ANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP

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1 ANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP ABSTRACT. If µ is a Gaussian measure on a Hibert space with mean a and covariance operator T, and r is a} fixed positive number, then the function ga, T = µ x < r satisfies the equation T ga, T = 1 a ga, T in a given sense. Moreover, its derivatives with respect to a and T can be expressed expicity. Introduction. Let µ be a Gaussian measure on a rea, separabe Hibert space H with mean vector a H and covariance operator T : H H. It is we-nown that a bounded, inear operator S : H H can be the covariance operator of a Gaussian measure on H if and ony if it is a sef-adjoint, positive operator with finite trace, which wi be caed a Gaussian covariance operator. Denote by L 1 H the Banach space of nucear operators on H, endowed with the nucear norm 1. Then the set Q of a Gaussian covariance operators on H is a cosed cone in L 1 H. Let now r > 0 be a fixed positive number. Put ga, T = µ x H : x < r } = µ U r 0 for a a, T H Q, where U r x denotes the open ba with radius r > 0 and center x H. It has been shown in [7] that g, T is Gâteaux differentiabe on suppµ 0,T for a fixed T Q Eventuay, it has been proved in case of Gaussian measures on arbitrary rea, separabe Banach spaces. Moreover, the derivative in the direction b H is ga, T b = im a ε 1 ga + εb, T ga, T ε 0 = λ 1 x, v µ 0,T dx b, v, 1 U r a where λ } are the eigenvaues of T, and v } is the corresponding orthonorma system of eigenvectors of T cf. [7]. Differentiabiity of the function g0, T with respect to T was investigated in [9]. Simiar questions were treated in [8] without proofs, but instead of the nucear norm the so-caed conica norm was considered, which in fact fais to be a norm, so usage of the notion of differentiabiity was not cear. 1

2 The aim of this paper is to examine differentiabiity properties of the function g, and to show that g satisfies the heat equation in a given sense. Preiminaries. By H we aways denote a separabe Hibert space H with norm and scaar product,. By U r x we denote the open ba in H with radius r > 0 and center x H. Let LH denote the Banach space of bounded, inear operators on H with the usua operator norm, and et L 1 H be the Banach space of nucear operators on H, endowed with the nucear norm 1. We sha appy the foowing resuts: LEMMA 1. Let T n, T be compact operators on H, and et T n T in the topoogy of LH. Let λ m T be an enumeration of the non-zero eigenvaues of T, each repeated according to its mutipicity. Then there exist enumerations λ m T n of the non-zero eigenvaues of T n, with repetitions according to mutipicity, such that im n λ m T n = λ m T for m 1, the imit being uniform in m cf. [4, XI.9.5]. LEMMA. Each continuous inear functiona on L 1 H can uniquey be represented in the form T = TrAT for T L 1 H, where A LH. Besides, 1 = A cf. [1, Chapter 11, Theorem 11]. LEMMA 3. The function T DetI + T of the Banach space L 1 H into R is Fréchet differentiabe on the set T L1 H : 1 σt } where σt denotes the spectrum of T, and its derivative at the point T L 1 H with 1 σt is the inear functiona S DetI + T Tr I + T 1 S cf. [4, XI.9.3]. LEMMA 4. Let a H. Then the function T I + T 1 a, a of the Banach space L 1 H into R is Fréchet differentiabe on the set T L1 H : 1 σt } and its derivative at the point T L 1 H with 1 σt is the inear functiona S I + T 1 SI + T 1 a, a. PROOF. For S, T L 1 H with 1 σt and for sma enough ε we have 1 σt + εs and ε 1 I + T + εs 1 a, a I + T 1 a, a = I + T + εs 1 SI + T 1 a, a. Consequenty, ε 1 I + T + εs 1 a, a I + T 1 a, a + I + T 1 SI + T 1 a, a = I + T 1 I + T + εs 1 SI + T 1 a, a = εi + T + εs 1 SI + T 1 SI + T 1 a, a I + T + εs 1 I + T 1 S 1 a ε 0,

