An approach to the Biharmonic pseudo process by a Random walk

Transcription

1 J. Math. Kyoto Univ. (JMKYAZ) - (), An approach to the Biharmonic pseudo process by a Random walk By Sadao Sato. Introduction We consider the partial differential equation u t 4 u 8 x 4 (.) Here 4 x 4 is called the biharmonic operator. The fundamental solution of this equation is q t x dλe ixλ 8 λ 4 t (.) As shown by Hochberg[H], this function is not positive valued. Hence the usual probabilistic method is not applicable. However, there are several probabilistic approaches to this equation. Krylov[K],later Hochberg[H] and Nishioka[N] considered a signed finitely additive measure on a path space, which is essentially the limit of a Markov chain with the signed distribution q t x. Krylov obtained a path continuity of the biharmonic pseudo process. Note that there does not exist the σ-additive measure on a path space realizing a stochastic process related to the biharmonic operator. We can consider the biharmonic pseudo process only through a limit procedure from a finite additive measure. Our aim of this paper is to propose a new limit procedure for the biharmonic pseudo process. Nishioka considered the first hitting time to and obtained the joint distribution of the first hitting time and the first hitting place. His remarkable result is that there appears the differentiation at for the distribution of the first hitting place(see 99 Mathematics Subject Classification(s). N/A Additional infomation Communicated by N/A Additional infomation(s) Revised Date Revised Dates Received Date Tokyo Denki University

2 Sadao Sato section 6). However, since the jump distribution q t x is spread on the entire of R, it is difficult to understand this fact intuitively. In this paper, we define a random walk, which has the only finite jumps with signed measure. And we will show Nishioka s result by a scaling limit. We can get short and elementary proofs for his result. And we can understand that the differentiation at is caused by the hitting measures on the two boundary points, which are infinite as a limit and have opposite signs (see Proposition 5.). We can also get the distribution of the first exit time and place for finite intervals(section 8).. Definition of a biharmonic random walk Let Y i be the i.i.d. random variables with the signed distibution defined by P Y p P Y q P Y r (.) where each of probabilities should be prescribed later. Consider the random walk X n X Y Y n Let f be a measurable function and we shall write E f X n k f k P X n k and Also we shall write By the Taylor expansion, we have E x f X n E f X n X x P x X n k P X n k X x E x f X f x p q r f x q 4r f x If we set that is then we have p q r q 4r p 6r q 4r q 4r 3 f x E x f X f x r f x (.) Here, number r must be negative, since we match (.) to (.). Now we compute the characteristic function of Y. M µ def E e iµy p qcosµ r cosµ 4r cosµ

3 & An approach to the Biharmonic pseudo process by a Random walk 3 Since r is negative, we get max µ M µ and min µ M µ 6r. Let r! 8, and we have " M µ #". Since E x e iµx n e iµx M µ n this quantity converges to zero as n $ if not cos µ. Now we have Let %'& p 4 q M µ cosµ n( n) be the filtration generated by % X n( n). Since p q r 3 3 the total variation of the measure restricted on & n is equal to n. Note that & n is a finite set, which consists of 5 n atoms, and the mean E x is defined for any event of n. Set p n k P X n k Then we have p n k e ikµ M µ n k Thus p n k is the Fourier cofficient of the right-hand. Therefore p n k M µ n e ikµ dµ (.3) and we have lim n* p n k (.4) Remark. We can take any r in 8 for our purpose. For example, if we take r + 6, then we have " M " and this situation looks like better for some cases. However, we point out that the value 8 seems to be best for the numerical simulation to be rapid. 3. Scaling limit to continuous time and space For ε, Then we have, set x kε t nε 4 p ε t x def p n k M µ t- ε 4 e ixµ- ε dµ - ε M εy t- ε 4 e ixy - dy ε ε

