Introduction to stochastic variational problems

Victor L. Berdichevsky Mechanical Engineering, Wayne State University, Detroit MI 48202 USA Abstract The lectures provide an introduction to the Chapters on stochastic variational problems from the author's book Variational Principles of Continuum Mechanics, Springer, 2009.

Reminder from complex analysis

The major goal of these lectures is to explain how to compute probabilities in some stochastic variational problems. The lecturers can be used as an introduction to the Chapters of my book Variational Principles of Continuum Mechanics, Springer-Verlag, 2009, which are concerned with stochastic variational problems. Before proceeding to stochastic variational problems we have to learn how to compute probability in much simpler cases, like, for example, how to find probability distribution of a sum of independent random variables. Unfortunately, these issues are discussed in engineering probability courses at the time when the students do not have enough mathematical background, and a simple and beautiful nature of the classical results of probability theory, like the central limit theorem or Gauss distribution, remain unrevealed. Therefore, I will spend the first part of the course to cover these issues.

To do the calculations we will use complex analysis. I would assume with a great deal of certainty, that not everyone in this room was exposed to properly taught complex analysis. In high school you, perhaps, were taught that the complex unity i is a square root from -1:

i = √ -1.
What does that mean? Of course, you can solve the equation x 2 + 1 = 0 in terms of i and write x = ±i. So what? Well, you can then write the solution of any polynomial equation in the form α + iβ, where α and β are real numbers, besides, the number of roots is equal to the power of the polynom. This is nice indeed. This was the way in which the complex numbers were introduced in XVI century by Italian mathematicians Cardano and Bombelli. This way penetrated in modern text books without much change. Unfortunately, one key word is missing in such a treatment, the word without which the real understanding of complex numbers is hardly possible. I begin with an explanation of what the complex numbers are (note that complex numbers are not numbers!) and why they are needed for the problems under consideration. Besides, I will review the basics of complex analysis. Then we spend two lectures for classical results of probability theory, and then go on to stochastic variational problems.

Complex numbers. We know two basic operations with vectors: We can multiply vector, a, by a number, λ; if a i (i = 1, ..., n) are the components of a vector a in some basis êi 1 , a = a i êi , then vector λ a has the components λa i : λ a = λa i êi .

We also can sum vectors; if a i and b i are the components of vectors a and b, then vector a + b has the components a i + b i :

a + b = a i + b i êi .
The latter definition corresponds to the parallelogram rule (Fig. 1). Now we wish more. We wish to operate with vectors as we do with numbers. We wish to introduce multiplication of vectors, a and b, in such a way that the product of vectors is a vector, and, as for numbers,

a • b = b • a, a + b • c = a • c + b • c, a • b • c = a • b • c .
Besides, we would like to be able to divide vectors, i.e., for each vectors a and b, we should be able to compute their ratio, a vector c, c = a / b.

For given a and b, vector c must be determined uniquely from the equation b • c = a.

1 In all formulas summation over repeated indices is assumed, e.g.

a i êi ≡ n i=1 a i êi .
Figure 1. The definition of a sum of two vectors and multiplication of a vector by number.

Clearly, the products of vectors which are introduced in vector analysis, the vector product, a × b, and the dot product, a • b, do not fit: a × b is antisymmetric a × b =b × a , the dot product is a scalar, not a vector. The multiplication operation we are looking for must be a new one. It turns out that such product can be introduced only in three spaces: two-dimensional, four-dimensional and eight-dimensional. The vectors of the corresponding two-dimensional space are called complex numbers, four-dimensional space quaternions and eight-dimensional space octonions. Quaternions were first conceived by W.R. Hamilton, octonions by A. Caley (Caley numbers). Only vectors of two-dimensional space, complex numbers, possess all the features of usual numbers; quaternions and octonions do not. Now we have to define what is the product of two vectors in twodimensional space. To this end, it is enough to define the products of the basic vectors. Then the product of any two vectors can be computed from the relation

a 1 ê1 + a 2 ê2 • b 1 ê1 + b 2 ê2 = a 1 b 1 ê1 • ê1 + a 1 b 2 + a 2 b 1 ê1 • ê2 + a 2 b 2 ê2 • ê2 .
Let us denote the two basic vectors by the symbols ê1 = 1, ê2 = î. Then any vector z has the form z = x • 1 + yî. The products of basic vectors are defined by the following rule:

1 • 1 = 1, 1 • î = î • 1 = î, î • î = -1.
Figure 2. Notations in complex plane.

z 1 = x 1 • 1 + y 1 î, z 2 = x 2 • 1 + y 2 î, their product is z 1 • z 2 = (x 1 x 2 -y 1 y 2 ) 1 + (x 1 y 2 + x 2 y 1 ) î.
One can prove that such a product has all the features of the product of usual numbers. The tradition is not to write the hat at the basic vector î : one writes for this basic vector i ≡ î. Moreover, without confusion the basic vector 1 in the expression x 1 + iy can be dropped. So the complex number takes the form z = x + iy.

Such form defines the components of vector z, x and y, uniquely. They are called real and imaginary parts of z and denoted by Rez and Imz, respectively. The complex numbers with zero imaginary part are called real. The two-dimensional space with such defined vector product is called complex plane. We arrive at the usual definition of complex numbers, which you can find e.g. in Wikipedia: complex numbers are the numbers of the form x + iy, where x, y are real numbers while i 2 = -1. What is missing in this definition is the key word: vector. Complex numbers are not numbers, they are vectors. E x e r c i s e s. Complex conjugate of z = x + iy is, by definition, the number z = xiy. Show that z z is a real number equal to the squared length of vector z; the length of z, |z| is defined as |z| = x 2 + y 2 (see Fig. 2). Find the number 1/z. Functions of complex variables. An advantage we gain working in complex plane is that we can operate with vectors in the same way we do with numbers. In particular, all functions we used in calculus, make sense being applied to vectors. For example, for numbers, function e x can be defined as a sum

e x = 1 + x + 1 2 x 2 + 1 6 x 3 + ... = ∞ n=0 1 n! x n .
In the same way it can be defined for vectors because the sum of vectors and the powers z n = z...z n times are well defined:

e z = 1 + z + 1 2 z 2 + ... = ∞ n=0 1 n! z n . (1)
Of course, one has to complement this by a notion of convergence, but this is done in a natural way: sequence

z n → 0 as n → ∞, if |z n | → 0 as n → ∞. By multiplying two series, ∞ 0 1 n! z n 1 and ∞ 0 1 n! z n 2 ,
one can check that e z1+z2 = e z1 e z2 .

(2)

An immediate consequence of (1) and ( 2) is Euler formula e iy = cos y + i sin y.

(3) Indeed, plugging in (1) z = iy, and using that

i n = {.(-1) k i if n = 2k + 1(-1) k if n = 2k,
we get

e iy = even n 1 n! z n + odd n 1 n! z n = ∞ k=0 (-1) k (2k)! y 2k + i ∞ k=0 (-1) k (2k + 1)! y k . (4)
Euler formula follows from the Taylor expansion of sin and cos :

cos y = ∞ k=0 (-1) k (2k)! y 2k , sin y = ∞ k=0 (-1) k (2k + 1)! y k .
Denote by θ the angle between the vector z and positive real axis. We use the convention, that angle θ increases in counter clockwise direction and changes within the limits -π, π. According to Euler formula one can write z in the polar form: z = |z| e iθ .

(5)

Another important function of complex variable to be encountered further is ln z. It is defined as inversion of exponential function, i.e.

e ln z = z.

(6)

Since, according to (5), z = e ln|z|+iθ , we can rewrite ( 6) as e ln z = e ln|z|+iθ from which ln z = ln |z| + iθ.

Construction of functions by means of a series makes a class of functions defined by a converging Taylor series especially important; they are called analytic functions. More precisely, f (z) is an analytic function in a region D if it can be presented in D by a converging series. We drop further all mathematical details, they can be found in numerous sources, and focus only on the basic ideas.

Derivative of a function of complex variable is defined in terms of its Taylor series 2

f (z) = ∞ k=0 a k z k , as f ′ (z) = ∞ k=0 ka k z k-1 .
This definition is equivalent to the usual one

f ′ (z) = lim ∆z→0 f (z + ∆z) -f (z) ∆z
2 Here we consider a function which is analytical in a vicinity of the point z = 0. If a function is analytical in a vicinity of some point z 0 , Taylor series is written with respect to the shift zz 0 :

f (z) = ∞ k=0 a k (z -z 0 ) k .
provided the convergence of all series, which are involved.

