Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

Подождите немного. Документ загружается.

346 14 Mechanical Systems with Random Perturbations

By Lemma 6.28, since all Γ

(t, x(·)) and Γ ˜a

(t, x(·)) satisfy (14.21) with

the same Q, one can show that the set of measures {μ

} is weakly compact and

so there exists a subsequence converging weakly to some probability measure

μ on (

Ω,

F). For the sake of convenience we use the same notation and say

that μ

itself is the converging subsequence. Denote by v(t) the coordinate

process on (

Ω,

F,μ).

The fact that the process v(t) −



f(s, v(s))ds is a martingale on

(

Ω,

F,μ) with respect to

is proved by analogy with Lemma 8.47.

Choose an orthonormal basis in T

M. Then the vectors in T

M are

considered as coordinate columns. If X is such a vector, the transposed row

vector is denoted by X

∗

. Note that for a column X and a row Y

∗

the product

∗

with respect to matrix multiplication is a matrix. Linear operators from

M to T

M are represented in coordinates as n ×n matrices, where the

symbol ∗ denotes conjugate transposition.

Consider the sequence a

(t, m, X)ofε

-approximations of A(t, m, X)that

point-wise converges to the Borel-measurable selector a(t, m, X)(seethebe-

ginning of this proof). One can easily see that a

(t, m, X)(a

(t, m, X))

∗

point-

wise converges to a(t, m, X)(a(t, m, X))

∗

. Then by analogy with the proof of

Lemma 8.48 one can show that

v(t)v(t)

∗

−



Γ ˜a(s, x(·))(Γ ˜a(s, x(·)))

∗

is a martingale on (

Ω,

F,μ) with respect to

P and so



(x(t + Δt) − x(t))(x(t + Δt) − x(t))

∗

−

t+Δt



Γ ˜a(s, x(·))(Γ ˜a(s, x(·)))

∗

ds |



=0. (14.23)

Using the standard Girsanov technique one can derive from the above

arguments that on (

Ω,

F,μ) there exists a Wiener process w(t), adapted to

, such that v(t)on(

Ω,F,μ) satisﬁes the equality

v(t)=C +



f(s, v(·))ds +



Γ ˜a(s, v(·))dw(s) (14.24)

(see [83]). Then, taking into account the construction of

f and the operators

S and Γ , one can easily see that the process ξ(t)=Sv(t) satisﬁes the equation

14.3 Set-Valued Forces. Langevin Type Inclusions 347

ξ(t)=S

⎛

⎝



Γf



s, ξ(s),

ξ(s)



ds +



Γa



s, ξ(s),

ξ(s)



dw(s)+C

⎞

⎠

(14.25)

Since f(t, m, X) ∈ F(t, m, X)anda(t, m, X) ∈ A(t, m, X)andl>0isan

arbitrary number, this completes the proof. 

In some cases we can prove the existence of a strong solution of the

Langevin inclusion (14.16). Let us present an example of such an existence

theorem.

In what follows we use [0,l], B,

Ω, F and P

as introduced in the proof of

Theorem 14.28.ByB

we denote the Borel σ-algebra on [0,t]fort ∈ [0,l].

We introduce the notation compZ for the space of compact subsets in the

metric space Z. Thus, we say that the set-valued force vector ﬁeld B(t, m, X)

sends [0,l] × TM into comp TM if for any (t, m, X) ∈ [0,l] ×TM the image

B(t, m, X) ⊂ T

M is compact.

We recall several deﬁnitions.

Deﬁnition 14.29. A single-valued map β :[0,l] ×

Ω → R

is called {P

progressively measurable if for every t it is measurable with respect to B

×P

Deﬁnition 14.30. A set-valued map B :[0,l] ×

Ω → compR

is called {P

progressively measurable if {(t, ω) ∈ [0,l] ×

Ω | B(t, ω) ∩C = ∅} ∈ B

×P

for

every closed set K ⊂ R

Deﬁnition 14.31. We say that a set-valued vector force ﬁeld B :[0,l] ×

TM → comp TM:

(i) is dissipative if for all t ∈ [0,l], m ∈ M, X, Y ∈ T

M and U ∈

B(t, m, X), V ∈ B(t, m, Y ) the inequality X − Y,U −V ≤0 holds.

