Quadratic Wiener Functionals and Dynamics on Grassmannians

Keisuke Hara ^{∗} Dept. of Mathematical Science Ritsumeikan University

Abstract

We will discuss a relation between quadratic Wiener functionals and dynamics on Grassmannians and show some examples in our framework. Firstly, we will show that Lévy’s stochastic area formulae for the Brownian motion and Gaussian processes can be described in terms of the Plücker coordinates, which satisfy linear differential equations associated with the Jacobi fields along classical paths on Grassmannians. Secondly, we will calculate the quadratic functionals of Gaussian processes introduced by S. Kotani that are related with inverse scattering problems and solitons. We will also discuss about the standard expressions for their processes.

This is a joint work with N.Ikeda.

1 Framework

Let W_{0} = {w ∈ C([0, 1] → R^{n}; w(0) = 0} and let (W_{0}, P) be the n dimensional Wiener space and H the Cameron-Martin subspace of W = W_{0}. We will often consider the complexification of H and use the same symbol H. Then we can decompose L^{2}(W_{0}, P) into the direct sum of mutually orthogonal subspaces known as Wiener-Itô homogeneous chaos:

L^{2}(W_{0}, P) = ∑_{n=0}^{∞} ⨁ C_{n}.

Let HS(H) be the space of the symmetric Hilbert-Schmidt operators. We will consider quadratic functionals F [w] ∈ C_{2} in this talk. It is well known that for every F [w] ∈ C_{2} there exists an unique operator B ∈ HS(H) such that

∗Email: hara@theory.cs.ritsumei.ac.jp, supported by Grant-in-Aid for Encouragement of Young Scientists and ACCESS Co.Ltd.

F [w] = ∑_{j=1}^{2} {∫_{0}^{1} ∫_{0}^{t} F^{ij}(t, s)dw_{s}^{j}dw_{t}^{i} + ∫_{0}^{1} ∫_{0}^{s} F^{ij}(t, s)dw_{t}^{i}dw_{s}^{j}} .

and {F^{ij}(t, s)} is the kernel of B:

\frac{d(B[h]^{i})}{dt}(t) = ∑_{j=1}^{2} ∫_{0}^{1} F^{ij}(t, s)ḣ^{j}(s)ds,

for h =^{t} (h^{1}, h^{2}) and ḣ^{j} = \frac{d}{dt}h^{j}(t).

We can regard the operator B as a characteristic of the Quadratic functional F, and we would often write ⟨Bw, w⟩ for F (w). We would sometimes consider some conditional functionals, for example, functionals for the pinned Brownian motion. We can consider the functional as the corresponding action integral

F [ϕ] = E[ϕ] = ∫_{0}^{t} L(ϕ(s), ϕ′(s))ds

for ϕ ∈ H. We can naturally consider its critical path (the classical path, or the geodesics) ϕ_{cr}, which satisfies the Euler-Lagrange equation and consider the Jacobi Field and its Jacobi equation along the critical path ϕ_{cr}(s). We know the close relation between the stochastic oscillatory integrals and the Jacobi fields by Van Vleck-Pauli formula ([41], [29], [16], [17]). We assume here the space of the solutions for the Jacobi equation or the Euler-Lagrange equation has the base {ψ_{j}}_{1,...,N}. According to M.Sato ([37], [38], [39], [40]), we identify the equation itself and the space of solutions, which expressed by the m-frame and the Plücker coordinates made from the basis. In our case, it is realized by the Wronskians:

ξ_{l_{0}l_{1}...l_{N−1}} = det ( \begin{matrix}ψ_{1}^{(l_{0})}(s) & ψ_{2}^{(l_{0})}(s) & ⋯ & ψ_{N}^{(l_{0})}(s) \\ ψ_{1}^{(l_{1})}(s) & ψ_{2}^{(l_{1})}(s) & ⋯ & ψ_{N}^{(l_{1})}(s) \\ ⋮ & ⋮ & & ⋮ \\ ψ_{1}^{(l_{N−1})}(s) & ψ_{2}^{(l_{N−1})}(s) & ⋯ & ψ_{N}^{(l_{N−1})}(s) \end{matrix} )

and we reach the linear differential equation for Plücker coordinates, that is, the dynamics on Grassmannians.