Integral over a contour γ in z-plane,

γ f (z)dz,
is defined in the same way as for a contour in two-dimensional real plane.

Integral of an analytic function over a closed contour in a simply-connected region3 is zero. Indeed, let a function be analytic in a vicinity of the point z = 0. The integral over closed contour of z n is zero:

γ z n dz = d z n+1 n = 0.
Thus, integrating the Taylor series of an analytic function we get zero for any analytic function. Consideration of functions analytical in vicinity of non-zero points is similar. An example of non-analytic function in vicinity of z = 0 is the function f (z) = 1/z; it is analytic in any region with excluded point z = 0 and non-analytic in any vicinity of point z = 0. E x e r c i s e s. 1. Write down polar forms for 1/z, z, 1/z, -1, z n . 2. Find (1 + i) 5 , (1 + i) 100 .

3. Derive from the definitions of trigonometric functions of complex variables, sin z = 1 2i e ize -iz , cos z = 1 2 e iz + e -iz , that sin 2 z + cos 2 z = 1.

4. Hyperbolic sin and cos are defined by the formulas:

sinh z = 1 2 e z -e -z , cosh z = 1 2 e z + e -z .
Show that cosh 2 zsinh 2 z = 1.

5. Show that sin 2z = 2 sin z cos z. 6. Show that for functions f (z) with real coefficients of Taylor expansion

f (z) = f (z) .
7. Write Taylor series for the function (e z -1)/z.

Step function. The only reason why we need complex numbers in what follows is the presentation of the step function by means of an integral in complex plane. By the step function, θ (t) , we mean the following function

θ (t) = {.1 t > 00 t < 0. (8)
Usually, the value of this function at t = 0 is not essential. We will set for definiteness

θ (0) = 1 2 . ( 9
)
We need the following integral presentation of θ (t) :

θ (t) = 1 2πi a+i∞ a-i∞ e tz z dz, a 0, ( 10
)
where the integration contour is a straight line in the right half-plane z.

Since formula (10) will play a central role in further calculations, we outline here its derivation.

First of all we need to specify the meaning of the integral in (10). The integral has infinite limits, and, in general, one can write The role of absolute convergence follows from the inequality: for any contour γ,

γ f (z)dz γ |f (z)| |dz| .
This inequality is a consequence of the inequality

|z 1 + ... + z n | |z 1 | + ... + |z n | ,
which obviously follows from the triangle inequality,

|z 1 + z 2 | |z 1 | + |z 2 | .
If an integral with infinite limits converges absolutely, it does not matter in which way the infinite limits are approached. The integral in (11) does not converge absolutely. Indeed

Let us compute this limit. First of all, for t = 0

θ (0) = 1 2πi lim b→∞ a+ib a-ib 1 a + iy idy = 1 2πi lim b→∞ (ln (a + ib) -ln (a -ib)) .
From ( 7)

ln (a + ib) = ln a 2 + b 2 + i arctan b a ln (a -ib) = ln a 2 + b 2 -i arctan b a . Since lim b→∞ arctan b a = π 2 , we have θ (0) = 1 2π π 2 -- π 2 = 1 2 .
This is in accord with (9). Let now t < 0. We aim to show that the limit in ( 12) is zero. Function e tz /z is analytic in the shadowed region in Fig. 3a. Therefore, integral of e tz /z over the boundary of the shadowed region is zero. Hence, the integral of e tz /z over the straight segment [aib, a + ib] is equal to the integral over the contour γ. Let γ be an arc of a circle with radius R = √ a 2 + b 2 and the center at z = 0. Then the integral is bounded from above:

γ e tz z dz |e tz | |z| |dz| = θ -θ
e tR cos θ dθ.

Since t < 0, this integral does not exceed the integral,

J(R) = π/2 -π/2 e -|t|R cos θ dθ = 2 π/2 0 e -|t|R cos θ dθ.
We are going to show that function J(R) tends to zero as R → +∞. This is the first point in these lectures where we encounter the necessity to find the asymptotics of an integral; we will have several such problems later on.

If the function in the exponent, cos θ, were strictly positive everywhere on the integration interval, i.e. cos θ min cos θ = c > 0, then the integrand does not exceed e -|t|R min(cos θ) , and the integral tends to zero as R → +∞.

In our case, however, min cos θ = 0, (see Fig. 4a), it is achieved at θ = π/2.

We have to study the contribution to the integral of the vicinity of the point θ = π/2. Let us split the integral in two parts

1 2 J(R) = π/2-α 0 e -|t|R cos θ dθ + π/2 π 2 -α e -|t|R cos θ dθ. (13)
The first integral in (13) does not exceed

e -|t|R cos( π 2 -α) π 2 -α Figure 4. Graph of cos θ.
and tends to zero as R → +∞. For sufficiently small α, in the second integral we can replace cos θ by the first terms of Taylor expansion

cos θ = cos θ θ=π/2 + d cos θ dθ θ=π/2 θ - π 2 = π 2 -θ .
Then the second integral takes the form

π/2 π 2 -α e -|t|R( π 2 -θ) dθ = α 0 e -|t|Rξ dξ.
We can increase this integral replacing the upper limit by +∞. Then we get the integral which is computed analytically

∞ 0 e -|t|Rξ dξ = 1 |t| R .
Hence, J(R) → 0 as R → +∞, and θ (t) = 0 for t < 0 indeed. Consider now the case t > 0. In this case we introduce a region shadowed in Fig. 3b. Function e tz /z is analytic in a shadowed region, and integral over the boundary of the shadowed region is zero. Therefore, the integral over the segment [aib; a + ib] is equal to the sum of integrals over contours γ and γ ′ , γ being a circle of radius R. The integral over γ can be estimated from above as

γ e tz z dz = γ |e tz | |z| |dz| = 3π 2 +α π 2 -α e tR cos θ dθ = 2 π π 2 -α e tR cos θ dθ. ( 14
)
Here α is the angle shown in Fig. 3b. The integral in the right hand side of ( 14) can be written as a sum

π π 2 -α e tR cos θ dθ = π π 2 e tR cos θ dθ + π 2 π 2 -α
e tR cos θ dθ.

Function cos θ on the segment [π/2α, π] is shown in Fig. 4b. The first member of the sum coincides with J(R) considered above and thus tends to zero as R → ∞. The behavior of the second integral is not immediately clear. Let us estimate this integral using that R cos θ a :

π/2 π 2 -α e tR cos θ dθ e ta π/2 π 2 -α dθ = αe ta .
As follows from Fig. 3b, α → 0 as R → ∞. Hence, the second integral tends to zero as well. So, as b → ∞, the integral of e tz /z over the segment [aib; a + ib] tends to the integral over γ ′ , which, in turn, is equal to the integral over a circle of small radius r,

1 2πi e tz z dz = 1 2πi e tre iθ re iθ rie iθ dθ = 1 2π e tre iθ dθ|. r→0 → 1 2π dθ = 1, as claimed.
Sometimes, it is convenient to have in formula (10) the integral over a line in the left half-plane. The corresponding relation is obtained from (10), if we notice that the integral of e tz /z over the boundary of the shadowed region in Fig. 5 is zero.

Therefore, the integral of e tz /z over the segment [aib; a + ib] is equal to the sum of integrals over [-aib;a + ib] , γ + , γ -and γ ′ . The integral over γ + tends to zero as b → ∞ :

γ + e tz z dz a -a |e tz | |z| dx = a -a e tx √ x 2 + b 2 dx e ta 1 b • 2a.
Similarly, integral over γ -tends to zero. Integral over γ ′ is equal to 1. Hence,

θ (t) = 1 + 1 2πi -a+i∞ -a-i∞ e tz z dz, a 0. ( 15
)
Figure 5. To the move of integration contour to the left half-plane.

It is known, that derivative of the step function is δ-function:

dθ (t) dt = δ (t) . (16)
Differentiating ( 10) over t we formally obtain a presentation of δ-function

δ (t) = 1 2πi a+i∞ a-i∞ e tz dz. (17)
This relation, strictly speaking, does not make sense, because the integral in (17) does not converge. However, δ-function is used only in the form of integrals with smooth functions. In such cases, one can write

δ (t) ϕ (t) dt = 1 2πi a+ib a-ib ϕ (t) e tz dtdz, (18)
and, if the function of z, ϕ (t) e tz dt, decays fast enough as |z| → ∞, (18) holds true.