(ii) is maximal if for t, m, X, Y and V as in (i) the inequality X −Y,U −

V ≤0 is equivalent to the assumption that U ∈ B(t, m, X).

Denote by w(t) a one-dimensional Wiener process. Let F(t, m, X)and

G(t, m, X) be set-valued vector force ﬁelds on M as above. Then we can

consider the stochastic diﬀerential inclusion of Langevin type

ξ(t) ∈S





PΓF(τ,ξ(τ),

ξ(τ))dτ +



PΓG(τ,ξ(τ ),

ξ(τ ))dw(τ )+C



(14.26)

Inclusion (14.26) is a particular case of (14.17)sinceΓG(τ, ξ(τ),

ξ(τ ))dw(τ )

can be represented as ΓG(τ,ξ(τ),

ξ(τ ))(P dW (τ )) (P is the orthogonal pro-

jection onto the linear span of vectors ΓG(τ,ξ(τ),

ξ(τ ))).

Theorem 14.32 Let the set-valued vector ﬁelds F (t, m, X) and G(t, m, X),

F, G :[0,l] × TM → comp TM be Borel measurable, uniformly bounded,

dissipative and maximal. Then there exists a strong solution of (14.26),w

ell-

deﬁned for t ∈ [0,l], with initial conditions ξ(0) = m

and

ξ(0) = C for any

∈ M and C ∈ T

348 14 Mechanical Systems with Random Perturbations

Proof. Let (Ω,F, P) be a probability space admitting a one-dimensional

Wiener process w(t). Denote by P

the σ-subalgebra of F generated by

all w(s)for0≤ s ≤ t and completed by all sets of zero probability. Let

Y : Ω → Ω be a measurable map. From the properties of parallel translation

and the assumed hypothesis one can easily derive that the coeﬃcients

ΓF(t, ω, Y )=ΓF



t, SY (ω)(t),

SY (ω)(t)



and

ΓG(t, ω, Y )=ΓG



t, SY (ω)(t),

dtSY (ω)(t)



for ω ∈ Ω satisfy all the conditions of [183, Theorem 1] and so on (Ω,F, P)

there exists a continuous P

-progressively measurable process v(t)(v(0) = 0)

in T

M and L

-selectors f(t, ω)ofΓF(t, ω, v)andg(t, ω)ofΓG(t, ω, v)such

that a.s.

v(t)=



f(τ, ω)dτ +



g(τ,ω)dw(τ)+C. (14.27)

Consider the M-valued process ξ(t)=Sv(t) with v(t) satisfying (14.27). In

the same manner as in the proof of Theorem 14.28 we can construct Borel

measurable selectors f(t, m, X)ofF (t, m, X)andg(t, m, X)ofG(t, m, X)

such that a.s.

ξ(t)=S





Γf(τ,ξ(τ),

ξ(τ))dτ +



Γg(τ,ξ(τ),

ξ(τ))dw(τ )+C





14.4 Systems with Random Perturbation of Velocity

In the previous two sections we dealt with equations obtained from the or-

dinary version of Newton’s law by a stochastic perturbation of the vertical

component on the right-hand side, i.e., of the force ﬁeld (see Section 14.1).

Here we investigate the systems in which the horizontal part is subjected

to stochastic inﬂuence. This means that a random perturbation of velocity

arises. Such a situation can appear, for example, if a particle, subjected to a

deterministic force, in addition moves within a random media. Note that in

this model the perturbation is independent of the particle velocity.

We investigate the system (14.1) with ¯α independent of velocities, i.e., it

turns into



˙x(t)=v(t)

˙v(t)=¯α(t, x(t)).

(14.28)

14.4 Systems with Random Perturbation of Velocity 349

A particular example of such a force is −grad U in a conservative mechanical

system.