4 4 Sadao Sato We will compute the following limit def q t x lim ε* ε p ε t x Let δ x be Dirac s delta function and we clearly have q x δ x Since M εy is an even function, we consider ε p ε t x - ε - ε -. ε - ε -. ε - ε / 4 I I I 3 say - ε / 4 M εy t- ε 4 cos xy dy First, we get I '" - ε - ε -. ε -. ε cosεy t- ε 4 dy -. ε ε u t- ε 4 du cosεu t- ε 4 du -. ε e u t- ε du $ ε $ Second we estimate I. Note that M µ is monotone in the interval and M ε ε M ε 3-4 Therefore I " ε 8 ε3- t- ε 4 ε e t- 8ε 5/ $ Finally, in the interval of the integral I 3, we have M εy Now we obtain the following conclusion: 8 ε4 y 4 Theorem 3.. By the scaling x kε t nε 4, we have def q t x lim p n k n* ε e 8 t y4 ixy dy which is the fundamental solution of the equation u t 4 u 8 x 4 8 ε3-

5 An approach to the Biharmonic pseudo process by a Random walk 5 Next we consider the initial value problem for (.). Theorem 3.. Let f be a continuous function in L R. Suppose that f is the Fourier transform of a function ˆf in L R, that is Set f x u ε t x where denotes its integer part. Then there exists the limit e iλ x ˆf λ dλ (3.) def E3 x f εx3 t ε 4 ε (3.) u t x lim ε* u ε t x which satisfies (.) and u x f x Proof. Let l x ε6 n t ε 4 Noting that E l is the measure on a finite set, we have u ε t x E l e iλ εx n ˆf λ dλ dλ ˆf λ e iλ ε7 l8 k9 p n k k dλ ˆf λ e iλ εl M ελ n Since ˆf is in L R and M ελ n is bounded, Lebesgue s dominated convergence theorem implies u t x which completes the proof. 4. Hitting measure to dλ ˆf λ e iλ x e 8 λ 4 t dy f y dλe iλ 7 x y9 8 λ 4t For each s;:, the Green operator of % X n( n) is defined by G s f l s n E l f X n n<

6 6 Sadao Sato By the Parseval s equality, we have p n k k M µ n dµ p n =" which is monotone decreasing to zero as n goes to. Schwarz s inequality implies E l f X n and we obtain the following: p n k l f k ;"?> p n #@ k Proposition 4.. For every f of C, G s f is analytic in s':, and it holds that G s f l D" We define the first hitting time to the set Z FG as and we shall write since & p n l k σ min% n;x n : ( def P l X n k;n : σ6 n is a finite set, here the right-hand quantity is well-defined. Then we have p n l k P l X n k P l X n k;σ " n P l X n k n P i< H l σ m X σ i P i X n m k m< where the last equality is implied from the strong Markov property of % X n(. For l and k in Z, define g s l k G s k l B H i s l g s l k s n p n l k 5 n< s n P l σ n X σ i n< Note that g s l k and H i s l are analytic in si: 3, since the total variation of E x 3 restricted on & n is equal to n. From the definition, we get g s l k g s l k H i< H i s l g s i k (4.) Fix s;:, and we shall use the notation c n e inµ dµ c sm µ n (4.)

7 An approach to the Biharmonic pseudo process by a Random walk 7 Since (.) and (4.) imply that we have Especiallly, when k and, we have g s l k c k l (4.3) g s l k c k l H i< H i s l c k i (4.4) c l8 c H s l H s l B (4.5) c l8 H s l c H s l (4.6) Proposition 4.. For s;: and l J Z FG, Moreover H i s l are analytic in s:. H s l c l8 c c l8 c c (4.7) H s l c l8 c c l8 c c (4.8) Proof. We know that c l s are analytic in sk:. The first part of the proposition is clear from (4.5) and (4.6), if we can show that the denominator c c has no zero in s:. We have c cosµ dµ sm µ cosµ sm µ dµ sm µ So, we get Re c cosµ Re s M µ sm µ dµ, Similarly, we obtain Re c L,. Now the second part is immediate and the proof is complete. 5. Existence of the Hitting measure In this section, we will compute the limit lim s* H i s l which is the interpretation of the hitting measure P l σ m X σ i P l X σ i σ : m<

