c Pleiades Publishing, Inc., 2008. ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 1, pp. 53–71.  c A.I. Ovseevich, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 1, pp. 59–73. Original Russian Text 

LARGE SYSTEMS

Kalman Filter and Quantization1 A. I. Ovseevich Institute for Problems in Mechanics, RAS, Moscow [email protected] Received October 15, 2007 Abstract—We give an interpretation of the problem of filtering of diffusion processes as a quantization problem. Based on this, we show that the classical Kalman–Bucy linear filter describes a flow of automorphisms of the Heisenberg algebra. We obtain new formulas for the unnormalized conditional density in the linear case, a new interpretation of the Mehler formula for the fundamental solution of the Schr¨ odinger operator for a harmonic oscillator, and formulas for a regularized determinant of a Sturm–Liouville operator. DOI: 10.1134/S0032946008010055

1. INTRODUCTION The paper is aimed at a conceptually simple and intuitive approach to the Kalman filter. We deal with both the linear and nonlinear filtering, but final results concern the linear case. The structure of these notes and our arguments are roughly as follows. First we state the problem of filtering of a partially observable diffusion process, i.e., that of optimal estimation of an unobserved component based on observations. To avoid geometric problems irrelevant to our main topic, we confine ourselves with diffusions in a Euclidean space Rn . By using the Girsanov theorem, the problem of nonlinear filtering can be reduced to that of computing a path integral, which is related to quantization of a classical Hamiltonian system. The corresponding Hamiltonian system arises via a deterministic optimal control problem, which might be interpreted as that of finding a “most probable” unobservable trajectory compatible with observations. In a most favorable situation, computation of a path integral can be performed by using classical trajectories. Generally, computation of the above-mentioned path integral is performed by using several basic tools from quantum mechanics. The first of them is the Hamiltonian (or Schr¨ odinger) approach, which is equivalent to the Zakai equation of nonlinear filtering. This is a stochastic partial differential equation for the conditional distribution density of an unobservable component of a diffusion process when the observable component is known. In fact, the Zakai equation does not exactly describe this conditional density but rather its projective class, i.e., the density considered up to multiplication by a constant. This, of course, complies with the spirit of quantum mechanics since quantum states are projective classes of vectors. The Zakai equation is a generalization of the Kolmogorov equation for the (unconditional) distribution density of a diffusion process and reduces to the latter when the observation is absent or irrelevant. The second quantum tool is the Heisenberg (or Lax) equation for observed values. It can be efficiently applied in the linear situation where we take linear combinations of coordinates and 1

Supported in part by the Russian Foundation for Basic Research, project nos. 05-08-50226 and 06-0100441. 53

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momenta as observed values to be found. These linear combinations form a (2n + 1)-dimensional Heisenberg Lie algebra, and the Heisenberg equation determines a flow of automorphisms of the algebra. Moreover, in the linear situation it suffices to search for a conditional density in the class of Gaussian functions. The Gaussian functions are intimately related to the Heisenberg algebra via the fundamental correspondence f −→ Ann f, where Ann f is the annihilator of a Gaussian function f in the Heisenberg algebra of first-order linear differential operators. The annihilators can be characterized as maximal abelian subalgebras of the Heisenberg algebra with some extra conditions. The above flow of automorphisms of the Heisenberg algebra maps the manifold of these annihilators to itself. As well as Gaussian functions, their annihilators are characterized by two parameters: a symmetric covariance matrix and a vector of mean values. When rewriting the arising flow on the manifold of annihilators in the language of these two parameters, we come to classical equations of the Kalman filter: the Riccati equation for the matrix and a linear equation for the vector. The third quantum tool is direct computation of the Feynman path integral. Again, this can be performed in the linear case, leading, for example, to a new interpretation of the Mehler explicit formula for the fundamental solution of the Schr¨ odinger equation for a harmonic oscillator. Since the Mehler formula is now commonly regarded as a cornerstone for such a fundamental fact as the Atiyah–Singer index theorem, any new insight into the formula might be valuable. Note in conclusion that the Heisenberg–Lax equations underlie the theory of integrable nonlinear equations; the idea of using Lie algebras in filtering is well known (see, e.g., [1]); and a somewhat similar approach to the Kalman filtering via Zakai equation is indicated in [2] without detail. The very existence of an efficient filtering in the linear case might be attributed, in the spirit of the Galois theory, to the solvability of a Lie algebra canonically associated to a partially observable process. A brief exposition of a part of results of the present paper is published in [3]. 2. EQUATIONS FOR THE CONDITIONAL DENSITY We study a diffusion process (in X = Rn to avoid geometric complications) governed by dx = a(x, t) dt + b(x, t) dw,

x(0) = x0 ,

(2.1)

where w ∈ R is a Wiener process. Also, we observe a process y (in Y = Rm , again just to avoid geometric complications) related to x via dy = c(x, t) dt + dW,

y(0) = 0,

(2.2)

where W is another Wiener process independent of w and x0 . For prerequisites, including stochastic differentials and integrals in the sense of Itˆo and/or Stratonovich, we refer the reader to [4, 5]. Here and in what follows, all identities between differentials are understood as equalities of the corresponding integrals, while integrals are sometimes understood in the Itˆ o sense, and sometimes, in the Stratonovich sense. We are interested in the evolution of the conditional density p(t, x) for the process x. This density is defined via 

u(x)p(t, x) dx = E(u(x(t)) | Ft ),

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where the σ-algebra Ft is generated by y(τ ), τ ∈ [0, t]. In other words, if we denote by y t the process y restricted to [0, t], then 







t



u(x)p(t, x) dx v(y ) = E u(x(t))v(y t ) ,

E

(2.4)

where v(y t ) is any measurable functional. Here we do not try to indicate optimal conditions which guarantee that the formal manipulations above and below are correct. However, it is easy to see that this is the case if (2.1) and (2.2) are understood as Itˆ o equations, and the coefficients a, b, and c in (2.1) and (2.2) are Lipschitz functions with respect to the spatial argument x. The same is true if the governing equations are regarded as Stratonovich ones, while the coefficients a, b, and c are Lipschitz, as well as the gradient

∂b . ∂x

In particular, there are no foundational difficulties in the linear Kalman situation. Now we invoke a fundamental fact about dependence of the probability measure P c on paths [0, T ]  t → y(t) on the drift c. The measure P = P 0 is the Wiener measure. The Girsanov formula says that 

