Error calculus with Dirichlet forms

Bias: We compute the bias of uncertainties on the position of the receiver.



therefore, we find that the estimation of latitude/longitude is unbiased, thanks to the sym-metry of the problem, however the estimation of Y is negative biased.

This example shows the power of Gauss’ formalism when we consider an explicit smooth function of parameters afflicted by uncertainties. Unfortunately, the great part of model models is defined via an implicit form, e.g. integrals, solutions of differential equations, PDEs, SDEs, SPDEs or fixed point problems. In these cases the classical Gauss’ language cannot be applied and a refined theory is needed. A possible extension is proposed by Bouleau, in the next section we present a survey of this theory.

1.2 Error calculus with Dirichlet forms

The Bouleau’s intuition is based on the remark that the two operators, Γ andA, i.e. the quadratic error and the bias, have the same chain rules of two operators existing in Dirichlet forms theory.

Bouleau, in the paper [7], rewrite the Gauss’ intuition in the rigorous frame of the Dirichlet form language. The basic tool is the error structure.

Definition 1.1 (Error structure) An error structure is a term

Ω,e Fe, P,e D, Γ where

1.

Ω,e Fe, Pe

is a probability space;

2. D is a dense sub-vector space of L²

Ω,e Fe, Pe

;

3. Γ is a positive symmetric bilinear application from D × D into L¹

Ω,e Fe, eP

satisfying the functional calculus of class C¹ ∩Lip, i.e. if F and G are of class C¹ and Lipschitzian, ∀ u and v ∈D, we have F(u) and G(v) ∈D and

(1.7) Γ [F(u), G(v)] =F^′(u)G^′(v)Γ[u, v] ePa.s.;

4. the bilinear form E[u, v] = ¹₂Ee[Γ[u, v]] is closed, i.e. the space (D, k u kD) is a Banach space, where the norm kuk^D=q

kuk²L² +E[u, u].

5. The constant function 1 belongs to D, in this case the error structure is said Markovian.

For sake of simplicity, we will always write Γ[u] for Γ[u, u] andE[u] forE[u, u]. First of all we have to emphasize the following crucial remark.

Remark 1.2 (Chain rule)

The function calculus (1.7) of the operator Γ is the same of the quadratic error operator, see equation (1.2). Therefore we can use the operator Γ to extend the Gauss’ theory. Then, if U = (U1, ..., Un)belongs toDⁿ, the matrix Γ[U] = [Γ[Ui, Uj]]_{0≤i, j≤n}plays the role of the variance-covariance matrix of uncertainties on U given the value of U.

The bilinear form E, introduced in the last definition, is known in the literature as a local Dirichlet form on L²(Ω,e Fe, Pe), local means that for all U ∈ D, let F and G be two smooth functions with non overlapped supports then E[F(U), G(U)] = 0. The form E possesses a carr´e du champ operator Γ, see Bouleau and Hirsch [6] page 16. This theory rise in the 20th century after the seminal work of Beurling and Deny [4] as a tool of potential theory, but it has a probabilistic interpretation in Markov process theory, see for example Fukushima et al. [22] or Silverstein [42].

Under some additional assumptions, see Bouleau and Hirsch [6] chapter 1 or Ma and Rockner [31], we can also associate a unique strongly-continuous contraction semi-group (Pt)_t≥0 on L²(eP) with the error structure

Ω,e Fe, eP, D,Γ

via Hille-Yosida theorem, see Albeverio [1]. This semi-group is Markovian if and only if the property 5 in the definition1.1 is verified, in this sense the error structure is said Markovian.

This semigroup has a generator (A,DA), with DA ⊂ D, i.e. a self-adjoint operator that satisfies, for all F ∈ C² with bounded derivatives, U ∈ DA and Γ[U] ∈L²(eP), F(U) belongs to DAand

(1.8) A[F(U)] =F^′(U)A+1

2F^′′(U) Γ[U] eP−a.s.,

a similar result exists when the functionF :R^d→R, see Bouleau [8] page 33.