We will see that the Plücker coordinates and the dynamics are naturally related with our stochastic problems through the Van Vleck determinant of the Jacobi equation Firstly, we will show our framework in the simplest example, that is, the stochastic area for the two dimensional Brownian motion under some conditions.

2 Lévy’s stochastic area formula for Gaussian processes

Next, we will apply our framework to the Lévy’s area formula for Gaussian processes, which was calculated originally by Ikeda-Kusuoka-Manabe ([14]).

We define the quadratic Wiener functional F_{N}(w) on the two dimensional Wiener space (W_{0}, P) by the following:

F_{N}(w) = \frac{(−1)^{N}}{2} ∫_{0}^{1}\{X_{N}^{1}(t)dX_{N}^{2}(t) − X_{N}^{2}(t)dX_{N}^{1}(t)\},

where the Gaussian process X_{N}(t) = (X_{N}^{1}(t), X_{N}^{2}(t)) is defined by the following:

X_{0}(t, w) = w(t),
X_{N}(t, w) = I[X_{N−1}(·, w)](t), (N ≥ 1)

where

I[f](t) = ∫_{0}^{t} f(s)ds

for every f(·) ∈ W_{0}. We consider the following oscillatory integrals on the Wiener space:

C_{1}(z) = ∫_{W_{0}} e^{zF_{N} (w)}P (dw) or C_{2}(z) = ∫_{W_{0}} e^{zF_{N} (w)}δ_{x}(η(w))P (dw),

where the second integral is conditioned by a delta functional in some sense. In this case, our Jacobi equation and Euler-Lagrange equation is written in the same form:

\{(\frac{d}{ds})^{2(N+1)} − zJ (\frac{d}{ds})\} ϕ(t) = 0,  (2.1)

where J is

J = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

We can solve this equation explicitly and get the structure of Plücker coordinate and its differential equation of Grassmannians. By Van Vleck - Pauli formula, the quantities C_{1}(z), C_{2}(z) is described just in terms of the Plücker coordinate. Then the simple dynamics on Grassmannian control the behavior of C_{1}(z), C_{2}(z). Another aspect of this problem is the relation of the eigenvalue problem. By the basic formula for quadratic Wiener functionals,

∫_{W_{0}} e^{zF [w]}P (dw) = {det_{2}(1 − 2zB)}^{−1/2}  (2.2)

= {∏_{j=1}^{∞}(1 − 2zλ_{j})e^{2zλ_{j}} }^{−1/2} ,  (2.3)

for every z ∈ C such that 2|ℜz|σ(B) < 1, where σ(B) is the spectral radius of the operator B and det_{2} is the regularized determinant. Then, it is enough to solve the eigenvalue problem of the operator B, which has the same form of the Jacobi equation above. We can explicitly solve the equation again and we can write the eigen functions by the linear combination of the solutions. Then their boundary condition can be written by the finite dimensional determinant det D(z) = 0. Because the eigenvalues are zeros of det D(z), the infinite product and det D(z) have the same zeros. Therefore, if we can get the estimate of the growth of det D(z) for large |z|, we can identify the two entire functions by Hadamard theory. We remark that the essence of this reduction to the finite dimensional determinant is the following decomposition of B by Ikeda-Kusuoka-Manabe ([14], [15]):

B = B_{0} + B_{1},

where B_{0} is a finite range operator and B_{1} is a Volterra type one. By the decomposition above, we have:

det_{2}(I − 2zB) = det(I − 2zB_{1}(I − 2zB_{0})^{−1}).

3 Inverse scattering problem and Gaussian processes

S. Kotani recently showed a relation between quadratic functionals of Gaussian processes and inverse scattering problem ([25], [26]). We will calculate the so called n-soliton case in our framework and show the relation with Jacobi equations and Grassmannians. By Marchenko’s result, we can express the reflectionless potential q(x) by the spectre measure Σ = {(σ_{+}, σ_{−}) where σ_{±} is an non-negative measure on (−∞, 0] such that

∫_{(−∞,0]} e^{x\sqrt{−ξ}}σ_{±}(dξ) < ∞ (x > 0)

in the following form

q(x) = −2\frac{d^{2}}{dx^{2}} log det(I + G_{x}), G_{x} = G|_{L^{2}([0,x])}.

