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

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356 15 The Newton-Nelson Equation

analogous way to Nelson’s original version. Moreover, following the ideology

of equations with mean derivatives of Section 8.1 (see Deﬁnition 8.21)we

suggest an equation of Itˆo type for ﬁnding solutions of the Newton-Nelson

equation such that for two independent particles it splits into two independent

equations. Another (very simple) modiﬁcation is that we ﬁnd the diﬀusion

term of the process from a separate equation with quadratic mean derivative

and do not postulate its value apriori.

The intersection of the domains of applicability of stochastic mechanics

and ordinary (Hamiltonian) quantum mechanics is as follows: for the case of

forces for which the Schr¨odinger equation is well-deﬁned (i.e., for potential

forces, certain forces with friction and the magnetic ﬁeld), there exists a

natural interrelation between the processes of stochastic mechanics and wave

functions, the solutions of the corresponding Schr¨odinger equations (see, e.g.,

[187, 188, 233]). Below in Section 15.1.1 we describe this interrelation for the

case of potential forces. Note that in the paper [38] (see also [27, 192, 240]) this

interrelation and the methods of partial diﬀerential equations were applied to

obtain the existence of trajectories for non-relativistic stochastic mechanical

systems with potential forces in R

where the potential belongs to the so-

called Rellich class. It should be pointed out that the trajectories obtained in

[27, 38, 73, 240] are Markov diﬀusion processes, as was postulated in Nelson’s

pioneering works [187, 188].

15.1 Stochastic Mechanics in R

This section consists of two subsections. In Subsection 15.1.1 we present

the basic ideas of stochastic mechanics and, in particular, we describe the

Newton-Nelson equation and its relation to the Schr¨odinger equation.

In Subsection 15.1.2, following [102, 105, 128, 129], we prove the existence

of trajectories for non-relativistic stochastic mechanical systems under vari-

ous initial conditions and with forces in R

which are a sum of a force inde-

pendent of velocities and a force linearly dependent on velocities. We do not

assume the forces to be potential or gyroscopic; i.e., we include cases where

other methods of quantization are not applicable (e.g., where the Schr¨odinger

equation does not exist).

We apply the methods developed in Chapter 8.

15.1.1 Principal ideas of Nelson’s stochastic mechanics

Let ξ(t) be a stochastic process and assume that the mean derivatives (see

Section 8.1) exist for ξ(t).

15.1 Stochastic Mechanics in R

357

Deﬁnition 15.1. The vector

(DD

∗

D)ξ(t) is called the acceleration of

the stochastic process ξ(t).

Of course there exists a Borel vector ﬁeld on R

(the regression) such that

the above acceleration is the composition of that vector ﬁeld and ξ(t).

Direct calculations show that

(DD

∗

+ D

∗

D)ξ(t)=(D

− D

)ξ(t)=D

(t) −D

(t) (15.1)

(see Deﬁnitions 8.15 and 8.16).

In this Section we shall mainly deal with a process of type (8.15), i.e., of

the form ξ(t)=ξ



a(s)ds + σw(t). For such a process, from formulae

(8.24)and(8.25) it follows that

(t)=

∂

∂t

(t)+(v

(t) ·∇)v

(t) (15.2)

and

(t)=(u

(t) ·∇)u

(t)+

∇

(t). (15.3)

(Note that the right hand side of (15.2) has the form of an ordinary derivative

of a non-autonomous vector ﬁeld in the direction of itself in R

.) Thus, from

(15.1)–(15.3) we obtain for a ξ(t)oftype(8.15)that

(DD

∗

+ D

∗

D)ξ(t) (15.4)



∂v

(t)

∂t

+(v

(t) ·∇)v

(t)



−



∇

(t)+(u

(t) ·∇)u

(t)



Consider a Newtonian mechanical system in R

with the vector force ﬁeld

¯α(t, x, X). Here Newton’s law (11.2) takes the following trivial form:

¨x(t)=

¯α(t, x(t), ˙x(t)), (15.5)

where m is the mass of a particle, ¨x(t) is the acceleration vector and ˙x(t)is

the velocity vector of the curve x(t).