8 N 8 Sadao Sato However, we note that the latter quantity may not be convergent, because these terms are signed. At first, we study the c n s in (4.). Set z e iµ, and change a variable. The residue theorem implies c n e inµ s% dµ (5.) cosµ ( KM dz z n i z s% (5.) z 8 z ( M 8z n8 dz is 8 s z z (5.3) 4 8 s z N O Res n8 4v z 4 z z 4 (5.4) where P is the unit circle in the complex plain and v - 4 (5.5) s In the remainder of this section, we restrict s as a real number. So, we assume that : s : and v is positive. We consider zeros of the denominator f z 4v 4 z z 4. By an elementary calculation, we obtain that α iv vq v 4 4 v ivq v 4 4 v (5.6) is a zero point, which satisfies α ':. We note that α satisfies α iv α (5.7) And ᾱ is another zero point inside of P. Other two zeros are outside of P. Therefore In the following, we shall use c n 8 s α n8 f α ᾱ n8 f ᾱ 6 s Re α n8 α 8v 4 α α 4 α 4 sv 4 Re α n α α (5.8) c n Re α n α α sv 4 c n (5.9) instead of c n for simple notations. Note that v tends to zero as s tends to. Let v be

9 An approach to the Biharmonic pseudo process by a Random walk 9 small and Taylor expansion derives that α i v iv 4 i v 3 3 i v 5 (5.) log α i v i v i v 5 (5.) α i 4 v 6 i v i v 5 (5.) α i v α 4 6 i v i v 5 (5.3) α n n i v in v n 4n i v 3 (5.4) c n v 4n v n n 8 v 3 (5.5) 3 From (5.7) and (5.9), we obtain the following simple equalities c n c n8 Re α n α 5 (5.6) c n c α n8 Re n α (5.7) α v α Im n8 α (5.8) which are often used in our computations. By Proposition 4. and the aboves, we get and H s l H s l c l8 c l8 c c l8 c l8 c Re α l8 α Re α v l 3l 9 4 v 3 v 4v 3 $ as s $ H s l H s l c l8 c l8 c l8 c l8 c c α Im l8 α α Im α l 3 v3 l 3l v 4 $ l 3 as s $ 8 v5 v3 Thus we get the following proposition: Proposition 5.. For l J Z FG, H l l (5.9) H l l (5.)

10 Sadao Sato Corollary 5.. As v $, the followings hold: g l k SR 8 8 g s l k sv 3 (5.) 8 l 3 8k kl 4kl 4 3 l3 k T l 8 k 3 8l kl 4lk 4 3 k3 k " l (5.) Proof. (5.) is easily obtained from (4.3) and (5.8). The latter is obtained by an elementary but long calculation. We will omit the details. Now we can explain why the differentiation appears at. Let f x be a differentiable function and consider E l s σ f εx σ f ε H s l f ε H s l f ε f ε H s l H s l f ε f ε By letting s $, we get H s l H s l lim s* E l sσ f εx σ f ε f ε We take the same scaling as in Theorem 3., that is, set x zero, the right hand side of this equation goes to f ε f ε l 3 lε and fix it. As ε goes to f f x Therefore we obtain Q x f Z σ f f x for the biharmonic pseudo process % Z t (, that is the stochastic process obtained through the scaling limit in Theorem 3.. Here, Q x denotes the expectation for % Z t ( starting from x. However, there does not exist such measure in the usual sense. We will consider Q x only through a limit procedure for our random walk % X n(. In this discussion, we used double limit procedure. We will give more direct proof by using the scaling limit in the next section. 6. Scaling limit of the first hitting time and place We will compute the Fourier-Laplace transform of the first hitting place and time: Q x e iµz σ λ σ 6 which was first obtained by Nishioka[N]. We use the same scaling as in section 3. Let f x be a bounded C -function and x lε. We consider Q x e λ σ f Z σ def lim ε* E l e ε 4 λ σ f εx σ 6 (6.)