Here G is the operator whose kernel G(x, y) is given by

G(x, y) = \frac{1}{4} ∫_{−∞}^{0} \frac{1}{\sqrt{−ξ}} (e^{\sqrt{−ξ}(x+y)} − e^{\sqrt{−ξ}|x−y|}) σ_{+}(dξ)  (3.1)
+ \frac{1}{4} ∫_{−∞}^{0} \frac{1}{\sqrt{−ξ}} (e^{−\sqrt{−ξ}|x−y|} − e^{−\sqrt{−ξ}(x+y)}) σ_{−}(dξ). (3.2)

Kotani’s idea is that this expression can be rewritten by the following expectation:

q(x) = 4\frac{d^{2}}{dx^{2}} log E [exp \{−\frac{1}{2} ∫_{0}^{x} X(y)^{2}dy\}]

by using the Gaussian process {X(x)} whose covariance is G(x, y) and mean is 0. (Note that G is positive definite.) Though Kotani theory covers the general solutions of KdV equation, we restrict our calculations on so-called ”n-solitons”, whose kernel is given by

ρ(t, s) = ∑_{j=1}^{n} \frac{c_{j}^{2}}{2λ_{j}} (e^{λ_{j}(t+s)} − e^{λ_{j}|t−s|}) ,

where 0 < λ_{1} < … < λ_{n} and c_{j} > 0(j = 0, 1, …, n) are constant. Then, our Gaussian process has the following expression:

X_{t}(w) = ∑_{j=1}^{n} c_{j}e^{λ_{j}t} ∫_{0}^{t} e^{−λ_{j}s}dw^{j}(s).

(In general case, the finding such a expression is another interesting problem. see, [9], [12], [24], and [33]. We will show some simple calculations for our processes about this problem in this talk.) Now we have the quadratic Wiener functional F_{t}[w] = ∫_{0}^{t} |X_{s}(w)|^{2}ds and we also have the corresponding Jacobi equation or the decomposition of the operator B and we can apply our framework.

References

[1] P. Biane, J. Pitman, and M. Yor. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. To appear in Bull. AMS, 2001.

[2] P. Biane and M. Yor. Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist., 23:359–377, 1987.

[3] R. P. Boas. Entire functions. Academic Press, 1954.

[4] R. H. Cameron and W. T. Martin. The Wiener measure of Hilbert neighborhoods in the space of real continuous functions. Jour.Math.Phys., 23:195–209, 1944.

[5] R. H. Cameron and W. T. Martin. Evaluations of various Wiener integrals by use of certain Sturm-Liouville differential equations. Bull.Amer.Math.Soc., 51:73–90, 1945.

[6] R. H. Cameron and W. T. Martin. Transformations of Wiener integrals under a general class of linear transformation. Bull.Amer.Math.Soc., 51:73–90, 1945.

[7] T. Chan. Indefinite quadratic functions of Gaussian processes as least action paths. Ann.Inst.H.Poincaré, Probab.Statist., 27(2):239– 271, 1991.

[8] C. L. Dolph and M. A. Woodbury. On the relation between Green’s functions and continuity of certain stochastic processes and its applications to unbiased linear prediction. Trans.Amer.Math.Soc., 72:519–550, 1952.

[9] J. L. Doob. Stochastic Processes. John Wiley and Sons, 1953.

[10] P. Erdös and M. Kac. On certain limit theorems of the theory of probability. Bull.Amer. Math.Soc., 52:292–302, 1946.

[11] W. Fulton. Young tableaux. Cambridge Univ. Press, 1997.

[12] T. Hida. Canonical representation of Gaussian processes and their applications. Mem. College of Sci. Univ. of Kyoto, 33:109–155, 1960.

[13] R. Hirota and J. Satsuma. A variety of non-linear network equations generated from the Bäcklund transformations of the Toda lattice. Prog. Theor. Phys. Suppl., 59:64–100, 1976.

[14] N. Ikeda, S. Kusuoka, and S. Manabe. Lévy’s stochastic area formula for Gaussian processes. Comm. Pure and Applied Math., XLVII:329– 360, 1994.

[15] N. Ikeda, S. Kusuoka, and S. Manabe. Lévy’s stochastic area formula and related problems. Proc.Pure.Math.AMS, 57:281–305, 1995.

[16] N. Ikeda and S. Manabe. Asymptotic formulae for stochastic oscillatory integrals. In K. D. Elworthy and N. Ikeda, editors, Proceeding of the Taniguchi International Symposium on Asymptotic Problems in Probability Theory, number 284, Sanda and Kyoto, Pitman Research Notes in Math. Series, 1993. Longman, Essex.