1 dP c (y) = exp − dP (y) 2

T

|c| (s) ds +

T

2

0



c(s), dy(s) ,

(2.5)

0

where c(s) is an abbreviation for c(x(s), s). It does not matter whether we interpret the stochastic integral c, dy in Itˆo’s or Stratonovich’s sense because of the independence of the Wiener processes w and W . Denote the right-hand side of (2.5) by ϕ(T ) = ϕ(T, xT , y T ), where xT and y T are trajectories of our diffusion processes in the time interval [0, T ], and denote by P the measure of the process x. One immediately sees that the conditional density p(t, x) = p(t, x, y t ) is defined by the Bayes formula



u(x)p(t, x) dx =

u(x(t))ϕ(t, xt , y t ) dP(xt ) . ϕ(t, xt , y t ) dP(xt )

(2.6)

Indeed, denote by R(y t ) the right-hand side of (2.6). Then, in notation (2.4), we have 

E{u(x(t))v(y t )} =

u(x(t))ϕ(t, xt , y t ) dP(xt )v(y t ) dP (y) 

=

 t

t

t

t

t

R(y )ϕ(t, x , y )v(y ) dP (y) dP(x ) =

R(y t )v(y t ) dP c (y),

which exactly means that R(y t ) is the conditional expectation E(u(x(t)) | Ft ). First, we note that if we know the conditional density p(t, x) up to a scalar factor λ(t), then we can reconstruct p(t, x) because of the normalization p(t, x) dx = 1. Second, we note that one can replace P(dxt ) in both the numerator and denominator integrals in (2.6) with P(dxT ), where T is any time ≥ t. Indeed, the drift coefficient corresponding to times ≥ t does not affect x(t) and y t and therefore the conditional probability p(t, x). Therefore, it suffices to find the conditional density up to a constant; thus, it suffices to find ρ(t, x) defined by   u(x(t))ϕ(t, xt , y t ) dP(xT ),

u(x)ρ(t, x) dx = because p(t, x) = ρ(t, x)/λ(t), where λ(t) = is equal to 



1 u(x(t)) exp − 2

t



ϕ(t, xt , y t ) dP(xT ). In other words,

|c| (s) ds +

t

2

0

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(2.7)

u(x)ρ(t, x) dx



c(s), dy(s) dP(xt ).

0

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(2.8)

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In a sense, the preceding formula gives an explicit expression for ρ(t, x), which is however rather impractical. There are at least two ways to deal with (2.8): (1) rewrite it as a functional integral obtained by quantizing a classical Hamiltonian system and then try to compute it in classical terms; (2) find an evolution equation for ρ(t, x) and solve it. In fact, both approaches are intimately related. 2.1. Functional Integrals In our probabilistic setup, functional integrals are heuristic expressions for the measure dP(xT ) or the filtering measure (2.8) for the process x(t). The basic ingredient of the functional integral is a classical Hamiltonian system. The latter is defined via the following deterministic optimal control problem: x˙ = a(x, t) + b(x, t)u,    (2.9) 1 2 1 2 |u| + |c| dt − c, dy → min . I(u) = 2 2 At that point, limits of integration and boundary conditions are irrelevant. It is well known that optimal trajectories are governed by the Pontryagin Hamiltonian function H(x, p). Sometimes it is more convenient to deal with the corresponding differential form   1 2 1 H(x, p) dt = max p, a + bu − |u| dt − |c|2 dt + c, dy u 2 2   1 2 1 ∗ 2 |b p| + p, a − |c| (x) dt + c(x), dy. (2.10) = 2 2 One might regard equations (2.9) (respectively, (2.10)) as giving a Lagrangian (respectively, Hamiltonian) description of the same classical mechanical system. It is not always possible to find a Lagrangian function L(x, x) ˙ corresponding to the system, because the function p → H(x, p) is convex but not necessarily strictly convex. Still, if the matrix b is invertible, the Lagrangian function (form) exists:  

2 1 1

b(x)−1 (x˙ − a(x)) + |c|2 (x) dt − c(x), dy(t). (2.11) L(x, x, ˙ t) dt = 2 2 In the case of the absence or rather irrelevance of observations, when c ≡ 0, we obtain a similar functional  1 2 |u| dt (2.12) I0 (u) = 2 and the Hamiltonian   1 2 1 (2.13) H0 (x, p) = max p, a + bu − |u| = |b∗ p|2 + p, a. u 2 2 Now, the Feynman expression for the measure P is dP(xT ) = exp(−I0 (u)) dλ(u),

(2.14)

while the filtering measure (2.8) is 

1 exp − 2

T 0

|c|2 ds +

T



c, dy dP(xT ) = exp(−I(u)) dλ(u),

(2.15)

0

where we use notation (2.9), integration in I(u) is over [0, T ], and the nonexistent “Lebesgue measure” dλ(u) is defined as  du(t) . (2.16) dλ(u) = (2π)/2 t∈[0,T ] In view of (2.15), the minimization problem (2.9) might be interpreted as the search for the “most probable” trajectory of the unobservable process x(t) which is compatible with observations. PROBLEMS OF INFORMATION TRANSMISSION

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2.2. The Zakai Equation Now we proceed with the evolution equation for ρ(t, x). It is governed by a “quantum analog” H = Op H of the Hamiltonian (2.10) dρ = H∗ ρ dt, (2.17) where ∗ stands for the adjoint operator. Basically, the operator H = Op H is obtained from H(x, p) by substituting the operator

∂ for the momentum p: ∂x ∂

H = H x,

∂x

.

