The Renormalization of Self Intersection Local Times of Fractional Brownian Motion

Transcription

1 International Mathematical Forum, 2, 27, no. 44, The Renormalization of Self Intersection Local Times of Fractional Brownian Motion Anis Rezgui Mathematics epartement INSAT centre urbain nor B.P Tunis Tunisia anis.rezgui@fsb.rnu.tn Abstract We present a new approach to treat the problem of self intersection local time of a -imensional Fractional Brownian motion base on the property of chaotic representation an the white noise analysis. This approach coul be generalize to general Gaussian processes. Mathematics Subject Classification: 6H4, 6J65 Introuction Let Ω, F,IP ) be an abstract probability space which coul be specifie later an {B t : t } a -imensional fractional Brownian Motion fbm) with Hurst parameter H =H,,H ) ], [. It is known that t B t = K H t, s)w s where {w t : t } is a stanar Brownian motion an K H =K H,,K H ) is a kernel, see for instance [6] for the proof). We are intereste, in this paper, in computing, when it exists, the selfintersection local time of the fbm. More precisely we stuy the existence of the limit when ɛ goes to zero, of the following sequence of processes where L ɛ T = T t P ɛ B t B s ) sts P ɛ x) = x 2 e 2ɛ, x IR. 2πɛ

2 262 Anis Rezgui Many questions arise here ) Fin out a relation between N the number of chaos to be subtracte from L ɛ T, an H such that N L ɛ T n= L ɛ T,n amits a limit in an appropriate space when ɛ. 2) Uner which conition on an H there is a blow up of the expectation IEL ɛ T ) when ɛ? 3) Suppose we are in the blow up case, fin out a renormalization factor r,h ɛ) such that ) r,h ɛ) L ɛ T IEL ɛ T ) is boune in L 2 Ω) when ɛ. 4) Fin out the limit in L 2 Ω) or in law, when it exists, of ) r,h ɛ) L ɛt IELɛT ) when ɛ. In view of what has been one in the case of the classical Brownian motion, see for instance [3][8] [9] [][7] an [8] we shoul compute explicitly the chaos of the fbm, for this we nee some tools from the white noise analysis.. Tools from White Noise Analysis We quote some white noise analysis concepts as introuce in [3], referring to [] for a systematic presentation. Consier a white noise space S IR), B,μ), where B is the weak Borel σ-algebra of S IR), an μ is the centere Gaussian measure whose covariance is given by the inner prouct of L 2 IR), in the sense that the vector value white noise has the characteristic function Cf) =IEe i ω,f )= μ[ω]e i ω,f = e 2 f,f, ) S IR) where ω, f = j= ω j,f j an f j SIR, IR). Then a realization of a vector of inepenent fractional Brownian motions B j,j=,,, is given by B j t) = ω j,k Hj = t K Hj s, t)ω j s)s. 2) We recall the explicit formula of the kernel of a one imensional fbm with Hurst parameter h ], [, see for instance [] h</2 K h t, u) =C h { t 2 )h /2 t u) h /2 3)

3 Fractional Brownian motion 263 h /2)u /2 h t u } r u) h /2 r h 3/2 r [,t] u), where C h is some constant. h>/2 t K h t, u) =C h u /2 h r u) h 3/2 r h /2 r [,t] u) 4) u where C h is some constant. Hence we consier inepenent -tuples of Gaussian white noise ω = w,,ω ) an corresponingly, -tuples of test functions f =f,,f ) SIR, IR ), an use the following multi-inex notation: F n, f n = n! = f, f = n i! 5) i= n tf n t,,t n ) tfi 2 t) 6) i= f n i i t,,t n ) 7) an similarly for : ω n :,F n where for -tuples of white noise the Wick prouct : : see []) generalizes to : ω n := i= : ω n i i :. 8) The Hilbert space L 2 )=L 2 μ) 9) is canonically isomorphic to the -fol tensor prouct of Fock spaces of symmetric square integrable functions: L 2 ) SymL 2 IR k,k! t)) k = F. ) k= For a general element ϕ of L 2 ) this implies the chaos expansion the norm of ϕ is given by ϕω) = : ω n :,F n, ) n= ϕ 2 L 2 ) = n! F n 2 2,n 2) n

