (x) = 1/ x n−κ , f ∗m (x) = ' f (λm−1 − λm ) f (λm ) dλ2 ...dλm ; % t λ Sm,2 (λ) = e−2 ( % "% f ∗m (λ) f ∗m (λ) = = 9 " * + "% % 2 Cm (c (n, κ))m . m!2 " ' ' f ∗m (λ) = κ−n λ − λ2 R × R λ2 − λ3 R κ−n λm−1 − λm mκ−n = K (κ, m) λ K (κ, m) = π ' " λm 2κ−n . κ−n κ−n dλm dλm−1 ...dλ2 Γ κ2 Γ n−κ 2 (m−1) mκ−n , m−1 m−1 Γ iκ 2 i=1 Γ n−iκ 2 Γ n−(1+i)κ 2 Γ (1+i)κ 2 . ( f ∗m (λ) = K (κ, m) λ Sm,2 (λ) = e−2 B + , 0 < mκ < n. = k (κ, κ) k (κ, 2κ) ...k (κ, (m − 2) κ) k (κ, (m − 1) κ) λ ' " * λm )n−κ dλm = k (κ, κ) λm−1 ( λm−1 − λm λm )n−κ R ' f ∗m (λ) dλ2 ...dλm λ2 − λ3 ... λm−1 − λm ( λ − λ2 − R f (x − λ2 ) f (λ2 − λ3 ) ... $ Sm,2 − ' f (λ − λ2 ) f (λ2 − λ3 ) ...f (λm−1 − λm ) f (λm ) dλ2 ...dλm − R ) ' $ ! : R ) C ,! 4 λ % λ mκ−n ) mκ−n K (κ, m) 2 Cm (c (n, κ))m . m! %"# ' %"% Cum (Xm (t, x1 ) , Xm (t, x2 ) , Xm (t, x3 )) = c3 hγ , γ∈Γ 3 Cm (c (n, κ))m/2 , m! " ' = ( %"% &% + m = 2k. c3 = Γc3,m ) ' Γc3,m = ∅ % ' C @! " m = 2k + 1. hx (λ1 , ..., λm ) = ei(x,λ {n1 , n2 , n3 } = {m, m, m} , m +...+λ )− t λ +...+λ ( λ1 ... λm )(n−κ)/2 :2 * * * - G * : : " γ ∈ Γc3,m , m = 2k, " ) ' hγ = 9 9* G : * 9: K (γ) = {(ki , kj )} ) ,D E hγ hx (λ1 , ..., λ2k ) hx (λ2k+1 , ..., λ4k ) hx (λ4k+1 , ..., λ6k ) R 3k × δ λk + λk dλk . i=1 (k ,k )∈K(γ) " " γ ∈ Γc3,m ) "< ( ( Γc3,m = ((2k)!/k!)3 . : % ( ( ' " # ( ' hx (λ1 + ... + λk − λk+1 − ... − λ2k ) hγ = R ×hx (λk+1 + ... + λ2k − λ2k+1 − ... − λ3k ) ×hx (−λ1 − ... − λk + λ2k+1 + ... + λ3k ) dλ1 ...dλ3k ei[(x = ,Σ λ )+(x ,Σ λ −Σ λ )+(x ,−Σ λ −Σ λ +Σ λ )] R − t ×e / Σ λ −Σ λ ( + Σ ) λ −Σ λ + −Σ λ +Σ λ dλ1 ...dλ3k . ( λ1 ... λ3k )n−κ < Σki=1 λi − Σ2k i=k+1 λi = ω 1 3k Σ2k i=k+1 λi − Σi=2k+1 λi = ω 2 C =! ′ Σ3k i=2k+1 λi = λ3k Σpi=1 λi = λ′p , Σpi=k+1 λi = λ′p , Σpi=2k+1 λi = λ′p , p = 1, 2, ..., k − 1 p = k + 1, k + 2, ..., 2k − 1 p = 2k + 1, 2k + 2, ..., 3k − 1, λi = λ′i , i = 1, k + 1, 2k + 1 λk = ω 1 + ω 2 + λ′3k − λ′k−1 λ2k = ω 2 + λ′3k − λ′2k−1 ,B " ( " R × R ei(x # λi = λ′i+1 − λ′i , i ∈ / {1, k + 1, 2k + 1, k, 2k} , ) ' −x ,ω )+i(x −x ,ω ) − t( ω e + ω ) + ω +ω k−1 f (λ1 ) − R i=2 f (λi − λi−1 ) f (ω 1 + ω 2 + λ3k − λk−1 ) dλ1 ...dλk−1 2k−1 × f (λk+1 ) − R i=k+2 