(2.18) ∂

do not commute. It is This substitution is, however, ambiguous, because the operators x and ∂x important to recall now that the ambiguity is also present in the governing equation (2.1) for the process x(t). These stochastic differential equations might be understood, say, in the Itˆ o or Stratonovich sense. Now we clarify the correspondence H → H in the Itˆ o and Stratonovich cases separately. In any case, 

H(x, p) dt =



1 1 ∗ 2 |b p| + a, p − |c|2 (x) dt + c(x), dy. 2 2



∂

2

is interpreted as ∂x

In the Stratonovich case the term b∗

   i

while in the Itˆ o case, as

j



∂ bji (x) ∂xj

bik bji (x)

i,j,k

2

,

(2.19)

∂2 . ∂xj ∂xk

(2.20)

All other terms are to be interpreted identically, so that in both cases 

a,

∂   ∂ = ai (x) . ∂x ∂x i i

Now, when the operator differential forms H dt and H∗ dt are formally defined, we still have to interpret the differential equation (2.17) as a stochastic differential equation: the presence of the term c, dyρ forces us to choose a correct scheme of stochastic integration. The following theorem of Zakai [6] gives a correct interpretation of (2.17). Theorem 1 [6]. The conditional distribution ρ satisfies the Stratonovich stochastic differential equation in partial derivatives 

dρ(t, x) =

H0∗



1 − |c(x)|2 ρ(t, x) dt + ρ(t, x)c(x, t), dy, 2

(2.21)

where H0 = Op H0 is the infinitesimal operator of the process x(t). In what follows, we denote the right-hand side of (2.21) by 

Zρ =

H0∗



1 − |c|2 ρ dt + ρc, dy, 2

in honor of M. Zakai. PROBLEMS OF INFORMATION TRANSMISSION

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(2.22)

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Proof. To prove (2.21), we compute the differential of the right-hand side of (2.7) 

d



u(x(t))ϕ(t, dP(xT ) =



du(x(t)) ϕ(t) dP(xT ) +

u(x(t)) dϕ(t) dP(xT ).

(2.23)

At that point, it does not matter whether we interpret the differential in Itˆ o’s or Stratonovich’s sense, but we accept Itˆo’s point of view here. The infinitesimal operator H0 is characterized by the property that Mt = u(x(t)) −

t

H0 u(x(t)) dt

(2.24)

0

o integral is a martingale for any test function x → u(x). The martingale Mt is equal to the Itˆ t  0

 ∂u (x(s)), b(x(s), s) dw(s) . ∂x

One can rewrite the right-hand side of (2.23) as 

H0 u(x(t)) dtϕ(t) dP(xT ) +

since the martingale part ϕ(t) dMt =

 ∂u



u(x(t)) dϕ(t) dP(xT ),

(2.25)



(x(t)), b(x(t), t) dw(t)

∂x does not contribute to the mathematical expectation. By using the fundamental identity dϕ(t) = ϕ(t)c(t), dy(t) for Itˆ o differentials, (2.25) can be rewritten as 

H0 u(x)ρ(t, x) dx dt +

Rn



u(x)ρ(t, x)c(x, t), dy dx,

(2.26)

Rn

which is apparently equivalent to the Itˆ o equation dρ(t, x) = H0∗ ρ(t, x) dt + ρ(t, x)c(x, t), dy.

(2.27)

It is clear that the Itˆ o equation (2.27) is equivalent to the Stratonovich equation (2.21).  Note that in the case c = 0 (absence of observations), the Zakai equation reduces to the direct Kolmogorov equation for the density p(t, x) of the random variable x(t): ∂ p(t, x) = H0∗ p(t, x). ∂t

(2.28)

The difference between the Zakai and Kolmogorov operators H and H0 is the potential term V , which is highly singular and involves a sample path of a white noise:  dy  1 . V (x, t) = − |c(x)|2 + c(x), 2 dt

One might, therefore, regard the Zakai equation (2.21) as an instance of the general Feynman–Kac formula. PROBLEMS OF INFORMATION TRANSMISSION

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3. KALMAN FILTER The problem resolved by Kalman and Bucy concerns filtering of Gaussian diffusion processes governed by linear equations dx = Ax dt + B dw, (3.1) dy = Cx dt + dW, where w and W are independent Wiener processes, and the matrices A, B, and C might depend on time. The initial values x(0), y(0) = 0 form a Gaussian random vector. The process y is observed, and one has to find the conditional distribution p(x) = p(t, x) of x(t) if the sample path y t of the process y up to time t is known. Since the combined process t → {x(t), y(t)} is Gaussian, the conditional distribution is Gaussian; i.e., 



 1 p(x) = C exp − R−1 (x − m), x − m , 2 

(3.2)

where C  = (det 2πR)−1/2 is a well-known (and so far irrelevant) constant, the conditional expectation m of x(t) is m = E(x(t) | y t ), and the conditional covariance matrix R is a positive definite symmetric matrix defined by 



Rξ, η = E x(t) − m, ξx(t) − m, η . All the involved quantities might depend on time t. In the linear case (3.1), the Hamiltonian function H(x, p) is quadratic with respect to all variables, which is clear from (2.10). This immediately implies that the Poisson brackets with H(x, p) determine an operator on the space of linear functions of x and p. Similarly, the operators H = Op H and H∗ from Section 2.2 have order 2, where the order is defined so that the operator Op x of mul∂

both have order 1. Again, this makes the operators tiplication by x and the operator Op p = ∂x H and H∗ very special, since they naturally operate on the Heisenberg algebra. 3.1. Heisenberg Algebra This is the Lie algebra h of differential operators h(ξ, η, λ), where ξ, η ∈ Rn and λ ∈ R, of the form 

h(ξ, η, λ) = Op(ξ, p + η, x + λ) = ξ,

∂  + η, x + λ, ∂x

(3.3)

which act on, say, smooth functions of x ∈ Rn . The commutator is given by 









h(ξ  , η  , λ ), h(ξ  , η  , λ ) = h 0, 0, ω (ξ  , η  ), (ξ  , η  ) ,

(3.4)

where ω is a standard symplectic form on the space R2n given by 



ω (ξ  , η  ), (ξ  , η  ) = ξ  , η   − ξ  , η  . The differential operators of order 2 

ξ,

∂   ∂  ξ , , ∂x ∂x

η  , xη  , x,



η  , x ξ  ,

∂  ∂x

(3.5)

normalize the Heisenberg algebra, meaning that if an operator A belongs to the list (3.5) and B ∈ h, then the commutator [A, B] ∈ h. This immediately implies that the Zakai operator Z = H∗ dt from Section 2.2 normalizes h. Assume that an operator A normalizes h. Define τ (A) : h → h by τ (A)B = [A, B] (the adjoint representation). One can easily give an explicit form of the operator τ (Z), where Z is the Zakai operator. PROBLEMS OF INFORMATION TRANSMISSION

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Namely, introduce the matrix



Z0 = and the linear functional

A BB ∗ ∗ C C −A∗



(3.6)

Zy : ξ −→ −ξ, C ∗ dy,

(3.7)

where the matrices A, B, and C and the differential dy of the observed process are taken from (3.1). Then (3.8) τ (Z)h(ξ, η, λ) = h(ξ  , η  , Zy (ξ)), where

(ξ  , η  )∗ = Z0 (ξ, η)∗ dt.