11 where x Here An approach to the Biharmonic pseudo process by a Random walk lε is fixed. The left-hand side of this equation has only symbolical meaning. σ inf% t, ;Z t : ( is the first hitting time to the set of the biharmonic pseudo process % Z t (. We consider the mean Q x for % Z t ( through the scaling limit of % X n(. Now we compute the right-hand side of (6.): ε E l e 4 λ σ f εx σ ε f ε E l e 4 λ σ ;X σ ε f ε E l e 4 λ σ ;X σ f ε H e ε 4λ l f ε H e ε 4λ l f ε f ε H H f ε f ε H H I I say Since s e ε 4λ, we have v e ε4 λ - 4 ε λ - 4 For the simplicity, we set By (5.), we have ν λ - 4 (6.) α l exp l v i O v (6.3) exp lεν i O ε (6.4) $ e νx7 8 i9 as ε $ (6.5) Recall (5.) and we have Thus we get H H Re α l α Re α Re α l i v v Re e νx7 8 i9 i as ε $ I On the other hand, we obtain f e νx cos νx sin νx as ε $ H H Im α l8 α α Im Thus it is implied that Im α l8 I α α v 4 Im e νx7 8 i9 v as ε $ f U e νx sin νx ν Therefore we have proved the following theorem by the new method, that is the scaling limit of the Markov chain % X n(.

12 Sadao Sato Theorem 6. (K.Nishioka). For every differentiable function f, we have Q x e λ σ f Z σ f e νx cos νx sin νx where ν λ - 4. In particular, it holds that Q x e iµz σ λ σ e νx cos νx sin νx Remark. Let λ $ in (6.5), and we obtain f ν e νx sin νx 5 (6.6) iµ ν e νx sin νx (6.7) which satisfies u x Q x f Z σ f f x u u f 5 u f U Here we find the differentiation of f x. This fact, which is due to K.Nishioka, seems to be remarkable for many probabilists. As was shown in [N], we can write the Laplace inverse of the above formula in an explicit form. From the Laplace inversion formula, a direct calculation shows that V e ν7 a8 ib9 i 4 when b '" u 3 e u4 t 8 e u 7 a b8 i7 a8 b9w9 e u 7 a8 b i7 a b9w9 du (6.8) a. Then we easily obtain the following functions V K t x YX Re e ν7 8 i9 x i (6.9) u 3 e u 4 t- 8 sinxu 4 cosxu e xu du (6.) V J t x X ν e νx sinνx (6.) u e u 4 t- 8 sinxu 4 cosxu e xu du (6.) which give us the density function of the first hitting time σ and the hitting place: Q x σ J dt Z σ J dy K t x dtδ dy J t x dtδ dy (6.3) 7. Scaling limit of the transition probability with killing In the previous section, we obtained g l k. Here, we consider the scaling limit of g s l k, which is the Laplace transform of the transition probability of the biharmonic pseudo process with the absorbing boudary. We can find the same result with that in Nishioka[N, section 9.]. Let lε x kε y and x y be positive. Noting that t ε 4, we consider Q x σ e λt def f Z t dt lim E ε* l e λ ε 4n f εx n ;n " σ ε 4 (7.) n<

13 An approach to the Biharmonic pseudo process by a Random walk 3 Hence Q x σ e λt δ y Z t dt E l e λ ε 4n k X n ;n " σ ε 3 n< g s l k ε 3 c k l H s l c k8 H s l c k8 ε3 where s e λ ε 4. On the other hand, since k : : l, we have p n l k P l X n k P l X n k ;σ : n n P j< H l σ r;x σ j P j X n r k r< Multiply the both side s n, and sum in n, we obtain Therefore where R c l8 k8 H j< H j s l c k8 8 j c k j H H H j< H j s l c k8 8 j g s l k c l8 k8 c k l R c k c k c k8 c k8 H H c k c k c k8 c k8 From (5.6),(5.7) and the equalities following them, we can easily get sv 4 R 4v Re α l8 i Im α k 4vIm α l8 Im α k as s $ v $. Then we obtain ε 3 R 4 ν 3 Im αl8 Im α k 4 ν 3 Im e 7 8 i9 νx Im e 7 8 i9 νy 4 ν 3 e νx sin νx sin νy as ε $ By remembering (5.) and (6.4), it is easily checked that We may therefore see that ε 3 c k l ν 3 Re e 7 8 i9 ν N x y N i e λt q t x dt ν 3 Re e 7 8 i9 ν N x N i