4 264 Anis Rezgui with kernel functions F in F an where 2,n is the norm in L 2 IR n,t). Given ξ SIR), let us consier the Wick exponential : exp ω, ξ : exp ω, ξ 2 ) ξ,ξ) 3) = n n! : ω n :,ξ n, ω S IR). 4) The S-transform plays an important role in the stuy of stochastic processes in particular the computation of their chaos expansion, see for example [?][]; we efine the S-transform of ϕ in L 2 )as Sϕξ) ϕ, : exp., ξ : = n ϕ n,ξ n ) 2,n. 5) In particular, for Hermitian operators A in L 2 IR), we can efine the secon quantization of A as an operator ΓA) inl 2 ) given by SΓA)Φ ) =SΦA ) 6) for Φ L 2 ). Generalize functions are obtaine via a Gel fan triple S) L 2 ) S) with S) the projective limit Hilbert spaces S) k S) projlim k S) k an S) k DΓA k )) where A is operating on L 2 IR, u) Afu) = 2 /u 2 + u 2 +)fu). 2 The kernels of the self-intersection local time of the fbm so L ɛ T = T t P ɛ B t B s )st P ɛ x) = x 2 e 2ɛ = 2πɛ 2π) L ɛ T = T t ts IR e ix ξ ɛξ 2 2 ξ e ibt Bs) ξ ɛξ 2 2π) 2 ξ IR

5 Fractional Brownian motion 265 in view of 2), for j =,,, B j t Bs j = ΔK j,ω j with ΔK j = K Hj u, t) K Hj u, s) ) ) S e ibt Bs) ξ f) = S e i ΔK j,ω j ξ j f j ) S e i ΔK j,ω j ξ j )f j ) = e f j = e f j 2 j= S IR) S IR) = e ξ 2 j ΔK j e iξ j Δk j,f j The last equality was by efinition of γ an so ) ξe ɛξ 2 2π) 2 S e ibt Bs) ξ f) = 2π) IR using this elementary equality e αx2 2π 2 +iβx = we obtain SP ɛ B t B s )f) = Finally we obtain = = = n= 2π 2π 2π 2π IR j= j= γ j ω j )e i ΔK j,ω j ξ j + f j,ω j γ j ω j )e i ξ jδk j if j IR j= β 2 α e 2α ΔKj ɛ exp{ f j, ΔK j 2 2 ΔK j ɛ)} e ξ 2 j 2 ΔK j 2 2 +ɛ) e iξ j Δk j,f j ξ j t s 2h j + ɛ exp{ f j, ΔK j 2 2 t s 2H j + ɛ) } j= n j = n= n j! 2 )n j t s 2H j + ɛ) n j+/2 f j, ΔK j 2n j n! 2 )n n= j= t s 2H j + ɛ) n j+/2 Proposition Let n IN an, L ɛ T = L ɛ T,2 n = 2π n! 2 )n : ω 2 n :,lt,2 n ɛ then l ɛ T,2 n = T t t s j= t s 2H j + ɛ) n j +/2 ) ΔK 2 n ) f 2 n, ΔK 2 n.

6 266 Anis Rezgui 2. The expectation of the self-intersection local time of the fbm Let us now compute the expectation of the self-intersection local time of the fbm, IEL ɛ T ), it is just the first chaos, so in view of the last proposition ) IEL ɛ T ) = = T t t s 2π T 2π j= T s s j= s2h j + ɛ) /2 t s 2H j + ɛ) /2 we use the change of variables s = ɛ 2H z = αɛ)z with H = j= H j IEL ɛ T )=αɛ)ɛ 2 2π T αɛ) T αɛ)z j= ɛ H j H z 2H j +) /2 we ivie the integral in two parts { IEL ɛ 2 T αɛ)z T )=αɛ)ɛ z 2π j= ɛ H j H z 2H j +) /2 + T αɛ) T αɛ)z } z j= ɛ H j H z 2H j +) /2 the first integral in braces is boune. Set π the secon integral in braces, so T { αɛ) z T H ɛ 2 π T = 2H H z H log T log ɛ H = 2H Proposition 2 Let T>an. If H lim IEL ɛ T )=. ɛ Moreover if H = IEL ɛ T ) const. log ɛ, an if H > IEL ɛ T ) const. ɛ 2 2H where the constants epen on T, H an. If H < there is no blow up lim ɛ IELɛ T )= 2π T s T s s H. z

