SOOCHOW JOURNAL OF MATHEMATICS Volume No pp Octob er ANALYSIS OF WHITE NOISE FUNCTIONALS BY HUIHSIUNG KUO Dedicated to the memory of Professor TsingHoua Teng Intro duction It has b een almost two decades since Professor T Hida intro duced the the ory of white noise in This theory has b een develop ed rather extensively and is now regarded as an innite dimensional calculus The b est source of information on this topic is the b o ok by T Hida H H Kuo J Pottho and L Streit White Noise An innite dimensional calculus Kluwer Academic Publishers In this set of lecture notes I have tried to give an easy intro duction of the white noise calculus to graduate students while at the same time provide a quick understanding of the theory for p eople working on other elds Basic ideas are explained and some elementary techniques are carried out in detail However most of the theorems are stated without pro ofs The sources of the pro ofs can b e easily identied from the content and the list of references In some sense this set of lecture notes is a continuation of my lecture notes given at So o chow University in see H H Kuo Lectures on white noise analysis Soochow J Math I have stated recent results on HitsudaSkorokho d integral sto chastic dierential equations and Fourier transform Ihave also discussed p ositive generalized functions and white noise kernel op erators Received March AMS Sub ject Classication F J G HUIHSIUNG KUO This set of lecture notes is rather incomplete Many imp ortant applica tions of white noise calculus eg Feynman integrals Dirichlet forms and quantum eld theory are not even mentioned They can be found in the HKPS book Other imp ortant applications are innite dimensional rotation groups and the variational calculus as Professor Hida describ ed in his b eauti ful lecture on June at Cheng Kung University We hop e that Professor Hida will complete the white noise building so on as mentioned in his lecture It is a great pleasure to give my deep est appreciation to Professor Y J Lee for arranging my visit to Cheng Kung University I would like to express my gratitude for the hospitality of the Mathematics Department of Cheng Kung University Iamvery grateful for the nancial supp ort from the National Science Council of Taiwan White noise space We will construct the white noise space in two dierent ways namely by using the Minlos theorem for nuclear spaces and by applying the Gross theorem for abstract Wiener spaces A Nuclear space Let S IR be the space consisting of all realvalued rapidly decreasing functions f on IR ie n k lim jx f xj n k jxj x x For instance C functions with compact supp ort e e are all functions in S IR For any n k dene a norm kk on S IR by nk n k kf k sup jx f xj nk xIR Then S IR fk k n k g is a top ological space In fact it is a nk nuclear space Let S IR b e the dual space of S IR ANALYSIS OF WHITE NOISE FUNCTIONALS Theorem Minlos Let E be a nuclear space with dual space E A complexvalued function on E is the characteristic functional of a prob ability measure on E ie Z ihxy i e d x y E y E if and only if it satises the fol lowing conditions is positive denite ie for any z y E j n any n j j n X z z y y j k j k jk is continuous Remark The measure is uniquely determined by Observe that E Thus when condition is not assumed then is a nite measure Example Let b e a function on S IR given by exp j j SIR where jj is the L IRnorm Then it is easy to check that conditions and are satised Tocheck condition note that Z j xj dx j j IR Z Z j xj dx j xj dx jxj jxj Z sup j xj jx xj dx x jxj jxj Z dx k k supjy y j x jxj jy j k k k k This shows that is continuous Therefore by the Minlos theorem there exists a unique probability measure on S IR such that Z ihx i e dxexp j j SIR S IR HUIHSIUNG KUO Denition The measure is called the standard Gaussian measure on S IR The probability space S IR is called a white noise space B Abstract Wiener space k Let b e the standard Gaussian measure on IR Then k Z k ihxy i e d x exp jy j y IR k k IR An obvious extension of this fact to an innite dimensional real separable Hilb ert space H is Z ihxhi e dxexp jhj h H H However such a measure do es not exist To see this take an orthonormal basis fv n g for H Then wehave n Z ihxv i n e dxe H Note that hx v i converges to zero as n for every x H Hence by the n Leb esgue dominated convergence theorem we get the ridiculous conclusion e Theorem Gross Let H be a real separable Hilbert space with norm j j Suppose B is the completion of H with respect to a measurable norm on H Then there exists a