3 when ε 0. Moreover, the inear mapping S I + T 1 SI + T 1 a, a of L 1 H into R is bounded, since I + T 1 SI + T 1 a, a I + T 1 S 1 a. Hence the assertion in view of Lemma. # and LEMMA 5. Let a H, T L 1 H and ϕ } be an arbitrary o.n.s. in H. Then T ϕ, ϕ T 1 a, ϕ a, ϕ T ϕ, ϕ a T 1. PROOF. The first statement is a specia case of [1, Chapter 11, Theorem 3]. For the proof of the second statement note that an operator T L 1 H has the poar decomposition T = j s j, u j v j, where s j } are the positive eigenvaues of T T 1/ with the property j s j = T 1 <, and u j }, v j } are o.n.s. in H. Thus we have a, ϕ a, ϕ T ϕ, ϕ s j a, ϕ a, ϕ ϕ, u j ϕ, v j j = s j a, ϕ ϕ, u j a, ϕ ϕ, v j j s j a, ϕ 1/ ϕ, u j 1/ a, ϕ 1/ ϕ, v j 1/ j a j s j = a T 1. Hence the assertion. # If S LH we denote by RS the range of S. of RS in H. By RS we denote the cosure Let Q be the cone of the sef-adjoint, positive operators in L 1 H. As is we-nown, every T Q has the spectra representation T = λ, v v, where λ } are the positive eigenvaues of T taing into consideration their mutipicity; it can be a finite or infinite sequence of positive numbers with λ < and v } is the corresponding orthonorma system of eigenvectors of T which is compete in RT. For a, T H Q we denote by µ the Gaussian measure on H with mean vector a H and covariance operator T Q, i.e. the Fourier transform of µ has the foowing form: ˆµ h = H e i x,h µ dx = exp i a, h 1 } T h, h 3

4 for h H. It is nown that suppµ 0,T = RT. For a, T H Q we denote by ν for a Bore subset B of R. ν B := µ x H : x B } the foowing measure on the rea ine: We sha use the foowing simpe facts: LEMMA 6. If a, T H Q such that T 0 and a suppµ 0,T = RT then the measure ν is absoutey continuous with respect to the Lebesgue measure and its characteristic function is ˆν t = 1 DetI itt exp it I itt 1 a, a } for t R. PROOF. Let us write x H in the form x = x 0 + x, v v where x 0 RT. Then x, v } are independent, normay distributed with respect to µ with mean a, v and variance λ, and x 0 = 0 µ -a.s. Thus using x = x 0 + x, v we have the foowing form for the characteristic function of ν : ˆν t = 1 it a, v } exp 1 itλ 1 itλ for t R. Invertabiity of the operator I itt LH and a suppµ 0,T = RT impy the form. The function ˆν is integrabe on R, so the measure ν is absoutey continuous. # Resuts. Let us now fix a positive number r > 0 and consider the rea-vaued function g : H Q [0, 1] defined by ga, T = µ U r 0 for a a, T H Q. In the first proposition we investigate differentiabiity of the function ga, for fixed a suppµ 0,T = RT. THEOREM 1. Let S, T Q such that T 0 and a suppµ 0,T. Denote the spectra representation of T by T = λ, v v. Suppose that RS RT. Then ga, T S = im T ε 1 ga, T + εs ga, T = TrA S, ε 0 where A LH is a sef-adjoint operator with the matrix representation A = A,, v v, 4

5 where A, = 1 [ ] itr itδ, ˆν t + it a, v a, v, it 1 itλ 1 itλ 1 itλ and δ, denotes the Kronecer symbo. PROOF. Using Lemma 6 the function ga, T can be expressed by the hep of inverse Fourier transformation: ga, T = µ x H : x < r } = ν x R : x < r } = = 1 itr Using Lemma 3 and 4 we concude that it 1 DetI itt exp it I itt 1 a, a }. 3 1 ga, T S = itr ˆν ta,s t, 4 T where A,S t = Tr I itt 1 S + it I itt 1 SI itt 1 a, a, if the integra on the right side of 4 converges uniformy in a neighbourhood of T. Consider now the decomposition A = A 1, where A1 = A1, v v and A = and A,, v v, where The inear operator A 1 + A A 1 = 1 itr ˆν t, 5 1 itλ A, = a, v a, v A 1 = sup it itr 1 itλ 1 itλ ˆν t. 6 is bounded, since for dimrt 3 we have A =1 1 + <, 7 4t λ 1/4 otherwise A 1 is a finite ran operator. Moreover we have TrA 1 S = A 1 Sv, v = 1 itr ˆν t = 1 itr ˆν ttr I itt 1 S, 5 Sv, v 1 itλ