(3.9)

It is noteworthy that the matrix Z0 and therefore the transformation (ξ, η) −→ (ξ  , η  ) are independent of the observed sample path y. We note also that the matrix Z0 is infinitesimally symplectic, meaning that ω(Z0 u, v) + ω(u, Z0 v) = 0,

for any u, v ∈ R2n .

Moreover, this is also the matrix of a linear Hamiltonian system with the Hamiltonian function H(ξ, η) =

1 1 ∗ 2 |B η| + Aξ, η − |Cξ|2 . 2 2

3.2. Heisenberg and Gauss Another aspect of the Heisenberg algebra important for us is that it is intimately connected with Gaussian functions. Consider the annihilator Ann f = {h ∈ h : hf = 0} of a function f in the Heisenberg algebra. Lemma 1. If f is any generalized function on Rn , then the annihilator Ann f is a Lie subalgebra of h such that 1. The intersection Ann f ∩ C with the center C ⊂ h is 0; 2. Ann f is an abelian Lie algebra; 3. The dimension of Ann f is not greater than n. Proof. The fact that Ann f is a subalgebra is obvious. The center C ⊂ h consists of multiplications by constants λ ∈ R, and a multiplication by a constant λ cannot annihilate f unless λ = 0. This proves statement 1. The commutator [x, y] of x, y ∈ Ann f belongs to Ann f ∩ C = 0. This proves statement 2. Statement 3 follows from 1 and 2 because the canonical map h → h/C = R2n embeds Ann f in R2n as an isotropic subspace with respect to the canonical symplectic form ω (cf. [7]).  Now consider f = p, where p is a Gaussian function (3.2). A trivial computation shows that h(ξ, η, λ)p is given by 



h(ξ, η, λ)p(x) = x, η − R−1 ξ + m, R−1 ξ + λ p(x).

(3.10)

In particular, h(ξ, η, λ)p = 0 is equivalent to two conditions ξ = Rη,

(3.11)

λ = −m, η, PROBLEMS OF INFORMATION TRANSMISSION

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which prove that the annihilator Θ = Ann p of p in the Heisenberg algebra is n-dimensional. Consider the quadratic form Q(h(ξ, η, λ)) = ξ, η on h. Then Q(h(ξ, η, λ)) = Rη, η is strictly positive definite on Θ. The following lemma asserts that these conditions distinguish the Gaussian functions. Lemma 2. Assume that Θ is an abelian Lie subalgebra of h such that 1. The intersection Θ ∩ C with the center C ⊂ h is 0; 2. The dimension of Θ is n; 3. The quadratic form Q is strictly positive definite on Θ. Then Θ = Ann p, where p is a Gaussian function (3.2). Moreover, if Θ = Ann f , then f is a Gaussian function. Proof. Indeed, it follows from conditions 2 and 3 that the projection h(ξ, η, λ) → η defines an isomorphism Θ  Rn . This proves that Θ is defined by equations of the form (3.11). Then the matrix R in the equations should be symmetric, because Θ is an abelian subalgebra, and strictly positive definite because of condition 3. This proves that Θ = Ann p, where p is a Gaussian function (3.2). Conversely, if Θ = Ann f , where f is a distribution on Rn , then after an affine transformation of the argument x → αx + β we may assume that the parameters R and m in (3.11) ∂f

+ xi f = 0 for i = 1, . . . , n, which implies are the unit matrix and zero vector. Then f satisfies ∂xi that f (x) = C exp(−1/2|x|2 ).  In view of the preceding lemma, it is natural to define a θ-subalgebra as a maximal abelian Lie subalgebra Θ in h such that under the natural homomorphism π : h → R2n = h/C, the algebra Θ is isomorphic to its image π(Θ). In other words, Θ ∩ C = {0}, and Θ is a lift of a Lagrangian subspace Λ ⊂ R2n . To specify the lift, one has to indicate a linear (lifting) functional Λ → C. Equations (3.11) mean that the annihilator Θ of p in h is a lift to h of the Lagrangian subspace Λ of R2n defined by the first equation (3.11), the lifting functional being defined by the second equation (3.11), and the annihilator is a θ-subalgebra. Define also a positive θ-subalgebra as a θ-subalgebra such that the quadratic form Q is strictly positive definite on Θ. It is clear that the evolution on h governed by the operator τ (Z) transforms θ-subalgebras to θ-subalgebras. A very mild extra effort in Section 3.3 proves that positive θ-subalgebras and sets (3.11) behave similarly. Since a positive θ-subalgebra and the corresponding Gaussian function up to a constant factor are in one-to-one correspondence, this gives a geometric solution of the Zakai equation via a flow of automorphisms of the Heisenberg algebra. 3.3. Heisenberg–Lax Equations Now our problem is as follows: We are looking for a solution ρ(t, x) of dρ = Zρ

(3.12)

with a Gaussian initial function ρ0 . We start with basic relations between the dynamics governed by (3.12) and the flow on the Heisenberg algebra defined by the Heisenberg–Lax equation dM = [Z, M ].

(3.13)

The map Ft : M0 → Mt is a flow of automorphisms of the Heisenberg algebra which maps Ann ρ0 to Ann ρt . There is nothing specifically Heisenbergian about these simple statements. Indeed, if M = Mt and N = Nt satisfy (3.13), then d[M, N ] = [dM, N ] + [M, dN ] = [[Z, M ], N ] + [M, [Z, N ]] = [Z, [M, N ]]. PROBLEMS OF INFORMATION TRANSMISSION

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If ϕ = M ρ, then one can easily check that dϕ = Zϕ. If, furthermore, ϕ0 = 0, we obtain that M ρ = 0 at any time t, which shows that the flow (3.13) maps Θ0 = Ann ρ0 to Θt = Ann ρt . Since ρ0 is Gaussian, Θ0 = Ann ρ0 is a positive θ-algebra according to Section 3.2. It is clear now that Θt , which is the image of Θ0 under a flow of automorphisms, is a θ-algebra too. Moreover, the flow (3.13) retains the positivity of Θ0 . Indeed, if M = h(ξ, η, λ) satisfies (3.13), then in notation (3.9) we have 



dQ(h(ξ, η, λ)) = ξ  , η + ξ, η   = |B ∗ η|2 + |Cξ|2 dt ≥ 0,

(3.14)

and therefore Θt is a positive θ-algebra. By the results of Section 3.2, this implies that ρt is a Gaussian function such that its parameters Rt and mt enjoy the following property: if Mt = h(ξt , ηt , λt ) is the solution of the Heisenberg–Lax equation (3.13) and ξ0 = R0 η0 ,

m0 , η0  + λ0 = 0,

(3.15)