E x e r c i s e s. 1. Let f (z) be an analytic function in a region D. Show that for a point z ∈ D and for any contour C, surrounding z,

f (z) = 1 2πi C f (ζ)dζ ζ -z .
2. Let f (z) = u(x, y) + iv(x, y) be an analytic function. Show that real and imaginary part of f , u and v, are linked by a system of equations (Cauchy-Riemann equations)

∂u ∂x = ∂v ∂y ∂u ∂y = - ∂v ∂x .
3. Derive from Cauchy-Riemann equations that u(x, y) and v(x, y) are harmonic functions, i.e.

∆u (x, y) = 0, ∆v (x, y) = 0, where ∆ is Laplace's operator

∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 .

Some facts about integrals

Gauss formula. Consider in some finite-dimensional space, R n , a quadratic form, (Au, u)

= A ij u i u j , ( 19
)
The form is assumed to be positive,

(Au, u) > 0 if u = 0.
Then the Gauss formula holds true:

Rn e -1 2 (Au,u) du = 1 √ det A . (20)
Here

det A ≡ det A ij , du = du 1 √ 2π . . . du n √ 2π .
The Gauss formula can be proved by changing the variables, u → ů,

u i = λ i j ůj , det λ i j = 1, ůj being the coordinates in which the tensor A ij is diagonal, (Au, u) = A ij λ i i ′ λ j j ′ů ′i ů′j = A 1 ů1 2 + . . . + A n (ů n ) 2 . ( 21
)
In the new variables,

Rn e -1 2 (Au,u) du = Rn e -1 2 A1(ů 1 ) 2 +...+An(ů n ) 2 dů = 1 √ A 1 . . . A n = 1 √ det A Here we used that 4 +∞ -∞ e -1 2 x 2 dx = √ 2π. ( 22
)
The Gauss formula admits the following generalization: for any linear function of u, (l, u) = l i u i ,

√ det A Rn e -1 2 (Au,u)+(l,u) du = e 1 2 (A -1 l,l) (23)
where A -1 is the inverse matrix to the matrix A. Formula (23) follows from (20) and the identity,

1 2 (Au, u) -(l, u) = 1 2 A u -A -1 l , u -A -1 l - 1 2 A -1 l, l . (24)
Plugging ( 24) in ( 23), changing the variables of integration, u → u + A -1 l, and using (20) we obtain the right hand side of (23). 4 The integral ( 22) can be found by a witty trick suggested by Poisson:

  +∞ -∞ e -1 2 x 2 dx   2 = +∞ -∞ e -1 2 x 2 dx +∞ -∞ e -1 2 y 2 dy = +∞ -∞ +∞ -∞ e -1 2 x 2 e -1 2 y 2 dxdy = +∞ -∞ +∞ -∞ e -1 2 (x 2 +y 2 ) dxdy = +∞ 0 2π 0 e -1 2 r 2 rdrdθ = 2π +∞ 0 e -1 2 r 2 d 1 2 r 2 = 2π.
E x e r c i s e. Let (Au, u) be a positive quadratic form of variables u = (u 1 , ..., u n ), and u be subject to s linear constraints, (l α , u) = c α , α = 1, ..., s; s < n.

Consider an integral,

J = Rn e -1 2 (Au,u) s α=1 δ (c α -(l α , u)) du 1 ...du n .
Show that

J = (2π) n det A (2π) s det A e -1 2 (A -1 c,c) ( 25
)
where A is the matrix with components

A αβ = A -1 l α , l β .
H i n t. Use the presentation of δ-function (17) and formula (20).

Laplace's asymptotics. Consider an integral which depends on a parameter, λ, in the following way:

I(λ) = V f (x)e λS(x) dV,
where V is a bounded region of n-dimensional space, f (x) and S(x) are some smooth functions. We wish to find the asymptotics of this integral as λ → ∞. Laplace suggested that the leading terms of the asymptotics of I(λ) are the same as that of the integral over the vicinities of the points where the function, S(x), has the maximum value. Then the asymptotics can be easily found. Indeed, let S(x) achieve its maximum value only at one point, x, this point is an internal point of V, and the matrix of the second derivatives, ∂ 2 S(x)/∂x i ∂x j is non-degenerated, i.e. its determinant, ∆, is non-zero. We can write,

I(λ) = f (x)e λS(x) V f (x) f (x)
e -λ[S(x)-S(x)] dV.

In a small vicinity of the point, x, we can replace S(x) -S(x) by the nondegenerated quadratic form,

S(x) -S(x) ≈ - 1 2 S ij (x i -xi )(x j -xj ), (26)
where S ij = ∂ 2 S(x)/∂x i ∂x j . Note that the quadratic form ( 26) is positive because x is the point of maximum of S(x). In a small vicinity of x we can replace f (x)/f (x) by unity thus obtaining

I(λ) ≈ f (x)e λS(x)
small vicinity of

x e -λ[-1 2 Sij (x i -x i )(x j -x j )] dV. (27) Since λ → ∞, the function, exp -λ[-1 2 S ij (x i -xi )(x j -xj )]
, decays very fast away from x. We do not pause to justify that the expansion of the integration region from a small vicinity of x to the entire space, R,causes only exponentially small corrections in ( 27). Thus, we can write:

I(λ) ≈ f (x)e λS(x) R e -λ[-1 2 Sij (x i -x i )(x j -x j )] dV.
The integral here, according to the Gauss formula (20), is equal to (2π) n /λ n |∆|.

Finally, the leading term of the asymptotics is

I(λ) ≈ (2π) n λ n |∆| f (x)e λS(x) . ( 28
)
As a more elaborated derivation shows, the error of the formula ( 28) is on the order of (e λS(x) /λ n/2 )/λ. If S(x) achieves its maximum at several internal points, one should sum the contributions (28) of all points. One can check that in the cases of the point of maximum lying on the boundary and/or degeneration of the quadratic form -1 2 S ij (x ixi )(x jxj ) the asymptotics remains qualitatively the same,

I(λ) ≈ prefactor(λ)e λS(x) , (29)
with the prefactor being a decaying power function of λ.

The prefactor is a constant independent on λ, if S(x) has maximum value on a set with non-zero volume. By Laplace's asymptotics we mean further the asymptotics of the form (29) where the prefactor changes slower than the exponential function of λ:

1 λ ln prefactor(λ) → 0 as λ → ∞.
Changing in the previous consideration S(x) by -S(x) we obtain the asymptotics,

V f (x)e -λS(x) dV ≈ prefactor(λ)e -λS(x) , ( 30
)
where x is the point of minimum of S(x).

In applications to the variational problems, we need also to know the asymptotics of integrals of the form (30) for complex values of λ. In this case we denote the parameter by z,

I(z) = V f (x)e zS(x) dV,
and consider the asymptotics of I(z) as |z| → ∞. Note first of all that I(z) is an analytical function of z at any finite point, z, if the integral, as we accept, converges absolutely, i.e.

V |f (x)| e RezS(x) dV < ∞.
The point z = ∞ can be, however, the singular point of I(z). Usually, the singularity is essential, the asymptotics of I(z) along different paths, z → ∞, are different. It turns out that Laplace's asymptotics,

I(z) ≈ (2π) n z n |∆| f (x)e zS(x) 1 + O 1 z , (31)
holds true for all paths, z → ∞, such that |Arg z| ≤ π/2-ε, for some small ε.

For other paths, this asymptotics does not hold. This is seen from studying the asymptotics when z → ∞ along the imaginary axis, z = iy, |y| → ∞.

It turns out that in this case the leading contribution to the asymptotics is provided by not only the point of maximum of S(x), but by all stationary points of S(x), in particularly, by all points of local maxima and minima. This asymptotics is called the stationary phase asymptotics; we do not dwell on it here since it will not be used further.

Reminder from probability theory

The random variables which we will be dealing with are the points, x, of some finite-dimensional region, V . Probability of the event that x belongs to a set A, A ⊂ V , is, by definition,

p(A) = A f (x)dx. Non-negative function f (x) is called the probability density. Since p(V ) = 1, V f (x)dx = 1.
Average value of function ϕ(x) is defined as

M ϕ = V ϕ(x)f (x)dx.
Here M stands for "mathematical expectation". This operation is often denoted by the symbol E (for expectation), but we reserve E to be the symbol of energy.