7 Fractional Brownian motion 267 Proposition 3 Suppose all H j = H. LetT>an. If H = H = IEL ɛ T )= ) 2H log ɛ +o). If H > ) IEL ɛ T )=C Hɛ 2H 2 +o), where C H = s s 2H +) /2. Remark If all H j = 2 an = 2 we are in the case H = an we obtain the Varahan renormalization term [23]. 3 The self-intersection local time of the fbm as a generalize function First of all let us recall Theorem. [9] Let Ω, F,m) be a measure space, an Φ λ a mapping efine on Ω with values in S). We assume that the S-transform of Φ λ ) is an m-measurable function of λ for any test function f SIR) 2) obeys to the following estimate SΦ λ f) C λ)exp{c 2 λ) A p f 2 2 } for some fixe p an for C L m), C 2 L m). Then Φ λ is Bochner-integrable in the Hilbert space S) q for q large enough, mλ)φ λ S) Ω an S mλ)φ λ )f) = mλ)sφ λ f). Ω Ω Using the last theorem an the following formula δ = e iξ x ξ 2π) IR

8 268 Anis Rezgui we coul efine δb t B s ) as a Bochner integral in S) an SδB t B s ))f) = exp{ f j, ΔK j 2 2π t s H t s } 2H j An so, again by the last theorem, if H < L T = T t j= st δb t B s ) is well efine in S). Suppose now that H. The iea is that if we subtract some of the first terms in the expansion of the exponential function in the expression of the S-transform of δb t B s ), we coul obtain an integrable function in factor of the remaining part, then the secon conition of Theorem will be satisfie. An so we coul efine a renormalization of the self-intersection local time in S). Let N IN an efine δ 2N) B t B s ) by its S-transform so Sδ 2N) B t B s )f) = 2π t s H n, n N Sδ 2N) B t B s )f) 2π t s H n, n N 2 )n we nee to estimate the L -norm of ΔK j, ΔK j for fixe j. ) We treat first the case when all H j > /2. Let h>/2, in view of 4) ΔK h = K h t, u) K h s, u) =C h u /2 h { [,s] u) + [s,t] u) t u 2 n t s j= j= } r u) h 3/2 r h /2 r f j, ΔK j 2n j 2 t s 2H jn j f j 2n j ΔK j 2n j t s 2H jn j r u) h 3/2 r h /2 r we obtain ΔK h C h {c T h /2 t s + c 2 t s h+/2 } suppose that t s is small enough, we get Sδ 2N) B t B s )f) t s 2n 2π t s H 2n t s 2H jn j n, n N 2π t s H +2N H ) exp{ 2 j= f j 2 } j= j= f j 2n j

9 Fractional Brownian motion 269 where H = max j H j. then Theorem 2. Let T> an N IN, suppose that L 2N) T 2N > H H T t is well efine as an element of S) an st δ 2N) B t B s ) lim ɛ L2N) ɛ = L 2N) in S) 2) When all H j < /2 we obtain a ba estimation of ΔK j an so we on t have a result. Remark When all H j =/2, the conition uner which L 2N) is well efine in S) is that 2N > 2, this correspon to a result obtaine in [9]. 4 Estimation of the L 2 -norms of the chaos of the fbm Suppose, in this section that we are in the blow up case i.e H an that H<. Now we state our main result 2 Theorem 3. Let T>, n an 2. Then, if H = log ɛ L ɛ T,2 n 2 has a finite non trivial limit when ɛ goes to zero an If H ], 3/2[, lim L ɛ T,2 n 2 2T 2 n)! ɛ bh,, n) /2. log ɛ 2π n!) lim ɛ L ɛ 2 n 2 2 n)! 2π n!) ch,, n) /2 3 2H)2 H) T 2 H.