14 V a a 4 Sadao Sato Thus, we conclude q t x y X q t x y;t : σ q t x y q t x y V 4 ν 3 e ν7 x8 y9 sinνxsinνy Since x q t x ν e νx sinνx and (6.) holds, we obtain the following result from the scaling limit of % X n(. Theorem 7. (K.Nishioka). q t x y q t x y q t x y J t y [Z x q t x Remark 3. The third term in the right hand of this equation shows an effect from the differential term of the first hitting place. For the Brownian motion, this term does not exist and the reflection principle holds. From the above, we also get lim e λt q t x y dt λ* lim λ* ν 3 Re e 7 8 i9 N ν x N y i Re e 7 8 i9 ν7 x8 y9 i 4e ν7 x8 y9 sin νx sin νy R 4x y 4 3 x3 y T x 4xy 4 3 y3 y " x which is the scaling limit of g l k. On the other hand, for the Brownian motion, it is well known that q t x y dt x y x y R x y T x y y " x 8. Hitting measure to the boundary of finite intervals In this section, we consider the first hitting time σ min% n;x n : or X n, L(I By an analogous argument as in section 4, we obtain g s l k c k l H j< H j s l c k j (8.) H L8 H L8 \] and ]^c c L8 c L8 3 c c L8 3 c L8 4 c L8 c L8 3 c c L8 3 c L8 4 c _` ` ]\ ]^H H H L8 H L8 _` ` ]\ ]^c l8 c l8 c L8 l c L8 l _` ` a (8.)

15 b b b b b b c c An approach to the Biharmonic pseudo process by a Random walk 5 We consider the new variables x H H H L8 H L8 y H H H L8 H L8 u H H H L8 H L8 w H H H L8 H L8 Then we get a simultaneous equation where a c c b c d c c b c x y c u w c h h h 3 h 4 a c c L8 c L8 3 c L8 4 Re α Re α L8 α 5 b c c L8 c L8 3 c L8 4 4v α Im α v α Im L8 3 α α d c c L8 c L8 3 c L8 4 Re α Re α L8 α 5 c c L8 c L8 4 v Im α L8 3 5 h c l8 c l8 c L8 l c L8 l h c l8 c l8 c L8 l c L8 l h 3 c l8 c l8 c L8 l c L8 l h 4 c l8 c l8 c L8 l c L8 l We set γ Re α L8 5 ρ Im α L8 Using (5.),(5.3),(5.6) and (5.8), we obtain Thus we get a v Re α L8 i v o v v γ ρ o v 5 b v 3 v Im α L8 i v o v 3 v 3 γ ρ o v 3 B d v γ ρ o v B c v ρ o v det def ab c v 4 ρ γ ρ o v 4 5 det def db c v 4 ρ γ ρ o v 4 (8.3) (8.4) B

16 6 Sadao Sato Moreover h Re α 5 α l8 α L8 l vre i 5 α l8 α L8 l o v 5 h v Im α αl8 α L8 l v Im α l8 α L8 l o v B h 3 vre i α l8 α L8 l o v 5 h 4 v Im α l8 α L8 l o v Therefore x b Z h c Z h det Re i 5 α l8 α L8 l 5 γ ρ Im α l8 α L8 l ρ ρ γ ρ o v 5 y c Z h a Z h det Re i 5 α l8 α L8 l ρ Im α l8 α L8 l γ ρ v ρ γ ρ o u b Z h 3 c Z h 4 det Re i 5 α l8 α L8 l 5 γ ρ Im α l8 α L8 l ρ γ ρ ρ o v 5 w c Z h 3 d Z h 4 det ρre i 5 α l8 α L8 l Im α l8 α L8 l γ ρ v γ ρ ρ o v v 5 Now we consider the scaling limit. We set x lε a Lε v εν where ν λ - 4. For the pseudo biharmonic process % Z t (, we consider σ in f % t;z t : or Z t, a(i Since we approximate the pseudo process by the scaling limit of % X n(, we define Q x e λ σ f Z σ def lim E ε* l e ε 4 λ σ f εx σ 6 (8.5) where x lε and a Lε are fixed and f is a bounded C -function. We shall evaluate