unique probability measure on B such that Z ihxhi jhj h H e dxexp B Fact Let T b e an injective Hilb ertSchmidt op erator of H Then kxk jTxj is a measurable norm Example Consider the Hilb ert space H L IR with norm jj Let H n b e the Hermite p olynomial of degree n n dened by n x n x H x e D e n x Let e b e the Hermite function of order n n dened by n p x n e x H xe n n n ANALYSIS OF WHITE NOISE FUNCTIONALS Obviously fe n gSIR Moreover this family is an orthonormal basis n for L IR Let A b e the following op erator d A x dx It is a fact that Ae n e n For any p IR dene n n p jf j jA f j f L IR p Note that jf j can b e expressed in another way p X p n f e jf j n p n where is the inner pro duct of L IR p Weshow that the op erator A is a Hilb ertSchmidt op erator of L IR if and only if p as follows X p p jA e j kA k n HS n X p jn e j n n X p n n Let S IR b e the completion of L IR with resp ect to the norm jj p p p It Then L IR S IR is an abstract Wiener space if and only if p p follows from the Gross theorem that the supp ort of the white noise measure is in S IR if p In fact the converse is also true ie if is supp orted p in S IR then p p WienerIto decomp osition theorem For simplicitywe will use L to denote the space L S IR over with norm kk It follows from the denition of that for each S IR the random variable hi is dened everywhere on S IR and is normally HUIHSIUNG KUO distributed with mean and variance j j Supp ose f is in L IR Take a sequence f g S IR converging to f in L IR It is easy to see that n fh ig is a Cauchy sequence in L Dene n hfi limit in L of h i n The limit is indep endent of the sequence f g Moreover the random variable n hfi is normally distributed with mean and variance jf j Dene the renormalized Hermite p olynomial of degree n b y x p p h x H n n n Let ff f g b e orthonormal in L IR Let n n n Dene k k b n b n k b b by f the multiple Wiener integral of f k b n b n k b b h hf i h hf i I f f n n k n k k The mapping I can b e extended linearly and continuously to the symmetric n n n c complex L space L IR onIR It can b e checked easily that p n c n jf j f L IR kI f k n WienerIto decomp osition theorem Any in L can be decom posed uniquely as fol lows X n c I f f L IR n n n n Moreover X kk njf j n n Example The generating function of the renormalized Hermite p olyno mials is given by n X t tx t e h x n n n ANALYSIS OF WHITE NOISE FUNCTIONALS Let g L IR jg j Replace x with hgi in the ab ove equation to get n X t thg i t e h hgi n n n n X t n I g n n n Now supp ose z and f L IRf Let g fjf j and t z jf j to get n X z z jf j n z hf i I f e n n n z hf i Notation Dene the renormalization of e by z hf i z hf i z jf j e e Wehave the following equality n X z n z hf i I f e n n n Furthermore from this equalitywe obtain n X jz j z hf i n k e k n jf j n n X n jz j jf j n n jz j jf j e Hence for any z f L IR wehave jz j jf j z hf i k e k e Gelfand triples A One dimensional space The Schwartz distribution theory on the space IR concerns with the following Gelfand triple S IR L IR S IR HUIHSIUNG KUO B Innite dimensional space For an innite dimensional distribution theory we need to construct a similar Gelfand triple The space IR with the Leb esgue measure is replaced by the white noise space S IR with the standard Gaussian measure Thus we have already the middle space L The question is howtointro duce a small space and a big space L In order to construct the Gelfand triple we need to reconstruct the spaces S IR and S IR in the Gelfand triple by another way The same pro cedure can then be used to construct the Gelfand triple Let A be the following op erator d A x dx For each p IR dene p jf j jA f j p As wesawinx the norm jj is stronger than the L IRnorm jj if p p and is weaker than jj if p Thus for p let S IR ff L IR jf j g p p On the other hand for p let S IR completion of L IR jj p p It is easy to see that the dual space S IR of S IR can be identied as p p S IR when we apply the Riesz representation theorem to L IR Thus we p have the following continuous inclusions S IR L IR S IR p p p Example Consider the delta function in S IR It is not in L IR t Its expansion in terms of the Hermite functions is given by X e e t t n n n X e te n n n ANALYSIS OF WHITE NOISE FUNCTIONALS Therefore for p wehave p j j jA j t t p X p e tn e n n n X je tj n p n n Fact For the supnorm of the Hermite functions wehave ke k O n n Hence j j if p IR for any p ie p Thus S t p t p We can check the following fact which gives another construction of the spaces S IR and S IR from the space L IR and the