6 since from 7 and Lemma 5 it foows that the integra converges uniformy: β n itr Sv, v β ˆν t 1 itλ 3 = t λ S 1/4 1 0, if, β or, β. =1 Consider now the operator A. A, For dimrt 5 we have the estimation t 5 = t λ 1/4 < a, v a, v. 8 Taing into account 8 and Lemma 5 and we have for arbitrary operator V L 1 H: TrA V = A, V v, v t 5 = t λ a V 1 <. 1/4 Thus the inear mapping V TrA V of the Banach space L 1H into R is bounded. From Lemma it foows that there exists a bounded, inear operator à LH such that TrA V = Trà V for a V L 1H. Appying this property for the operator V, =, v v L 1 H we have A, = A v, v = TrA V, = Trà V, = à v, v, thus the matrix representations of A Moreover, TrA S = 1 = 1 and à it itr ˆν t since the integra converges uniformy: β it ˆν itr t β coincide. Consequenty, A LH. a, v a, v Sv, v 1 itλ 1 itλ it itr ˆν t I itt 1 SI itt 1 a, a, n =1 =1 t 5 = t λ 1/4 a S 1 0, n a, v a, v Sv, v 1 itλ 1 itλ when, β or, β. In the end from 5, 6 and Lemma 5 we have the inequaity 1 β itr ˆν ta,s t it 1 β β 3 = t λ + t a 5 1/4 =1 1 + S 1. 4t λ 1/4 6

7 Hence taing into account Lemma 1 the integra in 4 converges uniformy in a neighbourhood of T. Therefore the Theorem is proved. # REMARKS 1. In view of Lemma the functiona S TrA S is inear and bounded on the space L 1 H. Thus T ga, T can be extended to an eement of L 1H the topoogica dua of L 1 H.. If a suppµ 0,T then we can consider the decomposition a = a 0 + a 1, where a 1 suppµ 0,T = RT and a 0 RT. Using µ = µ a1,t ε a0 where ε x denotes the Dirac measure, concentrated in x H and that suppµ a1,t = suppµ 0,T = RT we have ga, T = µ a1,t x H : x + a0 < r } = µ a1,t x RT : x < r a 0 } = µ a1,t x H : x < r a 0 }. Consequenty, if a 0 = a a, v r, then ga, T = 0, thus in that case T ga, T = 0. Moreover, if a } 0 < r then ga, T = µ a1,t x H : x < r, where r = r a 0. Thus in that case T ga, T S exists for S L 1H with RS RT and can be cacuated as in the case of a suppµ 0,T, using a 1 instead of a and r instead of r, respectivey. In the next proposition we give a new formua for the derivative of the function ga, for fixed T Q. THEOREM. Let T Q such that T 0 and a, b suppµ 0,T. representation of T by T = λ, v v. Then ga, T b = im a ε 1 ga + εb, T ga, T = B, b, ε 0 Denote the spectra where B = B v with B = 1 itr ˆν t a, v 1 itλ. PROOF. Using the expression 3 for the function g we have a ga, T b = 1 itr ˆν t I itt 1 a, b, 9 if the integra in 9 converges uniformy in a neighbourhood of a. have the estimation B 3 = t λ 1/4 a, v, 7 For dimrt 3 we