For any function ϕ, ϕ(x) is a random variable. Probability density of ϕ(x), f ϕ , is defined in terms of probability of the event {ξ ϕ(x) ξ + ∆ξ} for small ∆ξ :

f ϕ (ξ)∆ξ = Prob {ξ ϕ(x) ξ + ∆ξ} .
It is convenient to introduce the distribution function,

F (ξ) = Prob {ϕ(x) ξ} .
If the distribution function is smooth, then

f ϕ (ξ) = dF (ξ) dξ .
It is convenient to write the distribution function in terms of the step function

F (ξ) = M θ (ξ -ϕ (x)) = θ (ξ -ϕ (x)) f (x)dx. ( 32
)
This formula explains how the step function, to which we have paid already much attention, enters our consideration. Consider a random variable, which is a couple (x, y). Its probability density is a function of x and y, f (x, y). Let y take values only in some region B. What would be a probability density of x? It is natural to define probability density of x under condition that y ∈ B as

f (x) = B f (x, y) dy V B f (x, y) dxdy. Obviously, f (x)dx = 1.
In general, the conditional probability density of x depends on the choice of B. By definition the random variables x and y are statistically independent, if the conditional probability does not depend on B. This is possible only if f (x, y) is the product of two functions

f (x, y) = f (x)g(y) , V f (x)dx = 1, V g(y)dy = 1. ( 33
)
The notion of statistical independence is the central one in probability theory. All most important facts of probability theory are concerned with the sets of independent events. The major applications of probability theory are based to the possibility to identify the independent (or slightly dependent) events 5 .

We prepared everything to solve the central for applications problem of probability theory. Let x 1 , ..., x n be independent identically distributed random variables. Find probability distribution of the sum

ϕ (x 1 ) + ... + ϕ (x N )
where ϕ is a given function. Of course, we expect to get an analytical answer only in the limit of large N .

Let us find the probability distribution of the arithmetic average

E = 1 N (ϕ (x 1 ) + ... + ϕ (x N )) .
We have

F N (E) = M θ E - 1 N (ϕ (x 1 ) + ... + ϕ (x N )) .
We use for the sum the symbol E because in similar problems to be considered later, it has the meaning of energy. Using the presentation of the step function (10) we have

F N (E) = M 1 2πi a+i∞ a-∞ dz z e Ez-z 1 N ϕ(x1)-...-z 1 N ϕ(x N ) . ( 34
)
5 By the way, the recent financial crisis was caused in part by a wrong identification of independent events. The hedge fund traders believed that combining mortgage loans into large packages for sale to banks and pension funds reduces the risk of default. This is true if the defaults of individual loans were independent. However, as we have experienced, this is not always the case: there are rare catastrophic events when probabilities of defaults become strongly correlated. This is what happened in the recent economic crisis: simultaneous default of many mortgage loans along with other negative events drove the economy down, which, in turn, resulted in more mortgage defaults and bankruptcy of financial institutions.

It is convenient to change variable z by zN . Then

F N (E) = M 1 2πi a+i∞ a-i∞ dz z e N Ez-zϕ(x1)-...-zϕ(x N ) .
In such change, the constant a in the integral limit must be replaced by N a, but, since this constant is arbitrary, we keep the some notations for the integral limit. The operation of mathematical expectation is, in essence, integration. The order of integrals can be changed almost always; we do not pay attention to degenerated cases. Since the variables x 1 , ..., x N are statistically independent and identically distributed,

M e -zϕ(x1)-...-zϕ(x N ) = M e -zϕ(x) N ,
and we get

F N (E) = 1 2πi a+i∞ a-i∞ dz z e N Ez M e -zϕ(x) N = 1 2πi a+i∞ a-i∞ dz z e N Ez+N ln M e -zϕ(x) = 1 2πi a+i∞ a-i∞ dz z e N S(E,z) (35)
where we introduced a function of E and z

S (E, z) = Ez + ln Q (z) , Q (z) ≡ e -zϕ(x) f (x) dx. (36)
In physical applications, functions S has the meaning of entropy, and we will call S the entropy of the problem.

The integral (35) contains a large parameter, N . Therefore, probability distribution F N (E) for large N can be found by studying the asymptotics of the integral (35) as N → ∞. This idea is in the core of all further examples considered.

Usually, the integral in ( 35) can be differentiated over E. After differentiation we obtain formula for probability density of the normalized sum,

f N (E) = N 2πi a+i∞ a-i∞ e N S(E,z) dz. (37)
Note that the integrand, which is equal to

e N Ez e -zϕ(x) f (x) dx N , Figure 6. Plot of Q(z).
does not have singularities for finite z, and, if it decays as |Imz| → ∞, the line of integration can be moved to the left half-plane. Therefore, a in (37) can take both positive and negative values.

We consider examples in the next lecture.

The central limit theorem and the law of large numbers

We begin with the following example. Let x be a random number on the segment [0,1], which is homogeneously distributed, i.e. f (x) ≡ 1. Then Q (z) is computed analytically:

Q (z) = 1 0 e -zx dx = 1 z 1 -e -z .
Graph of Q (z) is shown in Fig. 6. Function S (E, z) is shown in Fig. 7. Function ln Q (z) is a convex function of z (we will prove it in a more general case later). Therefore, S = Ez + ln Q is also convex and has a minimum. To find the minimizer we have to solve the equation

∂S (E, z) ∂z = E + Q ′ (z) Q (z) = 0.
It can be written as

- Q ′ (z) Q (z) = 1 z - e -z 1 -e -z = E. (38)
Function -Q ′ (z) /Q (z) takes the values between 0 and 1 (again, we show this later in a general case). Therefore, for 0 E 1, equation ( 38) has a unique solution, which we denote β (E) . This notation is motivated again by physical problems: in similar physical problems β plays the role of inverse temperature. Function β (E) is shown in Fig. 8. If we tend z to zero in (38), we find

1 z - 1 -z + 1 2 z 2 1 -1 -z + 1 2 z 2 → 1 2 .
Therefore, the value of E at which β = 0, is 1/2. If E < 0, then S (E, z) has the minimum at z = +∞, if E > 1, S (E, z) has the minimum at z = -∞ (Fig. 9). Denote by S (E) the value of S (E, z) at the point of minimum over z,

S (E) = S (E, β (E)) .
The graph of function S (E) is shown in Fig. 10. E x e r c i s e. Show that

β (E) = dS(E) dE .
In the sum, x 1 + ... + x N , each member takes the values between 0 and 1. Therefore, the admissible values of the sum lie between 0 and N , while the values of E are between 0 and 1. Probability that E < 0 or E > 1 is zero. This fact can be derived directly from ( 35), ( 36), but we do not pause on this derivation and focus on the values of E from 0 to 1. We set a in (37) equal to β (E) . Formula (37) takes the form

f N (E) = N 2π +∞ -∞ e N S(E,β+iy) dy (39)
S (E, z) = Ez + ln 1e -z z .

Since f N (E) is real, the imaginary part of the integral (39) is zero. Consider function S (E, β + iy) for small y. Since ∂S (E, z) /∂z = 0 at y = 0, we have

S (E, z) = S (E) + 1 2 S ′′ (E) (iy) 2 = S (E) - 1 2 S ′′ (E) y 2 , (40)
S ′′ (E) ≡ ∂ 2 S (E, z) ∂z 2 z=β(E) = ∂ 2 ln Q (z) ∂z 2 z=β(E)
.

Function S (E, z) is strictly convex on real axis, thus S ′′ (E) > 0, and S (E, β + iy) has a local maximum on y-axis at y = 0. For finite y, ReS (E, β + iy) decays. Indeed, ln 1e -β-iy β + iy = ln 1e -β-iy

β 2 + y 2
= ln e -β e βe -iyln β 2 + y 2 .

The first term here is bounded, while the second one goes to -∞. So, the major contribution to this integral as N → ∞ is provided by a vicinity of the point y = 0. Replacing S (E, z) by ( 40) we have E) .

f N (E) = e N S(E) N 2π +∞ -∞ e -N 2 S ′′ (E)y 2 dy = N 2πS ′′ (E) e N S(
(41)

Here we made the change of variable y → y √ N S ′′ and used (22). Formula ( 41) is an asymptotic formula as N → ∞. The normalization condition

+∞ -∞ f N (E) dE = 1 0 f N (E) dE = 1 (42)
is satisfied asymptotically. Indeed, in the integral

1 -1 1 S ′′ (E) e N S(E) dE
the major contribution is provided by the vicinity of the point of maximum of S (E) , which is Ê = 1 2 . At this point S = 0; thus

1 -1 1 S ′′ (E) e N S(E) dE ≃ +∞ -∞ 1 S ′′ (E) e 1 2 N S EE ( Ê)(E-Ê) 2 dE. (43)
Here we denoted by S EE (E) the second derivative of S(E) :

S EE (E) = d 2 S(E) dE 2 .
The derivative S EE (E) is negative, because S(E) is a concave function. For the integral (43) we have in the leading approximation,

2π/N -S EE ( Ê) 1 S ′′ Ê . ( 44
)
From ( 38) and ( 40)

S ′′ Ê = d 2 S (E, z) dz 2 z=β( Ê) = d 2 ln Q dz 2 z=β( Ê) = - dE dβ z=β( Ê) .
Hence

S ′′ Ê • S EE ( Ê) = S ′′ Ê dβ dE Ê = -1. (45)
Combining ( 44), ( 45) and ( 41) we obtain (42).