ξt = Rt ηt ,

mt , ηt  + λt = 0

(3.16)

then at any time t. In Section 3.4 we will see that the classical Kalman–Bucy equations follow from (3.13) and (3.16) together with (3.8) and (3.9). We conclude this subsection with remarks on relations between the positivity improving property of the flow (3.13) on h and the controllability/observability. The positivity improving is the following property: Assume that we have a nonzero x ∈ h and that the canonical quadratic form satisfies Q(x) ≥ 0. Then Q(Ft (x)) is strictly positive for any positive t. We recall that a pair of matrices (A, B) is said to be controllable if one can reach any point from any other point by moving along the trajectories of the system x˙ = Ax + Bu with a control u = u(t). Similarly, a pair of matrices (A, C) is said to be observable if for the system x˙ = Ax, y = Cx, one can recover the trajectory t → x(t) from t → y(t). Lemma 3. The Heisenberg–Lax flow Ft is positivity improving if and only if the pair of matrices (A, C) is observable and the pair (A, B) is controllable. Proof. Indeed, it is clear from (3.14) and (3.8) that if Mt = h(ξt , ηt , λt ) changes in accordance with (3.13) and Q(Mt ) ≡ 0 for all t in an open interval, then ξ˙ = Aξ, and

η˙ = −A∗ η,

Cξ = 0

(3.17)

B ∗η = 0

(3.18)

in this interval. Equation (3.17) immediately proves the statement about the pair (A, C), while the statement about (A, B) follows from (3.18) and the Kalman duality (see [8]).  3.4. Riccati Equation Here we derive classical equations for the parameters R and m of the conditional expectation ρ. We have to differentiate the first equation (3.16) with respect to time t. This gives dξ = dR η + R dη, while

(3.19)

dξ = (Aξ + BB ∗ η) dt = (AR + BB ∗ )η dt, PROBLEMS OF INFORMATION TRANSMISSION

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and dη = (C ∗ Cξ − A∗ η) dt = (C ∗ CR − A∗ )η dt by virtue of (3.8) and (3.16). Combining the above equations, we get (AR + BB ∗ )η dt = (dR)η + R(C ∗ CR − A∗ )η dt for any η. In other words, R˙ = AR + RA∗ + BB ∗ − RC ∗ CR,

(3.20)

which is the famous Riccati equation. Its salient feature is that it does not depend on observations. Similarly, by differentiating the second equation (3.16), we get 0 = dm, η + m, dη + dλ = dm, η + m, (C ∗ CR − A∗ )η dt − ξ, C ∗ dy = dm, η + m, (C ∗ CR − A∗ )η dt − Rη, C ∗ dy, or, equivalently, dm = (A − RC ∗ C)m dt + RC ∗ dy.

(3.21)

Sometimes, another form of this equation is preferred: dm = Am dt + RC ∗ (dy − Cm dt),

(3.22)

because dy − Cm dt is the differential of a new Wiener process (innovation process). Equations (3.20) and (3.21) are the classical Kalman–Bucy equations from [9]. We have found the geometric meaning of these equations: they describe the evolution of a positive θ-subalgebra of the Heisenberg algebra under the Heisenberg–Lax flow of automorphisms. 4. EXPLICIT SOLUTION OF THE ZAKAI EQUATION Now we turn to a little bit more general problem than the classical Kalman filtering: We again consider processes x and y governed by (3.1), but this time we do not assume that the initial vector x(0) is Gaussian. Then the conditional density p(t, x) is no longer Gaussian, and the problem of determining p(t, x) reduces to the description of the integral kernel K(s, x; t, y) of the operator solving the Cauchy problem for the Zakai equation. In other words, we are talking about a solution K(t, y) of (2.21) for t ≥ s such that K(s, y) = δ(x − y), which is the integral kernel of the evolution operator for ρ:  ρ(t, y) =

K(s, x; t, y)ρ(s, x) dx.

(4.1)

It is clear that K(s, x; t, y) = ρ(s, x; t, y), where ρ(s, x; t, y) is the unnormalized conditional density corresponding to the initial condition x(s) = x. Moreover, the kernel ρ(s, x; t, y) is characterized as a fundamental solution of the Cauchy problem ∂ ∗ ∂ ρ = H y, ρ, ∂t ∂y ∂ ∂ ρ = −H x, ρ, ∂s ∂x ρ|t=s = δ(x − y), PROBLEMS OF INFORMATION TRANSMISSION

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(4.2)

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∂ where the operator H x, is defined in Section 2.2 and the differential equations are understood ∂x

as stochastic Stratonovich PDEs. According to (2.8), the kernel is characterized by 

⎡

t

K(s, x; t, z)f (z) dz = E ⎣f (x(t))e s

− 12 |Cx|2 dt+Cx,dy

⎤ ⎦,

where the process x is governed by (3.1), x(s) = x, and y is a sample path of the observed component. According to (2.15), one can symbolically represent K(s, x; t, y) in the form of a functional integral   

−

e

K(s, x; t, y) =

t

1 2

(|u|2 + 12 |Cx|2 )dt−Cx,dy

s

dλ(u),

(4.3)

Ω(s,x;t,y)

where Ω(s, x; t, y) = {x(s) = x, x(t) = y}, dλ(u) is the “Lebesgue measure” (2.16), and the trajectories x = x(u) and u are related via (2.9) as follows: x˙ = Ax + Bu.

(4.4)

In other words, we deal with a typical linear control system with the boundary conditions x(s) = x,

x(t) = y.