8 hence we have for arbitrary vector u H: B, u = B u, v 3 =1 1 + u, v 4t λ a, v 1/4 3 =1 1 + a u <. 4t λ 1/4 Thus the inear mapping u B, u of the Hibert space H into R is bounded. Consequenty, B H. Moreover, B, b = B b, v = 1 = 1 itr ˆν t a, v b, v 1 itλ itr ˆν t I itt 1 a, b, since the integra converges uniformy: β it n a, v ˆν itr b, v t 1 itλ =1 β t 3 =1 1 + a b 0, 4t λ 1/4 when, β or, β. the integra in 9, too. # From this inequaity we get the uniform convergence of THEOREM 3. Let T Q such that T 0 and a, b, c suppµ 0,T. Denote the spectra representation of T by T = λ, v v. Then ga, T b, c = im a ε 1 ε 0 a ga + εc, T b a ga, T b = C b, c, where 1 C = A LH is the sef-adjoint operator, described in Theorem 1. PROOF. Using 9 and Lemma 4 we concude that a ga, T b, c = 1 itr ˆν ta a,b,c,t t, 10 where A a,b,c,t = I itt 1 b, c + it I itt 1 a, b I itt 1 a, c, if the integra in 10 converges uniformy. Consider now the operator S =, b c L 1 H. Using 4 and Theorem 1 we have A b, c = A, b, v c, v = TrA S = 1 ga, T S = T 8 itr ˆν ta,s t.

9 By the definition of S we have Tr I itt 1 S = I itt 1 b, c and I itt 1 SI itt 1 a, a = I itt 1 a, b I itt 1 a, c. Consequenty, A,S t = A a,b,c,t t for a t R. From the proof of Theorem 1 it foows that the integra in 10 converges uniformy, thus a ga, T b, c = TrA S = A b, c. Hence the assertion. # REMARK. Using the same ideas as in the remars after Theorem 1 one can show that if a suppµ 0,T but b, c suppµ 0,T then aga, T b = a ga, T b, c = 0 in case of a 0 r, and aga, T b and a ga, T b, c can be cacuated using a 1 instead of a and r instead of r in case of a 0 < r. COROLLARY 1. The function g satisfies the heat equation T ga, T = 1 a ga, T in the foowing sense: If T Q such that T 0 and a suppµ 0,T and we consider the operator A LH, associated to T ga, T in Theorem 1 and the operator C LH, associated to a ga, T in Theorem 3 in a natura way then A = 1 C. Equivaenty, if b, c suppµ 0,T and we consider the operator S =, b c L 1 H then T ga, T S = 1 ga, T b, c. a REMARK. From the remars after Theorem 1 and 3 it foows that the same assertion is true in case of a suppµ 0,T, too. THEOREM 4. Let T Q such that T 0 and a suppµ 0,T. Denote the spectra representation of T by T = λ, v v. Then the quantities A,, A 1, A and B defined in Theorem 1, 5, 6 and Theorem, respectivey admit the foowing properties: i A, = 1 λ λ U r 0 x, v x, v µ 0,T dx 1 λ µ 0,T Ur a δ,. ii A, = A 1 δ, + A,. iii A 1 = 1 λ U r 0 e x r λ µ dx. iv If λ λ then A, = a,v a,v λ λ U r 0 9 e x r λ e x r λ µ dx.

10 v If λ = λ then A, = a,v a,v λ vi B = a,v λ U r 0 e x r λ µ dx. PROOF. 1. From Theorem 3 it foows that From 1 we have U r 0 x r e x r λ µ dx. A, = 1 C v, v = 1 im ε 0 ε 1 B a+εv,t, v B, v B = λ 1 = 1 im ε 0 ε 1 B a+εv,t B. U r a for a a suppµ 0,T. Thus we obtain A, = 1 1 im x, v µ 0,T dx λ ε 0 ε U r a = 1 1 im x, v µ 0,T dx λ ε 0 ε U r a x, v µ 0,T dx U r a+εv U r a x, v µ 0,T dx x + εv, v µ εv,t dx. 11 Taing into account v T H, the measure µ εv,t is absoutey continuous with respect to the measure µ 0,T, and the Radon-Niodym derivative is dµ εv,t x = exp ε µ 0,T T 1 v ε T 1 v, } T 1 x. 1 Cameron-Martin s formua; cf. [6, Chapter 3, Theorem 3.1] Combining 11 and 1, and appying the dominated convergence theorem we obtain A, = 1 x, v T 1 v, T 1 x µ 0,T dx δ, µ 0,T Ur a. λ λ Therefore i is proved. U r a. The proposition ii is trivia. 3. For the proof of iii, iv, v and vi we note that ˆν = R eitx ν dx, thus using the Fubini theorem we have A 1 = 1 e itx e itx r ν dx, R 1 itλ A, = a, v a, v it e itx e itx r R 1 itλ 1 itλ ν dx, 10