We have defined the carr´e du champ Γ as a bilinear operator like variance-covariance. This characteristic is crucial in the error theory but, frequently, makes the tool very awkward to

1.2. ERROR CALCULUS WITH DIRICHLET FORMS 13 perform computations. In the applications of classical Gauss’ theory in physics, a squared root of the variance is often used, as known as the standard deviation. Is it possible to define a linear standard deviation operator in error theory using Dirichlet forms? The answer to this problem is positive under some constraints, when we consider random variables with values in an Hilbert space and if the domain D of the carr´e du champ operator verifies the Mokobodzki hypothesis, see below. In this case, there are many linear versions of standard deviation of the error, called gradients, for more details see Bouleau [8] page 78. The existence of more than one gradient can be explain since the gradient is not intrinsic but is rather a derived concept depending on an exogenous space, as only Γ has an interpretation. However in the large class of gradients there is a preferable candidate called sharp.

Definition 1.2 (Sharp operator) Let

Ω,e Fe, eP,D, Γ

be an error structure and

Ω,b Fb,Pb

a copy of the probability space Ω,e Fe, Pe

. Under the Mokobodzki hypothesis, i.e. the space D is separable, there exists an operator sharp( )^# with these three properties:

• ∀u∈D, u^#∈L²(Pe×bP);

• ∀u∈D, Γ[u] =Ebh

u^#2i

, where Eb denotes the expectation under the probability bP;

• ∀u∈Dⁿ and F ∈ C¹∩Lip, (F(u1, ... , u_n))^#=Pn i=1

∂F

∂xi ◦u u^#_i .

This operator is especially useful in Wiener space applications, see section 1.4. We conclude this section with some examples.

Example 1.2 (first error structure on R )

We consider the probability space(R, B(R), N(0,1)), where B(R) is the Borelσ-algebra onR and N(0,1) denotes a reduced normal law.

We considerΓ[u]→(u^′)² as the carr´e du champ operator, its domain will beD=H¹(N(0, 1)), i.e. the first Sobolev space with respect to the measureN(0, 1).

The term

R, B(R), N(0, 1), H¹(N(0, 1)), (u^′)²

is an error structure, because the operator Γ[u] = (u^′)² is the carr´e du champ operator of the Ornstein-Uhlenbeck Dirichlet form on R. This identification supplies us the generator too, we have

A[u] = 1

2u^′′− 1 2I·u^′ DA =

u∈L²(N(0,1)) with u^′′−x u^′ belongs to L²(N(0, 1)) in the distribution sense where I is the identity map on R.

This example shows a powerful procedure in error theory using the language of Dirichlet forms. The Dirichlet forms theory is fifty-years-old, therefore, many Dirichlet forms have been identified. A classical approach in error theory is to recognize a known carr´e du champ to prove the closability of the related bilinear form, see hypothesis 4 in definition1.1.

Example 1.3 (error structure on an interval)

Let ([0,1], B([0, 1]), λ) be our probability space where λ is the Lebesgue measure. Let

Γ[u] = (u^′)²

D = H¹([0,1]) =

u and u^′ ∈L²([0, 1], λ) in the distribution sense be the carr´e du champ and its domain.

This term is an error structure, for the proof see Bouleau [8] pages 34-36, and the related generator is

A[u] = 1 2u^′′

DA =

u∈C²([0, 1]) with u^′(0) =u^′(1) = 0

More generally Hamza, see [26], has defined a necessary and sufficient condition, when the space is R, for a couple, probability law and bilinear operator, to generate a Dirichlet form.

However, this result cannot be generalized when the space has dimension bigger than one, see Fukushima et al [22] page 105.

Proposition 1.1 (Hamza 1975) Let(R, F,P)be a probability space onR. We defineΓ[u](x) = (u^′(x))² g(x), where g(x) is a positive integrable function. The term (R, F, P,D,Γ) is an error structure, with D suitable domain, if and only if

1. the measure g ·P is absolutely continuous with respect to the Lebesgue measure 2. and its density, denoted η(x) is worth zero a.e. on the set R\ R(η) with

R(η) =

x∈R such that ∃ǫ Z

[x−ǫ, x+ǫ]

dy

η(y) <∞

.