(4.5)

Of course, the kernel depends on the observed sample path y. A wonderful fact is that the integral (4.3) can be explicitly computed in terms of an optimal control problem for the system (4.4). More precisely, consider the linear-quadratic functional (2.9) involved in (4.3) t 

I(u) = s



1 2 1 |u| + |Cx|2 dt − Cx, dy 2 2

(4.6)

and its minimum value S(s, x; t, y) =

min

u∈L2 ([s,t])

I(ξ),

(4.7)

where u determines x via the system (4.4) and the boundary condition (4.5). More precisely, the above formula determines the action S(s, x; t, y) if there exists an admissible trajectory of (4.4) subject to the boundary conditions (4.5). Otherwise, we put S(s, x; t, y) = +∞. Now the main relation between the linear-quadratic problem (4.7) and the integral (4.3) is as follows: (4.8) K(s, x; t, y) = λe−S(s,x;t,y) , where λ = λ(s, t) does not depend on the spatial variables and observation y. In particular, K(s, x; t, y) = 0 if S(s, x; t, y) = +∞. In what follows we assume that the conditions of Lemma 3 are met, so that for the pairs of matrices (A, B) and (A, C) we have (A, B) is controllable, while (A, C) is observable.

(4.9)

This guarantees that the action S(s, x; t, y) is everywhere finite and is a quadratic form with respect to the arguments x and y such that its purely quadratic (even) part with respect to either x or y is positive definite. In fact, it is possible to handle the general case, but this requires a delicate refinement of most statements. PROBLEMS OF INFORMATION TRANSMISSION

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Since the Zakai evolution operator is defined by neglecting time-only-dependent factors, one cannot expect greater precision in the formula for the kernel K(s, x; t, y). It is remarkably easy to prove (4.8). We start with a “heuristic” computation of functional integrals. Then we show how to convert the heuristics into rigorous arguments concerning Gaussian measures. Indeed, the functional I(u) is strictly convex, continuous, and bounded from below in the space L2 ([s, t]). Therefore, its minimum is attained at a unique trajectory X0 = (u0 , x0 ). We can decompose any trajectory X = (u, x) subject to the boundary conditions (4.5) as X = X0 + X  , where X  = (u , x ) satisfies X  (s) = 0 = X  (t). This decomposition implies that I(u) = I(u0 ) + I2 (u ), where  t  1 2 1   2 |u | + |Cx | dt (4.10) I2 (u ) = 2 2 s

stands for the purely quadratic part of I, because u0 is a critical point of the quadratic functional I. Therefore, the right-hand side of (4.3) takes the form −I(u0 )

e





e−I2 (u ) dλ(u ),

J

(4.11)

Ω(s,0;t,0)

where the factor J arises as follows: Define R(u) = x(t) by solving (4.4) with the initial condition x(s) = x. This is an affine map from the Hilbert space H = L2 ([s, t]) → Rn , and its linear part F = ∂R/∂u defines n linear functionals Fi : H → R, which can be identified with vectors Fi ∈ H, i = 1, . . . , n. Then J = J(s, t) = (det 2πFi , Fj )−1/2 .

(4.12)

In other words, J dλ(u ) is the “canonical measure” on Ω(s, x; t, y) obtained from dλ(u) by the restriction from H to the submanifold defined by the equation R(u) = y. Since I(u0 ) = S(s, x; t, y) in (4.11), this “proves” (4.8), where 



e−I2 (u ) dλ(u ).

λ = λ(s, t) = J(s, t) Ω(s,0;t,0)

A rigorous version of the above arguments gives the following statement. Theorem 2. Under condition (4.9), identity (4.8) holds, where λ = μν with 

t

μ = det 2π

−1/2

Φ(t, σ)B(σ)B ∗ (σ)Φ∗ (t, σ) dσ

,

(4.13)

s

where Φ(t, s) is the fundamental matrix of the linear system x˙ = Ax, and 

ν = E exp −

t s



1 |Cz|2 dt , 2

(4.14)

where z is the Gaussian process associated with the Hilbert space H = {u ∈ L2 ([s, t]) : z˙ = Az + Bu, z(s) = z(t) = 0, has a solution}. PROBLEMS OF INFORMATION TRANSMISSION

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(4.15)

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Proof. First we note that x0 (t) = x(t) is a Gaussian vector given by the Cauchy formula t

x0 (t) = Φ(t, s)x +

Φ(t, σ)B(σ) dw(σ).

(4.16)

s

Therefore, the covariance matrix R of x0 (t) is given by t

Φ(t, σ)B(σ)B ∗ (σ)Φ∗ (t, σ) dσ

R=

(4.17)

s

and coincides with the matrix Fi , Fj  from (4.12), so that J = det(2πR)−1/2 . If g(x) is any test function, then −1/2

E g(x0 (t)) = det(2πR)



−1 (z−¯ z ),z−¯ z ) dt

g(z)e− 2 (R 1

dz,

(4.18)

where z¯ = Φ(t, s)x is the mathematical expectation of x(t). Denote by H0 the finite-dimensional Hilbert subspace in L2 ([s, t]) of optimal controls in the minimization problem (4.7) for a fixed initial value x(s) = x and an arbitrary terminal value x(t) = z. Denote by 1 2 dμ0 (u0 ) = e− 2 |u0 | dλ0 (u0 ) the canonical Gaussian measure on H0 , and by E0 , the corresponding mathematical expectation. An easy computation shows that if g(x) is any test function, then

E g(x0 (t)) = κ



g(x0 (t))e

1 2

t

|πu0 |2 dt

dμ0 (u0 ),

s

(4.19)

where u0 is the optimal control (4.4) steering from x0 (s) = x to x0 (t) = z, π : H0 → H is the orthogonal projection, and the constant κ = E exp

t s

u0 , u  dt is well defined since u0 is a

smooth function of time. By comparing (4.18) and (4.19) we deduce that κ dλ0 (u0 ) = det(2πR)−1/2 dz = μ dz.