Our asymptotic result converges to the exact one very fast. For N = 10, the exact and asymptotic results are shown in Fig. 11.

For N = 100 the exact and asymptotic results are practically indistinguishable.

As N increases, probability density converges to δ-function, concentrated at the point E = 1 2 . This value, E = 1 2 , is the mathematical expectation of each of the members of the sum. We obtain the law of large numbers:

1 N (x 1 + ... + x N ) → M x = 1 2 as N → ∞. (46)
Equation ( 46) can be interpreted in the following way. Let we do some experiments and the outcome of the experiment is a number, x, 0 x 1. All values of x on the segment [0, 1] are equiprobable. The outcomes of the experiments are independent. Then the arithmetic average of all outcomes for large N is approximately equal to the mathematical expectation of x. It is interesting to characterize the deviations of the arithmetic average from the mathematical expectation,

1 N (x 1 + ... + x N ) -M x = E ′ .
To estimate the order of deviations, let us find the mathematical expectation of E ′ 2 . We can do that using the probability density of E (41). From (41) we have

M E ′ 2 = M E - 1 2 2 = 1 0 E - 1 2 2 N 2πS ′′ (E) e N S(E) dE = N 2πS ′′ 1 2 +∞ -∞ E ′ 2 e N S EE ( 1 2 )E ′2 dE ′ = 1 N S ′′ 1 2 √ 2π +∞ -∞
x e -1 2 x 2 dx.

Here we used that S(E) has maximum at E = 1 2 and equal to zero at this point, besides, equation ( 45) was also employed.

We see that M E ′ 2 is of the order 1/N. Hence, E ′ is of the order 1/ √ N . This suggests that the scaled deviations,

ξ = √ N E ′ ,
are of the order of unity and can have a non-singular probability distribution. Let us find it. Denote the probability density function of ξ by g(ξ).

Since,

E = 1 2 + ξ √ N and f N (E) dE = g(ξ)dξ,
we have

g(ξ) = 1 √ N f N 1 2 + ξ √ N .
Plugging here (41) we find for large N g

(ξ) = 1 2πS ′′ 1 2 e 1 2 S EE ( 1 2 )ξ 2 .
Denoting S ′′ 1 2 by σ 2 and using (45) we obtain

g(ξ) = 1 √ 2πσ e -1 2σ 2 ξ 2 . ( 47
)
This is Gaussian distribution. The constant σ has the meaning of variance of ξ, M ξ 2 = σ 2 .

Formula (47) expresses the so-called central limit theorem. The law of large numbers and the central limit theorem are simple consequences of (41). Formula (41) provides much more information: it determines the probability of large deviations of the arithmetical average from the average value. This probability is exponentially small because S(E) is negative.

E x e r c i s e s. 1. Let u a be non-negative numbers, a = 1, ..., N . All points of the space {u 1 , ..., u N } are equiprobable. Note that we cannot introduce probability in a usual sense because the volume of the admissible values is infinite. Let u a be constrained by the condition

1 N N a=1 u a = 1.
This condition makes the volume of the admissible values finite. Show that in the limit N → ∞ the values of any two numbers (say, u 1 and u 2 ) become statistically independent, and each number has the probability distribution

f (u) = e -u .
2. Let a 1 , ..., a n , ... be an infinite sequence of numbers and x 1 , ..., x n , ... a sequence of independent identically distributed variables with mathematical expectation M x, variance M x 2 and probability density function f (x). Consider a random number ξ = a 1 x 1 + ... + a n x n + ...

show that M ξ = M x ∞ k=1 a k M ξ 2 = M x 2 ∞ k=1 a 2 k
while the probability density function of ξ, f ξ (y) is given by the integral

f ξ (y) = 1 2πi a+i∞ a-i∞ e zy-g(y) dz, g (y) = ∞ k=1 ln e -ya k x f (x) dx.
It is assumed that the sums and integrals in these formulas exist. A necessary condition for convergence of series is the decay a k as k → ∞.

Poisson distribution

Now we return to a more general case we have started with, when x was a point of some region distributed with some probability density f (x). As we will see all results we obtained for numbers are extended to this case. We have obtained already the formulas for probability density (37), (36). Consider the properties of S (E, z) on real axis. This function is a convex function of z. Indeed,

∂S (E, z) ∂z = E - ϕe -zϕ f dx e -zϕ f dx (48) ∂ 2 S (E, z) ∂z 2 = ϕ 2 e -zϕ f dx e -zϕ f dx - ϕe -zϕ f dx 2 e -zϕ f dx 2 . ( 49
)
Using Cauchy inequality,

f • gdx 2 f 2 dx g 2 dx,
we have

ϕe -zϕ f dx 2 = ϕe -1 2 zϕ f 1 2 • e -1 2 zϕ f 1 2 dx 2
ϕ 2 e -zϕ f dx e -zϕ f dx.

Therefore, ∂ 2 S (E, z) ∂z 2 0, and S (E, z) is a convex function of z. Hence, it may have only one local minimum at a finite z. It may have also minimum at z = +∞ or z = -∞.

Consider the case when minimum is achieved at a finite point, ž. According to (48), z is the solution of the equation

ϕe -zϕ f dx e -zϕ f dx = E. (50)
We assume that function ϕ (x) is piecewise continuous and bounded in V and has the minimum and maximum values, ϕ -and ϕ + . Since ϕ -e -zϕ f dx ϕe -zϕ f dx ϕ + e -zϕ f dx, the left hand side of ( 50) is within the limits

ϕ - ϕe -zϕ f dx e -zϕ f dx ϕ + .
Therefore, the solution of equation ( 50) exists only for the values of E belonging to the segment ϕ -E ϕ + .

According to Laplace asymptotics, the left hand side of ( 50) tends to ϕ -as z → +∞ and ϕ + as z → -∞. So, the picture is completely similar to that of the case of random numbers. Consider one special case which has a lot of applications. Let us choose ϕ (x) to be a characteristic function of some subregion B of volume V , i.e.

We replace N by ∞ in notation of distribution function because the right hand side of (53) is the limit as N → ∞. One can show that (53) can be differentiated over k. We get for probability density (55), ( 56). The minimum value of I(u) is always negative; in 1D case this is seen from Fig. 13. In physical problems the minimum value has the meaning of negative energy of the system.

dF ∞ (k) dk = f ∞ (k) = e -
If the properties of the system are random, so are the matrix (operator) A = (A ij ) and the vector (linear functional) l = (l i ) . We consider the simplest "probabilistic" question: What is the probability distribution of minimum values of I(u) (i.e. probability distribution of energy)? It is enough to discuss the finite-dimensional case; the results for continuum mechanics are obtained in the limit n → ∞.

If the matrix A does not depend on the event, ω, the problem is called weakly stochastic, otherwise the variational problem is called strongly stochastic. Many physical theories provide examples of such type of problems. The analytical results can be obtained mostly for weakly stochastic problems.

As is usual in the probabilistic approach, the probabilistic modeling is especially effective, if one can identify in the phenomenon to be modeled the statistically independent (or slightly dependent) events. Analytical investigation can be advanced considerably, if there are many statistically events involved. We focus here on a special case when (l, u) is a sum of small independent linear functionals. More precisely, there is a large number, N, of independent identically distributed random variables, r 1 , ..., r N , and a given random linear functional, (l 0 (r), u). Then the linear functional of the variational problem, (l, u) , is defined as an "empirical average" of N values of (l 0 (r), u):

Figure 3 .

 3 Figure 3. To justification of formula (10).

Figure 7 .

 7 Figure 7. Plot of ln Q(z), Ez and S(E, z) for E = 0.3.

Figure 8 .

 8 Figure 8. Dependence of inverse temperature β on energy E.

Figure 9 .

 9 Figure 9. A qualitative graph of S(E, z) as a function of z for E > 1 and E < 0.

Figure 10 .

 10 Figure 10. Dependence of S(E) on E.

Figure 11 .Figure 12 .

 1112 Figure11. Exact probability density of E compared with the asymptotic formula when N = 10; the exact and asymptotic results are hardly distin-

ϕF.