Now suppose that in (14.28) the right-hand side of the horizontal (i.e. ﬁrst)

equation is subjected to a random perturbation of the form A(t, x(t)) ˙w(t)

where ˙w(t) is white noise. Note that this perturbation is independent of the

velocity of the particle. In appropriate terms this means that the process

ξ(t) describing the motion of the particle satisﬁes the equality ξ(t)=ξ



v(s, ξ(s))ds +



A(s, ξ(s))dw(s) where the vector ﬁeld v(t, x) satisﬁes the

relation Dv(t, ξ(t)) = ¯α(t, ξ(t)). The formal equation of motion in terms of

forward mean derivatives then takes the form

⎧

⎨

⎩

Dξ(t)=v(t, ξ(t))

ξ(t)=A(t, ξ(t))A

∗

(t, ξ(t))

Dv(t, ξ(t)) = ¯α(t, ξ(t)).

(14.29)

We also suppose that A(t, x)and¯α(t, x)) satisfy the Itˆo condition

A(t, x) + ¯α(t, x) <K(1 + x) (14.30)

for some K>0.

Theorem 14.33 Let A(t, x) and ¯α(t, x) be jointly continuous in t, x and

satisfy (14.30). Then for every pair ξ

, v

∈ R

there exists a weak solution

of (14.29) with initial conditions ξ(0) = ξ

and v(0) = v

Proof. In C

([0,l], R

) introduce the σ-algebra

F generated by cylindrical

sets. By

denote the σ-algebra generated by cylindrical sets over [0,t] ⊂

[0,l].

Consider the map ¯v :[0,l] ×C

([0,l], R

) → R

deﬁned by the formula

¯v(t, x(·)) = v



¯α(τ, x(·)))dτ. (14.31)

By construction this map is jointly continuous in t ∈ [0,l]andx(·) ∈

([0,l], R

). In addition it is obvious that if x

(·)andx

(·) coincide on

[0,t]then¯v(t, x

(·)) = ¯v(t, x

(·)). This means that ¯v(t, x(·)) is measurable

with respect to

(see, e.g., [83]).

Taking into account (14.30) one can easily derive the inequality

¯v(t, x(·)) =





¯α(τ, x(·)))dτ



≤



¯α(τ,x(·)))dτ

≤ K



(1 + x(τ ))dτ

350 14 Mechanical Systems with Random Perturbations

≤ K



(1 + x(·)

)ds

≤ lK(1 + x(·)

where ·

is the norm in C

([0,l], R

Introduce A(t, x(·)) as A(t, x(·)) = A(t, x(t)). Notice that A(t, x(·)) is mea-

surable with respect to

and that from (14.30) it follows that A(t, x(·))≤

K(1 + x(·)

). So, both ¯v(t, x(·)) and A(t, x(·)) satisfy the Itˆo condition in

the form

¯v(t, x(·) + A(t, x(·)≤

K(1 + x(·)

)

with

K =max(K, lK).

Now the pair ¯v(t, x(·)) and A(t, x(·)) satisﬁes all the conditions of [83,

Theorem III.2.4], hence, the stochastic diﬀerential equation

x(t)=x



¯v(s, x(·))ds +



A(s, x(·))dw(s) (14.32)

has a weak solution on [0,l]. This means that there exist a probabilistic

measure μ on (C

([0,l], R

), F) and a Wiener process in R

, given on

([0,l], R

), F,μ) and adapted to P

, such that the coordinate process x(t)

on (C

([0,l], R

), F,μ)andw(t) satisfy (14.32). Let v(t, x) be the regression

v(t, x)=E(¯v(t, x(·)) | x(t)=x). This together with the construction of the

process ¯v(t, x(·)) completes the proof of the Theorem. 

The simple construction used in the proof of Theorem 14.33 can be gener-

alized so that it is applicable in more complicated settings. First we consider

the case where the force ﬁeld is set-valued, lower semi-continuous but not

necessarily convex valued.

Let F (t, x) be a lower semi-continuous set-valued map F : R ×R

 R

with closed images and A(t, x):R

→ R

be a ﬁeld of single-valued linear

operators jointly continuous in parameters t ∈ R and x ∈ R

. We suppose

that F (t, x)andA(t, x) satisfy the Itˆo condition, i.e., that there exists a

constant Θ>0 such that

F (t, x) + A(t, x) <Θ(1 + x) (14.33)

for all t ∈ R and x ∈ R

where A(t, x) is the operator norm and F (t, x) =

sup

y∈F (t,x)

y.