(4.20)

On the other hand, ⎡

−

E0 ⎣f (x0 (t))e

t

1 |Cx0 (τ )|2 2

t

⎤

s

⎦=

dτ + Cx0 (τ ),dy(τ )

s



f (z)e−S(s,x;t,z) dλ0 (u0 )

= κ −1 μ



f (z)e−S(s,x;t,z) dz,

(4.21)

since x0 (t) = z and 1 S(s, x; t, z) = 2

t

|u0 | dt +

t

2

s

s

1 |Cx0 (τ )|2 dτ − 2

t

Cx0 (τ ), dy(τ ).

s

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We want to prove that ⎡

−

E ⎣f (x(t))e

t

1 |Cx|2 dt+ 2

s

t

Cx,dy

s

⎤

⎡

−

⎦ = νκ E0 ⎣f (x0 (t))e

t

1 |Cx0 (τ )|2 2

t

⎤

s

⎦,

dτ + Cx0 (τ ),dy(τ )

s

(4.22)

where f (x) is any test function, and ν (which does not depend on the spatial variables) is given by (4.14). In view of (4.21), this is equivalent to (4.8). Now, 1 1 1 |Cx|2 dt − Cx, dy = |Cx0 |2 dt − Cx0 , dy + |Cx |2 dt − Cx , dy + Cx0 , Cx  (4.23) 2 2 2 and t





u0 , u  dt + Cx0 , Cx  dt − Cx , dy = 0

(4.24)

s

because u0 is an extremal point of the functional I. Put 

F (u0 ) = f (x(t)) exp −

t

1 |Cx0 |2 dt + 2

s



G(u ) = exp −

t



t



Cx0 , dy ,

s

1 |Cx |2 dt . 2

s

In view of (4.23) and (4.24), the left-hand side of (4.22) equals ⎡



E ⎣F (u0 )G(u ) exp

 t

⎤ u0 , u  dt ⎦ = κ E0 F (u0 ) E G(u ) 

(4.25)

s

with the constant κ = E exp

t s

u0 , u  dt , where E0 and E are mathematical expectations with

respect to the canonical Gaussian measures in the Hilbert spaces H0 and H , by basic properties of Gaussian measures. Recall that the Hilbert space H0 is defined right after (4.18), and the space H is defined in (4.15). From (4.21) it is clear, however, that the right-hand side of (4.25) is equal to 

μν

f (z)e−S(s,x;t,z) dz.



4.1. Determinants of the Sturm–Liouville Operators In fact, it is possible to give a rather explicit expression for 

ν = E exp −

t s

1 |Cz|2 dt 2



via the classical Hamiltonian system with the Hamiltonian H(x, p) of (2.10). Indeed, ν is equal to det(D)−1/2 , where D : H → H is a self-adjoint operator canonically associated with the quadratic form 2I2 of (4.10). In other words, 

Du, u  =

t

u, u  dt + Cx, Cx  dt.

s

In particular, D = 1 + T , where T is a nuclear operator, so that the determinant det(D) is well defined. PROBLEMS OF INFORMATION TRANSMISSION

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The first explicit computations of determinants of Sturm–Liouville operators were performed in [10, 11], and the existing technique [12] allows us to give the following recipe for computing det(D): Consider the flow Ψ = Ψ(t, s) for the Hamiltonian system with the Hamiltonian H(x, p). Here, Ψ(t, s) : R2n → R2n is the transformation from time s to t. It is an affine transformation,  is a symplectic matrix of the form and its linear part Ψ 



 = α β , Ψ γ δ

(4.26)

where α, β, γ, and δ are n × n matrices. Then an explicit form for ν follows from the identity ⎡ t ⎤−1  det(D) = det ⎣ Φ(t, σ)B(σ)B ∗ (σ)Φ∗ (t, σ) dσ ⎦ det β,

(4.27)

s

where Φ(t, s) is the fundamental matrix of the linear system x˙ = Ax. In other words, in notation (4.8) we have (4.28) λ2 = det(2πβ)−1 . A proof of (4.28) that does not rely on methods of [10–12] is given in Section 4.2. Note that it follows from (4.28) that det β > 0. It is possible to give an a priori proof of the latter inequality. Indeed, because of assumption (4.9) and Lemma 3, the quadratic form p → βp, δp is positive definite. Therefore, the matrix β is nondegenerate, so that det β = 0 at any time. Moreover, since  is symplectic, the matrix the map Ψ (4.29) δ∗ β = ϕ is symmetric and positive definite, so that det ϕ > 0. If the time t is close to the initial moment s, the matrix δ is close to the unit matrix 1, and it follows from (4.29) that det β > 0 for these time instants. Since det β = 0, we conclude that det β > 0 at any time t > s. Example 1. Consider an operator D associated with the simplest quadratic functional I =

1 t 2 (u + x2 ) dt, where x˙ = u. The corresponding Hamiltonian is H(x, p) = p2 /2 − x2 /2, and 20   cosh t sinh t  Ψ(t, 0) = . (4.30)

sinh t cosh t

The equation for eigenfunctions and eigenvalues of the operator D is of the form −λu˙ = −u˙ + x or (λ − 1)¨ x + x = 0. This, together with the boundary conditions x(0) = 0 = x(t), implies that 

∞  πns t2 t2 and λ = 1 + 2 2 , where n is an integer. Therefore, det(D) = 1+ 2 2 , t π n π n n=1   ∞  t2 1 − 2 2 for B(σ) ≡ 1, Φ(t, σ) ≡ 1, and (4.27) is equivalent to the Euler formula sin t = t π n n=1

x(s) = C sin

the sine.

4.2. WKB Solution There is another explicit expression for the factor λ: 

1 ∂2S λ = det − 2π ∂x ∂y 2



,

(4.31)

which can be regarded as a statement that the WKB approximation of the fundamental solution to (4.2) is exact in the linear case. Indeed, by (4.8) we have the WKB ansatz for the kernel K(s, x; t, y) = λ(s, t) exp(−S(s, x; t, y)). PROBLEMS OF INFORMATION TRANSMISSION

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It is well known and easy to prove that S satisfies the Hamilton–Jacobi equation 

∂S ∂S + H ∗ y, ∂t ∂y



= 0,

(4.32)

∂ ∗

where H ∗ is the Hamiltonian corresponding to the adjoint operator H x, . Moreover, S is ∂x a quadratic form with respect to the arguments x and y such that its purely quadratic part with respect to either x or y is positive definite. Note also that the purely quadratic parts with respect to momenta of the operators H and H ∗ coincide. The first equation (4.2) now takes the form 1  ∂2H 1 ∂2S ∂ log λ = − =− Hij Sij . ∂t 2 ij ∂pi ∂pj ∂yi ∂yj 2

(4.33)

This is the transport equation, which is well known in the WKB method [13, 14]. Now define the matrix ∂2S . (4.34) σij = ∂xi ∂yj By applying the operator

∂2 to the Hamilton–Jacobi equation (4.32) and taking into account ∂xi ∂yj

that S is a quadratic form with respect to the arguments x and y, we obtain n  ∂ σij + σi Hk Skj = 0. ∂t k,=1

(4.35)

 ∂ ∂ log det σ = − Hij Sij = 2 log λ. ∂t ∂t

(4.36)

This implies, in particular, that



We conclude that

λ2

 ∂2S = C det , where C is a constant. If we consider the limit behavior ∂xi ∂yj

of S and K as t → s and x → y, we obtain 

1 ∂2S λ ∼ det 2π ∂yi ∂yj

1/2



1 ∂2S ∼ det − 2π ∂xi ∂yj

1/2

.