11 and Using formuas we can compute that 0 B = a, v cos at 1 + t = e a, R e itx e itx r 1 itλ ν dx. 0 t sin at 1 + t = sgna e a } e itx e itx r = x r λ exp λ if 0 < x < r 1 itλ 0 if x r, and it e itx e itx r 1 itλ = x r r x exp if 0 < x < r λ 0 if x r, λ } and for λ λ it e itx e itx r } } 1 itλ 1 itλ = x r λ λ exp x r λ exp λ if 0 < x < r 0 if x r. Hence iii, iv, v and vi are proved. # REMARKS 1. The method of this paper can be used to examine higher derivatives of the function g, too.. Unfortunatey this method wors ony in case of Hibert space. In [] it has been shown that if E is a rea, separabe Banach space and PE denotes the space of probabiity measures on E equipped with the topoogy of wea convergence then the mapping T µ 0,T of the cone of Q Gaussian covariance operator equipped with the nucear norm into PE is continuous if E is of type and has the approximation property. Moreover this mapping is not continuous if E is not of type. Thus perhaps Theorem 1 can be extended for exampe to the spaces p with p <. 3. It is aso possibe to consider simiar questions for stabe measures. Let 0 < <. It is nown that a probabiity measure µ PE is symmetric and -stabe if and ony if there exists a finite, symmetric measure ϑ on S 1 = x E : x = 1 } such that the Fourier transform of µ has the form } ˆµa = exp x, a ϑdx 13 S 1 for a E. Moreover the measure ϑ spectra measure of µ is unique cf. [10, Theorem 6.4.4]. Let us denote by M S 1 the set of a spectra measures of symmetric, -stabe 11

12 measures on E, i.e. the set of finite, symmetric measures on S 1 such that the right side of 13 is the characteristic function of some probabiity measure µ PE equipped with the topoogy of wea convergence. For ϑ M S 1 we denote by µ ϑ the symmetric, -stabe measure on E with spectra measure ϑ. Then the mapping ϑ µ ϑ of M S 1 into PE is continuous if and ony if E has stabe type cf. [10, Proposition 7.5.6]. 4. In [5] it has been examined the function 1 a im ε 0 ε µ } x H : r ε < x < r + ε, of H into R. The above imit is caed Gaussian spherica mean, and can be aso expressed as the vaue of the density function of the norm with respect to µ at the point r. It was proved that the above function is differentiabe on H. It is aso possibe to examine the differentiabiity of the Gaussian spherica mean with respect to the covariance operator T. ACKNOWLEDGEMENT. I am gratefu to Professor Herbert Heyer for supporting my wor. REFERENCES 1. M. S. Birman and M. Z. Soomja, Spectra theory of sef-adjoint operators in Hibert space, D. Reide Pubishing Company, Dordrecht, Boston, Lancester and Toyo, S. Chevet, Compacité dans espace des probabiités de Radon gaussiennes sur un Banach, C.R. Acad. Sc. Paris , J. Dieudonné, Foundations of modern anaysis, Academic Press, New Yor and London, N. Dunford and J.T. Schwartz, Linear Operators. II, Interscience Pubishers, New Yor and London, A. Herte, Gaussian pane and spherica means in separabe Hibert spaces, Measure Theory Proc. Internat. Conf., Oberwofach, 1981, Lecture Notes in Math., Vo. 945, Springer-Verag, Berin and New Yor, 198, pp H-H. Kuo, Gaussian measures in Banach spaces, Springer-Verag, Berin and New Yor, W. Linde, Gaussian measure of transated bas in a Banach space, to appear in the Theory of Prob. and its App. 8. E. I. Ostrovsii, Gaussian semigroups in a Banach space, Theory of Prob. and its App , Gy. Pap, Dependence of Gaussian measure on covariance in Hibert space, Probabiity Theory on Vector Spaces III Proc. Internat. Conf., Lubin, 1983, Lecture Notes in 1

13 Math., Vo. 1080, Springer-Verag, Berin and New Yor, 1984, pp W. Linde, Infinitey divisibe and stabe measures on Banach spaces, Teuber-Verag, Leipzig, Department of Mathematics, University of Debrecen, H-4010 Debrecen, Pf.1, Hungary 13