The system of equations (14.30) is now replaced by the following analogous

inclusion

⎧

⎨

⎩

Dξ(t)=v(t, ξ(t))

ξ(t)=A(t, ξ(t))A

∗

(t, ξ(t))

Dv(t, ξ(t)) ∈ F (t, x(t)).

(14.34)

In what follows we consider R

and R with their Borel σ-algebras B

and B, respectively. Let x(·) be a continuous curve. Consider the set-

14.4 Systems with Random Perturbation of Velocity 351

valued vector ﬁeld F (t, x(t)) along x(·) and denote by PF (·,x(·)) the set

of all measurable selectors of F (t, x(t)), i.e., the set of measurable maps

{f : R → R

: f(x(t)) ∈ F (t, x(t))}. It is clear that, since condition (14.33)

is satisﬁed, all such selectors are integrable on any ﬁnite interval in R with

respect to Lebesgue measure. Denote by



PF (·,x(·)) the set of integrals with

varying upper limits of these selectors.

We recall some facts and notions which will be used shortly. Let l>0. In

what follows we denote by λ the normalized Lebesgue measure on [0,l], i.e.,

such that λ([0,l]) = 1.

Lemma 14.34 Let (Ξ,d) be a separable metric space, X be a Banach space.

Consider the space Y = L

(([0,l], B,λ),X)) of integrable maps from [0,l] into

X.Ifaset-valuedmapG : Ξ → Y is lower semicontinuous and has closed

decomposable images, then it has a continuous selector.

This is a particular case of the Bressan-Colombo Theorem (Theorem 4.9).

Denote by C

([0,l], R

) the Banach space of continuous maps from [0,l]

to R

(i.e., continuous curves in R

, given on [0,l]).

Theorem 14.35 As above, let F (t, x) be a lower semi-continuous set-valued

map F : R×R

 R

withclosedvaluesandA(t, x):R×R

→ R

be a ﬁeld

of single-valued linear operators jointly continuous in the parameters t ∈ R

and x ∈ R

.Letalso(14.33) be fulﬁlled. Then for any l>0, x

∈ R

inclusion (14.34) has a solution on [0,l] with initial position x

and initial

velocity v

Proof. In C

([0,l], R

) introduce the σ-algebra

F generated by cylindrical

sets. By

denote the σ-algebra generated by cylindrical sets over [0,t] ⊂

[0,l].

Consider the set-valued mapping B sending x(·) ∈ C

([0,l], R

)into

PF (·,x(·)). Since under condition (14.33) all selectors from PF (·,x(·)) are

integrable (see above), B takes values in the space L

(([0,l], B,λ), R

). It is

known (see, e.g., [155, Section 5.5]) that under the above-mentioned condi-

tions B : C

([0,l], R

) → L

(([0,l], B,λ), R

) is lower semicontinuous and

for any x(·) ∈ C

([0,l], R

)thesetPF (·,x(·)), i.e., the image B(x(·)), is de-

composable and closed. Thus, by Lemma 14.34, B has a continuous selector

b : C

([0,l], R

) → L

(([0,l], B,λ), R

For any t ∈ [0,l] introduce the map f

: C

([0,l], R

) → C

([0,l], R

)that

sends a curve x(·) ∈ C

([0,l], R

) into the curve

(τ,x(·)) =



x(τ)forτ ∈ [0,t]

x(t)forτ ∈ [t, l]

Obviously the map f

is continuous. Since f

(τ,x(·)) belongs to C

([0,l], R

the curve b(f

(τ,x(·))) ∈ L

(([0,l], B,λ), R

) is well-deﬁned. By construction

b(f

(τ,x(·))) ∈ F (τ, x(τ)) for almost all τ ∈ [0,t] and this selector continu-

ously depends on t in L

(([0,l], B,λ), R

352 14 Mechanical Systems with Random Perturbations

Consider the map v :[0,l] × C

([0,l], R

) → R

deﬁned by the formula

v(t, x(·)) = v



b(f

(τ,x(·)))dτ. (14.35)

By construction this map is jointly continuous in t ∈ [0,l]andx(·) ∈

([0,l], R

). In addition it is clear that if x

(·)andx

(·) coincide on [0,t]

then v(t, x

(·)) = v(t, x

(·)). This means that v(t, x(·)) is measurable with

respect to

(see, e.g., [83]).