  (x) = {.1 if x ∈ B0 otherwise. Then the sum, N = ϕ (x 1 ) + ... + ϕ (x N ) , has the meaning of the number of points which are in the region B. This number is random and takes the values 0, 1, ...N. We wish to find the probabilities that N has values 0, 1, .... To this end, we have to rewrite formula (35) in terms of probability distribution of the non-scaled sum, N = N E. Denotes its values by k. Repeating the derivation from (34) to (35) we have |B| + |B| e -z . (52)Note that region B is not necessarily simply connected and may consist of many pieces, but formula (52) contains only the volume of region B. We consider the "thermodynamic limit", when |V | → ∞, N → ∞, while the number of points per unit volume,In the limit N → ∞ e -zϕ(x) f (x) dx N = e -|B|n(1-e -z ) . |B|n(1-e -z ) .

  This integral can be computed analytically. Indeed, let us present exp (|B| ne -z ) as the series Figure 13. Minimum value of I(u) is negative; it has the meaning of negative energy.

						|B|n 1 2πi	a+i∞ e kz+|B|ne -z dz.
							a-i∞
	e |B|ne -z = 1 + |B| ne -z +	1 2	(|B| n)	2 e -2z + ...	1 s!	(|B| n)
	According to (17),
	1 2πi	e kz 1 s!	(|B| n)	s e -sz dz = δ (k -s)	1 s!	(|B| n)

s e -sz + ... s .

A region is simply-connected if any closed contour can be shrunk by a continuous transformation to a point.

So,

i.e. f ∞ (k) is a sum of δ-functions, concentrated at the points k = 0, 1, 2.... This means that the region B contains k points with probability

The sum of all probabilities is equal to 1,

as follows from Taylor expansion of exponential function

We arrived at the so-called Poisson distribution.

Stochastic variational problems

Many problems of physics and mechanics can be formulated as variational problems, i.e. as problems of minimization of some functional, I(u), on a set of elements, u. We will consider the simplest case, when I(u) is a quadratic functional, i.e. the functional of the form

By u one can mean a point of a multidimensional space, u = u 1 , ..., u n , (l, u) a linear function

and (Au, u) a quadratic function

In continuum mechanics problems, one considers the limit n → ∞, but a finite-dimensional truncation of continuum mechanics problems (for example, by the finite-element method) returns us to the finite-dimensional case

Consider a quadratic function of a finite number of variables (54). The minimum value of this function is

where A -1 is the inverse matrix for the matrix A. We wish to find probability distribution of energy,

Following the same path as for a sum of independent random variables we plug in (59) the presentation of δ-function (17),

It would be desirable to change somehow the operations of mathematical expectation and minimization: then we would arrive to some deterministic problem. This can be achieved by presenting exp [-z min I(u)] by an integral of some function of u over u. Since mathematical expectation is, in fact, also integration, the order of integrals can be changed, and we obtain an integral of the mathematical expectation of the function of u, which can be found explicitly in some cases. Now let us discuss precisely what this trick means.

According to (58) formula ( 23) can be also written as

We see that this relation reduces the computation of the minimum value to integration indeed. Since any quadratic functional in variational problems of continuum mechanics admits a finite-dimensional truncation, one can write formula (61) for a finite-dimensional truncation, and then consider the limit when the dimension of the truncation tends to infinity. In the limit, in the right hand side of ( 61) we obtain what is called the functional integral. We include √ det A in the definition of the "volume element" in the functional space, The notation, D A u, emphasizes that the volume element depends on the operator A. E x e r c i s e. Consider a variational problem with a set of constraints: minimize a quadratic functional,

on all u obeying to linear constraints

Show that

and A is a matrix with the components

H i n t. Use (25). In Section 5.12 of the above-cited book Variational Principles of Continuum Mechanics one can find various generalizations of (63) involving non-positive quadratic functionals and complex-valued functionals. We illustrate the idea using one of such generalizations,

If we change variable z → N z, then probability density f (E) (60) takes the form

where S(E, z, u) is a function that is independent on N, S(E, z, u) = Ez + z 2 (Au, u) + ln M e -z(l0,u) .

(66)

The functional integral (65) depends on a large parameter N. In many cases the asymptotics of this integral can be studied. It is determined by the stationary points of the entropy functional (66). Examples and further details can be found in the above-cited book.


Julien Rebotier
email: -julien.rebotier@univ--pau.fr

Understanding urban risks as encompassing socio-ecological phenomena. Territorializing landslides in Caracas and Quito

.

Without entering into the debate between realism and constructivism in environmental studies [START_REF] Demeritt | What is the 'social construction of nature'? A typology and sympathetic critique[END_REF][START_REF] Forsyth | Critical political ecology. The politics of environmental science[END_REF], we would like to propose an encompassing, dynamic and place--based framework that allows accounting for the different drivers of risk construction (both material and intangible, as well as context--sensitive). We do not look for a systematic and exhaustive grid for reading environmental situations. The main objective is not to address the entire and complex set of causal factors that shape risks. It rather consists in setting an approach broad enough in order to bear in mind the very different factors that influence environmental issues as critical environmental challenges in our present world. Such encompassing framework starts from the importance given to contexts and politics. It aims to articulate integrative approaches that have been proposed previously [START_REF] Thouret | Vulnérabilité aux risques naturels en milieu urbain : effets, facteurs et réponses sociales[END_REF]Wisner, Blaikie et al. 2004[START_REF]At Risk, natural hazards, people's vulnerability and disasters[END_REF]; D'Ercole and [START_REF] D'ercole | La vulnérabilité territoriale: une nouvelle approche des risques en milieu urbain[END_REF], putting the stress on structural framings (time--, space-- and society-- sensitivity), as well as on the epistemology of risks and the environment (representation, discourses and narratives) as they shape risk situations and its implications on people and places.

In that perspective, in order to highlight contexts and political dimensions in assessing risks, the core proposal relies on the idea of territory as it is defined in French social geography [START_REF] Di Méo | L'Homme, la Société, l'Espace[END_REF][START_REF] Jean | Lire les territoires[END_REF][START_REF] Di Méo | L'espace social[END_REF]. Territory is defined as a socio--spatial construction, both material and intangible, and historically identified and appropriated through practices and representations. It is a never--end process of interactions among societies, and that links societies to places. This way, it is pretty much different from Sachs's acceptation of territory and territoriality [START_REF] Sack | Human territoriality: its theory and history[END_REF]. It rather gives room to the work of articulation and situation that is critical at the time to assess (even part of) risk situations in a given place, time, and social context. Putting forward the social and political dimensions of contexts and articulating scales through the notion of places, territories or "terroirs" acknowledges connections between epistemic communities in social science perspectives (for instance it is obvious in western Africa: [START_REF] Bassett | Constructing locality: The terroir approach in West Africa[END_REF]). The territorialization of risks seems to be an accurate notion to address risks as a component of the social production of space, and a contribution from one of the epistemic communities to the challenging social science approach of risk assessment.

The following presentation is based on two PhD fieldworks where landslides, its identification, management and instrumentalization, have been studied. The first one concerns Caracas and is part of a broader research on urban risks [START_REF] Rebotier | Les territorialités du risque urbain à Caracas. Les implications d'un construit socio--spatial dans une métropole d'Amérique latine[END_REF], the other one relates with Quito's ravines and the politics of risk management [START_REF] Sierra | Gestion et enjeux des espaces urbains à risques d'origine naturelle. Les versants et les quebradas de Quito, Equateur[END_REF]. In these cases, the application of the territorialization of urban landslides puts the stress on the epistemology of risks and on the importance of their conceptualization by different actors in shaping the way they are handled. The first part sheds light on the ontology of risks and its political economic drivers. It shows how political and economic reasons did matter in producing landslides, mainly in Caracas. On the basis of such structural framework, the second part puts forward the importance of the epistemology of risks. The ways risks are conceptualized are critical in framing priorities of intervention and management. In addition, as risks are not only a result of humanizing the environment, but also part of the process of human -environment interaction, risks have critical implications that are accounted for in the third part. Finally, a model is proposed in order to articulate epistemology, narratives and discourses to other critical drivers that produce risk situations and vulnerability. The objective of the model is to acknowledge the recognition and assessment of the multiple factors involved in the territorialization of risk, and to take their respective importance into account. Such framework is consistent with a politically--, socially-- and culturally--contingent approach of risks. It underlines also the close relationship between epistemology and ontology by integrating social and human systems into human -environment interactions as the core research object for risk assessment. This way, the territorialization of risk as an encompassing approach of risks is part of a broader proposal for improving a situated and politicized assessment of current environmental challenges from a social science point of view.