[17] N. Ikeda and S. Manabe. Van Vleck-Pauli formula for Wiener integrals and Jacobi fields. In Itô’s Stochastic calculus and Probability theory, pages 141–156. Springer, 1996.

[18] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland / Kodansha, 2nd edition, 1989.

[19] K. Itô and M. Nisio. On the convergence of sums of independent Banach space valued random variables. Osaka J. Math., 5:35–48, 1968.

[20] M. Kac. On distributions of certain Wiener functionals. Trans. Amer. Math. Soc, 65:1 – 13, 1949.

[21] M. Kac. On some connections between probability theory and differential and integral equations. In Proc. 2nd Berkeley Symp. on Math. Statist. and Probab., pages 189–215. Univ. California Press, 1951.

[22] M. Kac. Integration in Function Spaces and Some of Its Applications. Academia Nazionale Dei Lincei Scuola Normale Superiore, Lezioni Fermiane, Pisa, 1980.

[23] M. Kac. Enigmas of Chance, an autobiography. Harper and Row, 1985.

[24] K. Karhunen. Über der Strucktur stationären zafalliger Funktionen. Ark. Mat., 4:141 –160, 1950.

[25] S. Kotani. Probabilistic approach for reflectionless potentials. In Symposium on Random Matrix and related topics. Tsukuba Univ., 2000. (in Japanese).

[26] Kotani, S.: Probabilistic approach for reflectionless potentials (in Japanese), Invited talk at Annual meeting of Math. Soc. Japan (2000)

[27] P. Lévy. Wiener’s random function and other Laplacian random functions. In Proc. 2nd Berkeley Symp. on Math. Statist. and Probab., pages 171–186. Univ. California Press, 1951.

[28] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. In K. Itô, editor, Proc. Internat. Sympos. SDE, pages 195–263. Kinokuniya (Tokyo) and Wiley (New York), 1976.

[29] Malliavin, P: Analyticité réelle des lois conditionnelles de fonctionnelles additives, C.R.Acad.Sci.Paris Sér. I Math. 302, 73–78 (1986)

[30] H. Matsumoto and S. Taniguchi. Wiener functionals of second order and their Lévy measures. preprint, 2001.

[31] T. Miwa, M. Jinbo, and E. Date. Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, 2000. (Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo, in 1993, and translated by M.Reid).

[32] C. Morette. On the definition and approximation of Feynman’s path integral. Phys.Rev., 81:848–852, 1951.

[33] Y. Okabe. On a stochastic differential equation for a stationary Gaussian process with T-positivity and the fluctuation-dissipation theorem. J. Fac. Sci. Univ. Tokyo, 28:169–213, 1981.

[34] R. E. C. Paley and N. Wiener. Fourier transforms in the complex domain. Colloq. Publ. Amer. Math. Soc., 19, 1934.

[35] J. Pitman and M. Yor. Infinitely divisible laws associated with hyperbolic functions. Prépublication du Laboratoire de Probabilités et Modéles Aléatoires No.616, 2000.

[36] K. Sato. Lévy Processes and Infinitely Divisible Distributions, volume 68. Cambridge University Press, Cambridge studies in advanced mathematics edition, 1999.

[37] M. Sato. Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. In Random Systems and Dynamical Systems, volume 439, pages 30–46. R. I. M. S., Kyoto Univ., 1981. (R. I. M. S. Kokyuroku).

[38] M. Sato and Y. Sato. Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. In Lect. Notes in Num. Appl. Anal., volume 5, pages 259–271. 1982.

[39] M. Sato. The KP hierarchy and infinite Grassmann manifolds. In Proc. Symp. in Pure Math. (AMS), volume 49, pages 51–66, 1989.

[40] M. Sato. Lecture taken by N. Umeda. R. I. M. S., Univ. of Kyoto, 1989. (in Japanese, R. I. M. S. Lecture note series No.5).

[41] J. H. Van Vleck. The correspondence principle in the statistical interpretation of quantum mechanics. Proc.Nat.Acad.Sci.U.S.A., volume 14, pages 178–188, 1928.

[42] S. Watanabe. Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab., 15:1–39, 1987.

[43] D. V. Widder. Laplace transform. Princeton University Press, 1941.