Indeed, for any positive definite quadratic form R(y) we have 

1 ∂2R det 2π ∂y 2

1/2 

e−R(y) dy = 1,

which explains the first asymptotic equality since 

λ(s, t) exp(−S(s, x; t, y)) dy →

as t → s, while the relation



δ(x − y) dy = 1

∂S ∂S ∼− ∂x ∂y

as x → y explains the second asymptotic equality. PROBLEMS OF INFORMATION TRANSMISSION

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Finally, we have



1 ∂2S λ = det − 2π ∂xi ∂yj

1/2

,

which agrees with (4.28) because of the remarkable identity β

∂2S = −1. ∂x ∂y

(4.37)

To prove (4.37), it suffices to consider the case where the Hamiltonian H is purely quadratic, so that 

∂S ∈ R2n ∂y  Because of the (already used) identity ∂S = − ∂S if x = y, is invariant with respect to the matrix Ψ. ∂x ∂y we obtain the matrix equality ⎞ ⎛ ⎞ ⎛   x y α β ⎜ ⎟ ⎜ ⎟ (4.38) ⎝ ∂S ⎠ = ⎝ ∂S ⎠ .

the Hamiltonian flow Ψ(t, s) : R2n → R2n is linear. Then the Lagrangian manifold

γ

In particular, αx − β

−

δ

∂x

y,

∂y

∂S ∂ = y. By applying the operator , we arrive at (4.37). ∂x ∂y

4.3. The Mehler Formula Armed with an explicit formula for the factor λ, we can give a new interpretation of the classical Mehler formula. In our setup, it corresponds to the simplest Kalman filtering problem: dx = dw,

x(0) = 0,

dy = Cx + dW,

y(0) = 0,

(4.39)

where the result of observation of y is identically zero (y(t) ≡ 0) and the matrix C is time-invariant. Theorem 2 together with (4.31) allow us to rederive an explicit formula due to Mehler for the fundamental solution ρC (t, x, y) = ρ(s, x; s + t, y) of (4.2), where H(x, p) = |p|2 /2 − |Cx|2 /2: 

M ρC (t, x, y) = det 2π sinh(M t)

1/2

exp (−St (x, y)) ,

where M = (C ∗ C)1/2 and St (x, y) is the minimum value of

(4.40)

 1 t  2 |x| ˙ + |Cx|2 over trajectories 20

x : [0, t] → Rn such that x(0) = x and x(t) = y. The most important part of the formula is the

factor det

1/2 M , which is responsible for the form of the index formula for the Dirac 2π sinh(M t)

operator on a compact spin-manifold [15]. In fact, the problem of an explicit determination of the kernel K(s, x; t, y) of (4.2) can also be solved via the classical Kalman–Bucy filtering, because y → K(s, x; t, y) is a Gaussian function. This gives yet another connection between the Kalman filter, deterministic linear-quadratic problem, and the Riccati equation. REFERENCES 1. Hazewinkel, M. and Marcus, S.I., On Lie Algebras and Finite-Dimensional Filtering, Stochastics, 1982, vol. 7, no. 1–2, pp. 29–62. 2. Brockett, R.W. and Clark, J.M.C., The Geometry of Conditional Density Equation, Analysis and Optimization of Stochastic Systems, Jacobs, O.R.L., et al., Eds., London: Academic, 1980, pp. 299–309. PROBLEMS OF INFORMATION TRANSMISSION

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3. Ovseevich, A.I., Kalman Filter and Quantization, Dokl. Ross. Akad. Nauk, 2007, vol. 414, no. 6, pp. 732–735 [Dokl. Math. (Engl. Transl.), 2007, vol. 75, no. 3, pp. 436–439]. 4. Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland; Tokyo: Kodansha, 1981. Translated under the title Stokhasticheskie differentsial’nye uravneniya i diffuzionnye protsessy, Moscow: Nauka, 1987. 5. McKean, H.P., Jr., Stochastic Integrals, New York: Academic, 1969. Translated under the title Stokhasticheskie integraly, Moscow: Mir, 1972. 6. Zakai, M., On the Optimal Filtering of Diffusion Processes, Z. Wahrsch. Verw. Gebiete, 1969, vol. 11, pp. 265–284. 7. Arnol’d, V.I., Matematicheskie metody klassicheskoi mekhaniki, Moscow: Nauka, 1989, 3rd ed. Translated under the title Mathematical Methods of Classical Mechanics, New York: Springer, 1989, 2nd ed. 8. Kalman, R.E., On the General Theory of Control Systems, in Proc. 1st Int. Conf. on Automatic Control, Moscow, USSR, 1960, vol. 1, pp. 481–492. 9. Kalman, R.E. and Bucy, R.S., New Results in Linear Filtering and Prediction Theory, Trans. ASME, Ser. D.: J. Basic Engrg., 1961, vol. 83, pp. 95–108. 10. Levit, S. and Smilansky, U., A Theorem on Infinite Products of Eigenvalues of Sturm–Liouville Type Operators, Proc. Amer. Math. Soc., 1967, vol. 65, no. 2, pp. 299–302. 11. Dreyfus, T. and Dym, H., Product Formulas for the Eigenvalues of a Class of Boundary Value Problems, Duke Math. J., 1978, vol. 45, no. 1, pp. 15–37. 12. Kirsten, K. and McKane, A.J., Functional Determinants by Contour Integration Methods, Preprint of Max Planck Institute f¨ ur Mathematik in den Naturwissenschaften, Leipzig, 2003, no. 57. 13. Guillemin, V. and Sternberg, S., Geometric Asymptotics, Providence: AMS, 1977. Translated under the title Geometricheskie asimptotiki, Moscow: Mir, 1981. 14. Maslov, V.P. and Fedoryuk, M.V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Moscow: Nauka, 1976. Translated under the title Semi-classical Approximation in Quantum Mechanics, Dordrecht: Reidel, 1981. 15. Getzler, E., A Short Proof of the Local Atiyah–Singer Index Theorem, Topology, 1986, vol. 25, no. 1, pp. 111–117.

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