The political economy of landslides in Caracas

In Caracas, it is hard to discriminate the distribution of landslides according to social stratification. The distribution rather depends on biophysical mechanisms, and above all, on the kind of land occupancy. Actually, residential settlements on landslides--prone areas do not concern only poor people (Figure 1). Landslides also affect wealthy urban sectors. The following assessment is focused on them. Landslides are far to be a marker of poverty or even of "social weakness". The drivers of vulnerability to landslides in wealthy sectors are embodied in political, economic and institutional dimensions of Caracas urbanization, which is a form of territorialization (i.e. it is a particular way for society to interact with the environment in a place). Let us figure out three characteristics of the urbanization that have been influencing risks of landslides up to present days.

The construction of Santa Monica residential area (sector 1 on figure 1, picture 1) has been allowed in the early 1970s through special urban ordinances. Regarding urban planning in Caracas, the use of "special ordinances" is the normal way for public authorities to regulate the urbanization since the 1960s. In spite of the unstable slopes geoscientists worried about, local urban planning services were used to deliver special permits for urbanizing such or such risky areas, for political or economic reasons. Intents of coordination of the urbanization definitely fail at the end of the 1990, when the OMPU -Oficina Metropolitana de Planificación Urbana (Metropolitan Office for Urban Planning) disappears. Santa Monica is not an invasion. It is rather a formal middle--class sector, but a residential sector that is actually barely legal.

Picture 1: Santa Monica residential area at the bottom of Bello Monte sector, on unstable slopes (Source: Noticias 24).

Landslides are also the results of market rules in the construction sector. Because of the hills, the site of Caracas needs excavations. The few flat lands are completely occupied during the 1970. By law, the material excavated ought to be evacuated out of the construction site. But most of the time, urban developers only push the material away, on the edge of the tops of the hills that have been flattened. In the case of Santa Inés, a middle--class residential sector, landslides are due to the material excavated and tipped on the surrounding slopes (Picture 2). Such practices mean more meter squares to sell for promoters in a context of land shortage. But some embankments of friable material are so thick that they cannot even be anchored in the bedrock. The pillars cannot be longer than 30 meters, because of technical restrictions due to the iron inside (Picture 3). Some embankments are thicker than this, causing problems for houses at the top and at the bottom of the slope. In addition, it is worth mentioning that still because of economic constraints, even wealthy sectors are exposed to bad quality constructions. In many cases, the sewage disposal of individual houses leaks, putting more weight on the friable slopes and worsening their susceptibility to slide down.

Finally, during the period of urbanization of the Caracas valley, biophysical mechanisms have been neglected, following economic, political or ideological logics. Humanenvironment interactions have not been accounted for [START_REF] Rebotier | Les territorialités du risque urbain à Caracas. Les implications d'un construit socio--spatial dans une métropole d'Amérique latine[END_REF]. Indeed, in the mid--1980, the huge interventions on urban landscape from the 1950s on, through excavation and embankments, represented more than 20 times the volume of material involved in pre--Hispanic landslides and debris--flows in the valley of Caracas [START_REF] Singer | Geomorfología, arqueología y aspectos de geología reciente[END_REF]. In that perspective, the underlying conception of risks and the environment appears to be critical with regards to the kind of urbanization at stake and to the manufacturing of vulnerability.

The framings of landslides, its identification and management

The framing of risks, its representation and management refer to principals and assumptions underlying social order, political debates and action [START_REF] Forsyth | Critical political ecology. The politics of environmental science[END_REF]. Assessing the different ways to manage landslides allows alternative rationalities among different actors to be put forward. These differences appear to have concrete implications on the regulation of urbanization, on socio--spatial order, or even on the urban landscape. The framing of risks and environment are part of humanizing the environment. The following examples that show such importance come from different urban scales, and concern diverse actors in Caracas and Quito.

The metropolitan district of Caracas (DMC) is made of 5 municipalities that have autonomy in terms of urbanism. Each municipality relies on its own Civil Defense and risk management services. Within the DMC, different ways of managing landslides are working - or not working - side--by--side. Coordination between municipal scales of risk management remains really hard to achieve. Chacao and Baruta municipalities have the most advanced systems of risk monitoring and assessment, but they work differently (Figure 2), and the other municipalities, like Libertador, are barely equipped with services of crisis management [START_REF] Rebotier | Les territorialités du risque urbain à Caracas. Les implications d'un construit socio--spatial dans une métropole d'Amérique latine[END_REF]. At the scale of the metropolitan district, the Japanese Cooperation (JICA) has been charged by public authorities to assess urban landslides in Caracas, but only in the northern part of the agglomeration, because of political constraints and arrangements (Figure 3). Obviously, the boundaries of the research area are politically dependent. In addition, the Japanese cooperation has focused on the biophysical diagnostic, sub-contracting local NGOs to assess the social vulnerability of landslides in so far as JICA has no particular skills in these society--related aspects of risks.

Finally, it is worth highlighting that the coordination between public institutions at different scales can be highly controversial because of critical discrepancies in viewing risks. Indeed, in 2006, the ministry of housing has built collective housing units as part of a social program in Plan de Manzano, western Caracas. But the sector had been classified as high--risk area by metropolitan Civil Defense only weeks before. Risk-- and environment--related priorities and conceptions are far to be equally shared among state institutions, giving rise to an important weakness at the time to regulate urbanization (and contribute to make it safer). The city of Quito - Ecuador, is also located in an Andean valley, at around 2800 meters high, at the bottom of important volcanoes. Like in Caracas, the slopes of the gullies (ravines) are landslides--prone areas, but they have different meanings according to viewpoints, periods and actors at stake. As for traditional belief, the gullies are bewitched, and people traditionally avoided them [START_REF] Sierra | Gestion et enjeux des espaces urbains à risques d'origine naturelle. Les versants et les quebradas de Quito, Equateur[END_REF]. But in modern thoughts that go together with the intensive urbanization of the valley, the gullies are seen first as a problem with regards to hygienist statements, in the 19 th century. Latter on, during the 20 th century, they are supposed to bring flood risk for the city, obstacles to transportation, and problems related with delinquency [START_REF] Metzger | Gestion de l'environnement urbain et risques "naturels". La problématique des quebradas à Quito[END_REF]. For all these reasons the ravines have been filled in over the past century (Figure 5). The wealthy northwestern parts of the city are particularly concerned by the transformation of the landscape in spite of the worried geoscientists [START_REF] Rode | Marges urbaines à Tours et à Quito. Essai de comparaison Nord --Sud[END_REF].

It is really significant to see how different actors identify and manage risk and the environment unevenly over time, according to different logics that cannot be reduced to supposedly neutral biophysical mechanisms. In these cases, one of the root causes of vulnerability is epistemological. Indeed, the ways to consider risks are not only markers of a context, different actors, periods or social order. They have also concrete implications for risk situations as they contribute to shape territories, and thus the distribution of vulnerability and its production.

Viewing urban landslides as drivers of urbanization

Particularly in times of crisis, risks can be used as a hegemonic category that shapes urbanization, enforces political choices, public policies, or leads to defend particular interests. Different conceptions of landslides do not only highlight social groupings and discrepancies giving rise to diverse risk framings. They are also performative and they contribute to shape the urbanization in Caracas and Quito, making risk and its management critical drivers and consequential processes, not only outcomes.