10 27 Anis Rezgui If H =3/2 L ɛ T,2 n 2 has a finite non trivial limit when ɛ goes to zero an log ɛ If H > 3/2 lim L ɛ ɛ T,2 n 2 T 2 n)! ch,, n) /2. log ɛ 2π H n!) lim ɛ 2 3 4H L ɛ 2 n 2 = ɛ 2π 2T 2 n! /2, fh,, n)+gh,, n)) n!2 n where the constants b, c, f an g are given in 2), 23), 9) an 22) Remark )The results of the last theorem generalizes the case of the classical Brownian motion i.e H =/2, in fact it was shown in [3] that the renormalization factor was log ɛ /2 when = 3 an ɛ 3 2 when 4. 2)It was shown in [2] that when = 2 an / < H < 3 there is no nee 2 to renormalize by multiplication, the result in our theorem coul be seen as a generalization for H</2an 2. 3)The case H>/2is more complicate because we obtain a specific singularity in ɛ for each chaos. Which is not so surprising because when H>/2 the fbm is smoother an, intuitively, its self intersection local time is worse. Proof of Theorem 3 For n IN the 2 nth chaos is given by see proposition ) L ɛ T,2 n = 2π n! 2 )n : ω 2 n :,lt,2 n ɛ l ɛ T,2 n = T t t s j= t s 2H j + ɛ) n j +/2 ) ΔK 2 n so IE{L ɛ T,2 n )2 } = 2π) n!) 2 2 )2n 2 n)! lt,2 n ɛ 2 L 2 IR 2n ) 7) In view of 7), we nee to estimate lt,2 n ɛ 2 IR = 2n lɛ T,2 n 2 2,2n, IR 2n 2n u T t T t tst s l ɛ T,2 n 2 2,2n = j= ΔK j t, s) 2n j t, s)δk j t, s) 2n j t,s ) t s 2H j + ɛ) t s 2H j + ɛ) ) nj +/2

11 Fractional Brownian motion 27 by using Fubini theorem we first get 2n u ΔK j t, s) 2n j t, s)δk j t, s) 2n j t,s )= IR 2n j= = = = = j= IR 2n j 2n j j= i= 2n j j= i= j= IR 2n j uδk j t, s) 2n j t, s)δk j t, s) 2n j t,s ) u j i ΔK jt, s)t, s)δk j t, s)t,s ) ) IE ΔB j t, s)δb j t,s ) t t 2H j + s s 2H j t s 2H j s t 2H j ) 2nj so l ɛ T,2 n 2 2,2n = T t T t tst s ) 2nj t t 2H j + s s 2H j t s 2H j s t 2H j ) nj 8) t s 2H j + ɛ) t s 2H +/2 j + ɛ) j= in view of the symmetry of the omain an the integran function it suffices to integrate only on T T2 where T = { <s <t <s<t<t} an T 2 = { <s <s<t <t<t}. Let us first integrate over T, we make the following change of variables x = t s y = t s z = s t where t is consiere as a parameter. Set l ɛ, T,2 n 2 2,2n resp. lɛ,2 T,2 n 2 2,2n ) the integral over T resp. T 2 ), we obtain j= l ɛ, T,2 n 2 2,2n = T t x+y+z t xyz x + z) 2H j +y + z) 2H j x + y + z) 2H j z 2H j ) 2nj) x 2H j + ɛ)y 2H j + ɛ) ) nj +/2

12 272 Anis Rezgui it is almost impossible to compute this integral at least for us) when all H j are ifferent, so let us suppose that all H j are equal to some H. Denote by θ t ɛ) = t, an make the following change of variables, x, y, z) = ɛ 2H ɛ 2H x,y,z ), we get T l ɛ, T,2 n 2 2,2n = ɛ 3 2H xyz x+y+z θ tɛ) x + z) 2H +y + z) 2H x + y + z) 2H z 2H ) 2n, x 2H + )y 2H +) by a symmetry argument in x, y) l ɛ, T,2 n 2 2,2n =2ɛ 3 2H T x+y+z θ t ɛ) x y θ tɛ) xyz x + z) 2H +y + z) 2H x + y + z) 2H z 2H ) 2n, x 2H + )y 2H +) enote by f H x, y, z) =x + z) 2H +y + z) 2H x + y + z) 2H z 2H an by ft, H,, n, ɛ) = x+y+z θ t ɛ) x y θ tɛ) xyz ) 2n f H x, y, z). x 2H + )y 2H +) First, note that f H x, y, z) x 2H an for H > x 4Hn xy <, x y x 2H + )y 2H +) this implies that for every z IR + x y f H x, y, z) 2n xy x 2H + )y 2H +)