Taking into account (14.33) one can easily derive the inequality

v(t, x(·)) =





b(f

(τ,x(·)))dτ



≤



b(f

(τ,x(·)))dτ

≤



F (τ,x(τ ))dτ ≤ Θ



(1 + x(τ ))dτ

≤ Θ



(1 + x(·)

)ds ≤ lΘ(1 + x(·)

where ·

is the norm in C

([0,l], R

Deﬁne A(t, x(·)) by A(t, x(·)) = A(t, x(t)). Notice that A(t, x(·)) is mea-

surable with respect to

and that from (14.33) it follows that A(t, x(·))≤

Θ(1 + x(·)

). So, both v(t, x(·)) and A(t, x(·)) satisfy the Itˆo condition in

the form

v(t, x(·) + A(t, x(·)≤

Θ(1 + x(·)

)

with

Θ =max(Θ, lΘ).

Now the pair v(t, x(·)) and A(t, x(·)) satisﬁes all the conditions of [83,

Theorem III.2.4], hence, the stochastic diﬀerential equation

x(t)=x



v(s, x(·))ds +



A(s, x(·))dw(s) (14.36)

has a weak solution on [0,l]. This means that there exist a probabilistic

measure μ on (C

([0,l], R

), F) and a Wiener process in R

, given on

([0,l], R

), F,μ) and adapted to P

, such that the coordinate process x(t)

on (C

([0,l], R

), F,μ)andw(t) satisfy (14.36). This together with (14.35)

completes the proof of the Theorem. 

To investigate this problem on a manifold we use the constructions of

Section 7.7.3. The assumptions here are more restrictive than in the case of

Euclidean space.

Let M be a stochastically complete Riemannian manifold on which a vector

force ﬁeld ¯α(t, m) independent of velocities is given. Thus the Newton law of

the mechanical system takes the form

˙m(t)=¯α(t, m(t)).

14.4 Systems with Random Perturbation of Velocity 353

We suppose that the random perturbation of velocity takes the form A(m)˙w(t)

where A(m):R

→ T

M is a smooth ﬁeld of linear operators. We suppose

in addition that A(m)A

∗

(m)=I where I is the unit operator in T

M.This

assumption can be interpreted as the fact that the Riemannian metric on M

is determined by the diﬀusion coeﬃcient generated by A(m). In particular it

means that we can apply the machinery of the equations with unit diﬀusion

coeﬃcient from Section 7.7.3.

The equation of motion for the system with random perturbation of veloc-

ity is given here in terms of mean derivatives on manifolds as introduced by

Deﬁnition 9.1 and formula (9.18), and by the covariant mean derivative in-

troduced by (9.15) in terms of stochastic parallel translation (see Sections 9.3

and 9.4). This equation takes the form

⎧

⎨

⎩

Dξ(t)=v(t, ξ(t))

ξ(t)=I

Dv(t, ξ(t)) = ¯α(t, ξ(t)).

(14.37)

Theorem 14.36 Let the force ﬁeld ¯α(t, m) be jointly continuous in t, m and

be uniformly bounded, i.e., ¯α(t, m) <K for all m ∈ M and t ∈ [0,l] ⊂ R

for some K>0. Then for every pair m

∈ M, v

∈ T

M there exists

asolutionof (14.37) with initial conditions ξ(0) = m

, v(0) = v

that is

well-deﬁned on the entire interval t ∈ [0,l].

Proof. The idea of the proof is analogous to that for Langevin equations. We

reduce (14.37) to the equation of velocity hodograph type in a single linear

space. Then we show that the latter has a weak solution and that its Itˆo

development satisﬁes (14.37). The diﬀerence is that here we use the velocity

hodograph equation in terms of stochastic parallel translation (unlike the

case of the Langevin equation where the ordinary parallel translation was

applied) and so we have to appeal to the constructions of Section 7.7.3.In

fact the proof uses the same argument as that of Theorem 7.95 and we refer

the reader to the latter for a detailed explanation.