f (λi − λi−1 ) f (ω 1 + λ3k − λ2k−1 ) dλk+1 ...dλ2k−1 & 3k−1 × f (λ2k+1 ) − R ei[(x = i=2k+2 f (λi − λi−1 ) f (λ3k − λ3k−1 ) dλ2k+1 ...dλ3k−1 dλ3k dω 1 dω 2 −x ,ω )+(x −x ,ω )] − t( ω + ω e + ω +ω ) R × f ∗k (ω 1 + ω 2 + λ3k ) f ∗k (ω 2 + λ3k ) f ∗k (λ3k ) dλ3k dω 1 dω 2 . R %") " ' ( %" C ,! $ ) (2k)! k! Sm,3 (ω 1 , ω 2 ) = c3 %" ' − t( ω =0} e 3 sym{ω ,ω ,ω :Σ ω ((2k)!/k!)3 , Xm (t, x) : + ω + ω +ω f ∗k (ω 1 + ω 2 + λ) f ∗k (ω 2 + λ) f ∗k (λ) dλ, × ' f ∗k (ω 1 + ω 2 + λ) f ∗k (ω 2 + λ) f ∗k (λ) R ( ω1 + ω2 + λ ω2 + λ A ' % R m = 2k + 1, Sm,3 = 0. ’ " 7 ' * +J " 0 < α < n, 0 < β < n, 0 < α + β < n, R x−z α−n x−y β−n k (α, β) = π n/2 %- 0 1 I,! Γ Γ "% < dz = k (α, β) x − y ' ,! λ )kκ−n dλ R c3 (2k!/k!)3 = (C2k /k!)3 (c (n, κ))3k . 9 α 2 n−α 2 Γ Γ β 2 n−α−β 2 n−β Γ α+β 2 2 Γ α+β−n ) m = 2k, R = {K (κ, k)}3 ' , ( :2 * * A * - G * : : 9 9* G : * 9: ,I &&% 0 1 7 4 C $ ) )% 0 1 &&% 0 1 9 $ 0 1 ) ( % " " % "% &% / # % m1 , ..., mp ) ( ! ( " ( % " " ( " V % "% " "% : " 2 % %"% "% / " " ( ) "% "% # ( % (m1 , ..., mp ) Γ Γ' m1 +...+mp = M " V = {(1, 1) , ..., (1, m1 ) , (2, 1) , ..., (2, m2 ) , ..., (p, 1) , ..., (p, mp )} = 4 ,! :; " / ' p ) Wj , j=1 ' Wj = {(j, l) : 1 ≤ l ≤ nj } j Γ, 1 ≤ j ≤ p; , (j2 , i2 ) 6 Γ ( )! ( $ " " ) " E ! ( (j1 , i1 ) ) w = ((j1 , i1 ) , (j2 , i2 )) , j1 = j2 , . ( Γ (m1 , ..., mp ) " (m1 , ..., mp ) . ) K (γ) " γ ∈ Γ (m1 , ..., mp ) . / " v ∈ V, ' ' v 4 ,! % (1, 1) , (1, 2) @ ( $ (p, mp ) M. w = ((j1 , i1 ) , (j2 , i2 )) ∈ K (γ) ) % w = (k1 , k2 ) , ' k1 ( $ (j1 , i1 ) k2 ( $ (j2 , i2 ) 4 ,! "γ ( " ) $ ( : % %") γ |K (γ)| = M/2. " ( {Wj , j = 1, ..., p} ) ' %) ) # hi ∈ L2 (Rnm ) , i = 1, ..., p, p h (λ1 , ..., λM ) = hi λM − +1 , ..., λM , i=1 ' % Mi = m1 + ... + mi , i = 1, 2, ..., p, M0 = 0 $ ( < $ m ′ h1 (λ1 , ..., λm ) Cum R 4 @! hp λ1 , ..., λm R γ∈Γ (m ,...,m ) "% W (dλi ) i=1 δ λk + λk dλk h (λ1 , ..., λM ) R ' m ′ W (dλi ) , ..., i=1 = Mp = M. (k ,k )∈K(γ) , % ,H ' GPO B 8" %" !S 2434, B ( " " γ, 7 U7 8" δ( ) 7 M S QLD 4001, A < 7 !S CF24 4YH, UK 8" (m1 , ..., mp ) , K (γ) % " γ M , Q ,C U $ U $ %, S % T % R %, ,C 7 < ! D M 7 < , K% $ U $ % (N ), K% $, 01033,