13 Fractional Brownian motion 273 exists. On the other han for α ], 2 2H[, z α f H x, y, z) 2n ecreases to zero when z tens to infinity, then f H x, y, z) 2n fh,, n) = z xy < IR + x y x 2H + )y 2H +) an so that lim ft, H,, n, ɛ) =fh,, n). 9) ɛ Therefore we obtain, if H ], 3/2[ an, if H 3/2 Suppose now H = an then where an lim ɛ lɛ, 2 n 2 2,2n = lim 3 ɛ ɛ 2H l ɛ, 2 n 2 2,2n =2TfH,, n). l ɛ, T,2 n 2 2,2n =2ɛ 2H T ft, H,, n, ɛ) θ t ɛ) xy x y θ tɛ) ft, H,, n, ɛ) log ɛ ah,, n) = bh,, n) = sup sup t T ɛ> So we obtain Let us now treat l ɛ,2 2 n 2 2,2n, x y x t ft, H,, n, ɛ) x 4Hn x 2H + )y 2H +) ah,, n) θ t ɛ){ + bh,, n)}, log ɛ xy xy log ɛ x y θ tɛ) x 4Hn 2) x 2H + )y 2H +) x 4Hn x 2H + )y 2H +) log ɛ lɛ, T,2 n 2 2,2n T 2 ah,, n) { + bh,, n)}. log ɛ T l ɛ,2 2 n 2 2,2n =2ɛ 3 2H t gt, H,, n, ɛ). 2)

14 274 Anis Rezgui where gt, H,, n, ɛ) = x+y z θ t ɛ) z x y θ tɛ) xyz ) 2n g H x, y, z) x 2H + )y 2H +) an g H x, y, z) =x z) 2H +y z) 2H x + y z) 2H z 2H we have g H x, y, z) 2x 2H an so z z 4Hn+2 z x y x y ) 2n g H x, y, z) xy = x 2H + )y 2H +) xy g H x, y, ) ) 2n z 2H x 2H + )z 2H y 2H +) is well efine on IR + an for H > 3/2, one can choose α ], 2H 2[ such that when z + ) 2n g H x, y, z) z α xy z x y x 2H + )y 2H +) an then IR + z z x y xy ) 2n g H x, y, z) x 2H + )y 2H +) = gh,, n) < 22) so that lim gt, H,, n, ɛ) =gh,, n). ɛ Suppose H =, the same computation as in the case of l ɛ, 2 n 2 2,2n leas to an finally log ɛ lɛ,2 T,2 n 2 2,2n 22n T 2 ah,, n) { + bh,, n)} log ɛ log ɛ lɛ T,2 n 2 2,2n 222n T 2 ah,, n) { + bh,, n)}. log ɛ

15 Fractional Brownian motion 275 Suppose now <H<3/2, we have θtɛ) gt, H,, n, ɛ) zz 4Hn+2 θtɛ) zz 2 2H = 2 2n ch,, n). x y θ t ɛ) z x y xy g H x, y, ) xy zx) 2H + )zy) 2H +) 2 2n x 4Hn x 2H y 2H So that then gt, H,, n, ɛ) 2 2n ch,, n). 23) l ɛ,2 2 n 2 2,2n 2 2n ch,, n) 3 2H)2 H) T 4 2H we know that lim l ɛ, ɛ T,2 n 2 2,2n =, then lim l ɛ ɛ 2 n 2,2n exists. Finally let us suppose that H =3/2 gt, H,, n, ɛ) + θtɛ) l ɛ,2 2 n 2 2,2n =2 z z T z x y θ tɛ) z x y θ tɛ) xy t gt, H,, n, ɛ) ) 2n g H x, y, z) xy x 2H + )y 2H +) ) 2n g H x, y, z). x 2H + )y 2H +) The first term is boune by 2 2n xy x 4Hn =2 2n eh,, n), x y x 2H + )y 2H +) the secon term is boune by θtɛ) z 2 2n x 4Hn ) z n+/2 =2 x 2n ch,, n) log θ t ɛ), 2H y 2H so log ɛ lɛ,2 2 n 2 2,2n 2 x y log ɛ 22n eh,, n)t + 22n H T 222n ch,, n)t + log tt. log ɛ

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