In Caracas, multiple landslides caused several deaths in December 2010. The crisis has been the opportunity for President Chavez to strengthen the new orientations of the government's housing policy (Figure 4). As for President Chavez, poor people stricken by landslides should find shelters, and then permanent housing solutions among the many vacant houses and apartments in the city, particularly in wealthy sectors. In a period of environmental crisis, landslides were a way to enforce the coming law on vacant lands and buildings. Indeed, since the adoption of an organic law of emergency for lands and housing at the end of January 2011, public authorities can take over private housing units under specific circumstances, buy them, and reassign them according to the priorities of the moment. The state of emergency due to "natural hazards" is one of these circumstances. But landslides do not always strike poor people, rich people are not always safe, and managing a landslide crisis cannot be reduced to addressing housing issues. The problems remain the same and the landslide crisis is instrumentalized. In Quito, the powerful local company of water supply and sewage plays a critical role in the urbanization of the capital city. The gullies are a suited place to install sewer pipes, causing nevertheless many troubles in terms of floods and water drainage. As a consequence of recurrent floods in Quito in the late 1990s, public authorities have chosen the wealthy northwestern part of the city as a priority area of intervention in spite of similar risk conditions elsewhere in the city [START_REF] Sierra | Gestion et enjeux des espaces urbains à risques d'origine naturelle. Les versants et les quebradas de Quito, Equateur[END_REF], reinforcing this way a selective representation of risks. Indeed, during a yellow alert before a potential volcanic eruption in 1998, the most cited urban sectors in the media regarding risky situations are clearly concerning the wealthy northwestern part of the city (Figure 6). By framing this way problems of urban floods and landslides, hegemonic discourses also shed light on a hypothetical responsibility of the invasions of the slopes on top of wealthy northwestern sectors. Actually, it is well known that the combination of intensive rains, high slopes, and friable material are the main criteria that cause such environmental accidents [START_REF] Rode | Marges urbaines à Tours et à Quito. Essai de comparaison Nord --Sud[END_REF]. But a link is made between the invasion of slopes by marginal people on the one hand, and triggering--off mechanisms of floods and debris--flows on the other hand. At the end, selective risk identification is closely linked with selective risk management. Both of them are socially contingent, strongly sensitive to the unequal urban order, and they both shape Quito's urbanization. In addition, such discriminated risk--oriented interventions in Quito are in line with an environmental conservation program of the slopes of the Pichincha volcano, with view to prevent deforestation. Above a line that limits the altitude of the urbanization in the neighborhood of the wealthy sectors, it is forbidden to build anything inside a national park that has been created following "green belt" ideas in the mid--1970s [START_REF] Sierra | Gestion et enjeux des espaces urbains à risques d'origine naturelle. Les versants et les quebradas de Quito, Equateur[END_REF]. Previous environmental arguments of the 1970s have been substituted today by risk--(and climate--?) related priorities in enabling dominant discourses as levers for concrete -and selective -interventions or regulations.

Across the instrumentalization of the environment--and risk--related discourses in Quito, three points that have already been mentioned allows putting the stress on the contingency of the current rationality in managing risks of landslides or debris--flows, and its potential intentionality: 1--nothing is said about the way the sewage company handles the intervention on ravines nor about the relation between the gullies and the water disposal, in spite of the crucial implication of sewage equipment in triggering off or lessening floods; 2--in the 1970s the reasons to keep "green" the slopes of the Pichincha above the wealthy sectors were ecology and environment, not risk of landslides and debris--flows. At last, the problem framing is different, but the target remains the same: keeping invaders and poor people away from wealthy sectors; and 3-nothing much is done elsewhere in the city, where vulnerability to landslides is critical, though mostly misrecognized. From the 1960s on, urbanization is heading south, and the ravines are also filled in, causing similar troubles in a more popular part of the city (Figure 5).

Most of the time

, what is at stake in identifying and managing risks cannot be reduced to supposedly neutral biophysical mechanisms. It is highly political and context--sensitive. Epistemological framings are one of the many drivers that contribute to the production of the humanized environment by allocating blames and priorities within society, and by performing discriminated social relations. Territorializing risks, as a framework for risk assessment, allows considering the different -and sometimes contested -ways of viewing risks and the environment, as well as its implications.

Territorializing landslides and viewing risks as encompassing socio-ecological phenomena

In a constructivist view, the assessment of vulnerability to environment--related risks focuses on the complex set of causal--factors. Biophysical mechanisms are only one of the many drivers that contribute to shape concrete situations identified as risks. But risks are social productions and need to be politicized. Thus it is worth analyzing the epistemology of risks in addition with its ontology. The epistemology of risk corresponds to the ways of viewing it, its framing, its conceptual boundaries, the language, metaphors or dominant explanations that are used to qualify it, or even the recognition of discriminated responsibilities. Considering the epistemology of risks leads to considering them as place--, time-- and society--contingent. They are part of the process of humanizing the environment, as an outcome, but also as a driver. They contribute to shape landscape, spatiality, land uses, or mobility. And the way to view risks and to conceptualize them is critical in the process of territorialization, i.e. in producing and making sense of situated human -environment interactions.

Territorializing risks accounts for both ontological and epistemological production of the humanized environment. Such process is rooted in a socio--spatial context. Time, space and society frame the many interactions and mechanisms at stake. It articulates social and natural systems. In the case of landslides, biophysical mechanisms do matter, but political economic drivers leading to the production of risks (Part 1), or even the ways they are identified by different actors (Part 2) are critical to understand the causes of landslides as socio--environmental problems in unequal urban contexts, as well as its role in humanizing the environment (in that case, in shaping urbanization -Part 3).

The model of territorializing risks allows articulating different kinds of explanations and causal mechanisms that shape the humanized environment (Figure 7). The main objective of such model is not to systematically identify the many drivers of environmental situations (it would be a never--ending crossway!). It is rather to recognize the contingencies of environment--and risk--related issues, to acknowledge their interactions (social framing of environmental problems does influence their assessment), and to put the stress on different entry point (according to the situation at stake, to the information available or to the skills of the researcher) without despising the other causal factors involved in the production of the humanized environment.

Discussion of the territorialization of risks as an encompassing approach

Territorializing risk offers a framework for considering the large scope of risk drivers that are involved in human -environment interactions. These interactions are deeply rooted in time, in a place and in society (for instance, Modern paradigms that define the relation to Nature are strongly framing the humanized environment in specific places today). The "humanized environment", with its context, its ontological as well as its epistemological aspects, constitutes the core research object in assessing risk and the environment in terms of political ecology. The territorialization offers a politicized and encompassing framework to account for risk as a co--production (co--evolving with political or economic logics and interests) and its implication in a broader production of space. The discriminated ways of viewing risks and conceptualizing them are only one of the many drivers involved in the production of the humanized environment. Indeed, even if the social framing of landslides as urban environmental issues has concrete implications, biophysical mechanisms must not be neglected as they also contribute critically in shaping urban risks of landslides.

Obviously, other entry points -than epistemology, problem framings and instrumentalization -can be chosen in risk and vulnerability assessment to address environmental issues. Still, the framework of the territorialization of risks remains a way to bear the complex set of drivers in mind without necessarily addressing all of them. Indeed, whether the researcher is more sensitive to the cultural school of landscape, to political economic analysis, to analysis of discourses, biophysical or ecological mechanisms or to the construction of knowledge, different entry points will be chosen at the time to assess risk. Whatever the entry point can be, it is important to bear in mind that risk is part of the human -environment interactions as an outcome and as a determinant. Many different risk drivers do matter, though unevenly. The difficulty consists in prioritizing and putting weight on these different causal factors.

Finally, as parts of a broader production of humanized environment, risks are also a window of opportunities [START_REF] Pelling | Adaptation to climate change. From resilience to transformation[END_REF]. The framework of the territorialization allows envisioning risks as a lever for social transformation putting the stress on socially rooted logics though without neglecting biophysical mechanisms. A further step would

Figure 1 :

 1 Figure 1: Distribution of landslides (in blue) and barrios -invasions (in yellow) in Caracas agglomeration. Wealthy sectors (examples in squares 1, 2 and 3) are also affected by landslides (Source: Rebotier according to JICA 2004).

Picture 2 :

 2 Landslide threatening Santa Inés residential sector, at the bottom of the slope (Source: Picture of the author, November 2008). Picture 3: Intent of stabilization of a sliding slope surrounding the sector of Santa Inés. Pillars reinforced with iron are anchored in the bedrock (Source: Picture of the author, November 2008).

Figure 2 :

 2 Figure 2: Risk assessment follows municipal boundaries in Caracas agglomeration (Source: Rebotier 2008, and web pages of municipal risk management services of Baruta and Chacao)

Figure 3 :

 3 Figure 3: Caracas urban area assessed by the Japanese cooperation (Source: JICA 2004)

Figure 4 :

 4 Figure 4: Screenshot of the online version of El Universal (among the main national newspapers) after important landslides and debris--flows in eastern Caracas: « Chavez: we are going to take over abandoned buildings » (Source : El Universal web page -http://www.eluniversal.com/2010/12/05/pol_video_chavez:--vamos--a--toma_05A4818413.shtml).

Figure 5 :

 5 Figure 5: Historical stages of urbanization and filling of the ravines in Quito -From North to South. The darker are the sectors and the ravines, the earlier they have been occupied or filled in (Source: Metzger and Peltre 1996).

Figure 6 :

 6 Figure 6: Sectors « at risk » according to the number of citations in national press in Ecuador during the period of yellow alert of Pichincha eruption in 1998. The darker is the color, the more cited is the sector (Source: Rode and Sierra 2008).

Figure 7 :

 7 Figure 7: A model of territorialization of risk putting human -environment interactions at the center of the assessment.