17 An approach to the Biharmonic pseudo process by a Random walk 7 E l e ε 4 λ σ f εx σ : E l e ε 4 λ σ f εx σ f ε H v l f ε H v l f a ε H L8 v l f a ε H L8 v l f ε x y u w f x ε y u w f a ε x y u w f a x ε y u w f ε f ε f a ε f a x ε f ε f ε f a ε f a ε f ε f ε f a ε f a u ε f f a x ε f f a f ε f ε f a ε f a ε w y f f a u ε f f a y w where we set s e ε 4λ We define two new functions C x e νx cos νx B S x e νx sin νx (8.6) It is immediate to check that those functions satisfy g 4ν 4 g 8λg Since α L8 e 7 8 i9 νε7 L8 9 α l8 e 7 8 i9 νx α L8 l e 7 8 i9 ν7 a x9 e 7 8 i9 νa we get γ C a 5 ρ S a B Re i α l8 α L8 l C x S x C a x S a x 5 Im α l8 α L8 l S x S a x B Finally, we obtain the following theorem:

18 8 Sadao Sato Theorem 8.. Let f be a bounded function in C, and set def w λ x Q x e λ σ f Z σ C x f f a S x C a x S a x C a S a S x S a x S a S a C a S a f C x f a S x C a x S a x C a S a S x S a x S a S a C a S a f C x f a S x C a x S a x S a S x S a x C a S a ν S a C a S a C x f f a S x C a x S a x S a S x S a x C a S a ν S a C a S a Then w λ x satisfies and 4 w 8 x 4 λw w λ f B w λ a f a w x λ f B w x λ a f a Remark 4. (i) As λ goes to zero, we obtain u x which satisfies def Q x f Z σ f f a f f a 4 a f f a x a x a a f f a x a 3 x 3 a x a 3 x 6 a x a u u f 5 u a f a B ud f 5 ud a f U a (ii) When a goes to infinity, for any fixed values f a and f a, we obtain Q x e λ σ f Z σ f 5 C x S x which naturally coincides with Theorem 6.. f ν S x Acknowledgement: The author would like to thank Professor K. Nishioka for his friendship and many valuable discussions.

19 An approach to the Biharmonic pseudo process by a Random walk 9 DEPARTMENT OF NATURAL SCIENCES TOKYO DENKI UNIVERSITY HATOYAMA-MACHI, SAITAMA HIKI, , JAPAN References [B] B.Benachour,B.Roynette and P.Vallois: Explicit solutions of some fourth order partial differential equations via iterated Brownian motion. Progress in Probability, vol.45,39-6(999) [F] T. Funaki: Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad.,Ser.A.55,76-79(979) [H] [K] K.J. Hochberg: A signed measure on path space related to Winer measure. Ann. Prob.,6, (978) Y.Yu.Krylov: Some properties of the distribution corresponding to the equation u t q8 q u x q.soviet Math.Dokl.,,76-763(96) [N] K. Nishioka: The first hitting time and place of a half-line by a biharmonic pseudo process. Japanese J. of Math. vol.3,no.,35-8 (997) [Na] T. Nakajima: Joint distribution of the first hitting time and first hitting place of a random walk. Kodai Math. J.,9-(998) [NS] T.Nakajima and S.Sato: On the joint distribution of the first hitting time and first hitting place to the space-time wedge domain of a biharmonic pseudo process. Tokyo J. Math.,vol.,No.,399-43(999)