Consider the space

Ω = C

([0,l],T

M) with the σ-algebra

F generated

by cylinder sets and a Wiener measure ν on

F. On the probability space

(

Ω,

F,ν) the coordinate process ˜w(t, x(·)) = x(t) is a Wiener process adapted

to the family of σ-subalgebras P

that for each t is generated by cylinder sets

with bases on [0,t] and is completed by all sets with ν-measure zero.

Since M is stochastically complete, the Itˆo development R

˜w(t) is well-

deﬁned on [0,l]forν-a.s. all curves in

Ω and the parallel translation along

ν-almost all sample paths of R

˜w(t) is also well-deﬁned (see Section 7.6).

Thus we can apply the operator Γ of parallel translation along R

˜w(·)from

each R

˜w(t)toR

˜w(0) = m

Introduce the process β(t, x(·)) =



Γ ¯α(s, R

x(s))ds in T

M.Fromthe

properties of parallel translation and of R

it follows that β(t) is uniformly

bounded by the constant lK and that it is non-anticipative with respect to

. In addition the density

354 14 Mechanical Systems with Random Perturbations

ρ (x(·)) = exp





β (t, x(·)) , d˜w(t)−



β (t, x(·))



(14.38)

satisﬁes the equality



ρ dν =1, (14.39)

and so the measure μ on (

Ω,

F) deﬁned by the relation dμ = ρdν is a prob-

ability measure.

Then the coordinate process ¯v(t) on the probability space (

Ω,

F,μ) satis-

ﬁes the equation

¯v(t)=v



β(s, ¯v(s))ds + w(s) (14.40)

where w(t) is a Wiener process in T

M, given on (

Ω,

F,μ), that is non-

anticipative with respect to P

.TheItˆo development ξ(t)=R

¯v(t)hasthe

same sample paths as R

˜w(t) and so it is well-deﬁned on the entire interval

[0,l].

Introduce the vector ﬁeld v(t, m) as the regression

v(t, x)=E(Γ

0,t

β(t) | R

¯v(t)=x)

where Γ

t,0

is the operator of parallel translation along R

¯v(·)fromR

¯v(0) =

to R

¯v(t). Taking into account the properties of the Itˆo development

and of parallel translation as well as the construction of the covariant mean

derivatives (9.15)and(9.18), one can easily show that ξ(t)andv(t, m) satisfy

(14.37). 

Chapter 15

The Newton-Nelson Equation

The Newton-Nelson equation is a version of Newton’s law formulated in terms

of mixed symmetric second order mean derivatives. It describes the motion

of a quantum particle in the framework of stochastic mechanics.

Nelson’s stochastic mechanics is a subject based on the ideas of classical

physics but giving the same predictions as quantum mechanics. Stochastic

mechanics can be considered as a third method of quantization diﬀering from

the well-known Hamiltonian and Lagrangian (path integrals) methods. Each

method has its own domain of applicability, and these domains have a large

intersection (where the results are equivalent), but none of them includes any

other completely.

The history of stochastic mechanics is presented, for example, in [45, 136,

187, 188, 190]. Apparently, the idea was ﬁrst suggested by I. Fenyes [73], but

it only became widely known, and obtained a natural form, following the

appearance of Nelson’s independently developed work [187, 188]. At present,

among other things, a description of spinning particles, relativistic particles,

the uncertainty principle and some parts of quantum ﬁeld theory have been

given in the language of stochastic mechanics (see, e.g., [45, 55, 136, 141, 142,

237, 238]). Nelson’s book [190] surveys the main developments of the theory

up to 1985.

In the version of stochastic mechanics pioneered by Nelson, the trajectory

of a particle was assumed to be a Markov diﬀusion process. It is important to

point out that in the works of V.P. Dmitriev and of H. Grabert, P. Hanggi and

P. Talkner (see, e.g., [52, 137] and the references therein) physical reasoning

is used to conclude that such a trajectory must be a non-Markov process.

In addition, in his later work, E. Nelson discovered that in the framework of

his “Markov” approach some higher momenta of two independent particles

could be dependent.

We develop another approach to stochastic mechanics where the equation

of motion is that of Nelson (the Newton-Nelson equation) but the trajectory is

allowed to be an Itˆo diﬀusion type process that may not be a Markov process.

Nevertheless we show that this version is related to quantum mechanics in an

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

355