In stochastic mechanics the trajectory of a particle is assumed to be a

stochastic process, not a deterministic curve. More precisely, we have the

following:

Deﬁnition 15.2. A diﬀusion type process ξ(t)inR

is called a stochastic-

mechanical trajectory of a particle with mass m > 0 under the action of the

force ﬁeld ¯α(t, x, X)ifitsatisﬁesthesystem

⎧

⎨

⎩

(DD

∗

+ D

∗

D)ξ(t)=

¯α(t, ξ(t),v

(t))

ξ(t)=



(15.6)

358 15 The Newton-Nelson Equation

where  =

2π

, h is Plank’s constant and I is the unit n × n matrix. In this

case we also say that the stochastic-mechanical system with force ¯α(t, x, X)

is given. The equality (15.6) is called the Newton-Nelson equation.

From the second equation of (15.6) it follows that ξ(t) is a process of type

(8.15) with σ such that



If σ = 0, the process ξ(t) from Deﬁnition 15.2 turns into a deterministic

curve and (15.6) becomes the ordinary Newton law (15.5). Without loss of

generality we assume m = 1.

Remark 15.3. Equality (15.6) for the Euclidean space R

was ﬁrst obtained

by Nelson in [187]. It was also shown there that among all possible deﬁnitions

of the acceleration of a stochastic process which are symmetric in time (i.e.,

well-deﬁned physically) and coincide with the ordinary deﬁnition for smooth

trajectories, only Deﬁnition 15.1 gives the correct result for some particular

examples in quantum mechanics. Later equation (15.6) (for potential forces

and in the form where the right-hand side is transformed according to (15.4))

was derived from some variational principles (see [190]).

Note that in stochastic mechanics one deals with the “quantization” of

Newton’s second law, while in the ordinary quantization procedures some

other equations of motion (Euler-Lagrange or Hamilton) are involved.

The correspondence between stochastic mechanics and ordinary (Hamil-

tonian) quantum mechanics was established for potential forces (see, e.g.,

[187, 188, 190]) and for certain forces with friction [233] where both the

Schr¨odinger equation and Newton-Nelson equation (15.6) are well-deﬁned.

We illustrate this correspondence with an example of potential forces. The

arguments here are close to those in [187], but since we apply Lemmas 8.17

and 8.18 instead of classical results for Markov diﬀusions, we show that the

correspondence is also valid under the assumption that the trajectories are

Itˆo processes of diﬀusion type.

Let the force ﬁeld ¯α of the mechanical system be a potential, i.e., it does

not depend on velocity and ¯α = −grad V ,whereV is the potential energy.

Let ξ(t) be a trajectory of the stochastic-mechanical system as in Deﬁnition

15.2 with this force. Recall that for the osmotic velocity u

(t)=u

(t, ξ(t))

the vector ﬁeld u

(t, x) is always described in the form u

= σ

grad R,where

R =

log ρ

(t, x)(see(8.18)). Let us suppose that for the current velocity

(t)=v

(t, ξ(t)) the vector ﬁeld v

(t, x) is also a gradient v

= σ

grad S for

some real function S(t, x). Note that S(t, x) is deﬁned to within the functions

depending only on t, i.e., whose gradient is zero. Consider the complex-valued

function Ψ on M of the form Ψ(t, x)=exp(R +iS).

Theorem 15.4 Ψ satisﬁes the Schr¨odinger equation

∂Ψ

∂t

∇

Ψ − i



VΨ. (15.7)

15.1 Stochastic Mechanics in R

359

Proof. From Lemmas 8.17 and 8.18 it follows that

∂u

∂t

= −

grad div v

−

grad (v

˙u

). From (15.4)and(15.6)for¯α = −grad V it follows that

∂v

∂t

−grad V −(v

·∇)v

+(u

·∇)u

∇

. Then straightforward calculations

show that

∂Ψ

∂t



∂R

∂t

∂S

∂t



Ψ. (15.8)

In order to ﬁnd (

∂R

∂t

∂S

∂t

)wenotethatgrad(

∂R

∂t

∂S

∂t

(

∂u

∂t

∂v

∂t

)and

so, by the above expressions for

∂u

∂t

and

∂v

∂t

,sinceu

and v

are gradients,

we obtain the formula

grad



∂R

∂t

∂S

∂t



= −grad div v

−

grad (v

· u

) −

grad V

−

·∇)v

·∇)u

∇

=grad



−

div v

−

· u

) −

V −

2σ

)

2σ

)

div u



=grad



(div u

+idivv

2σ

+iv

)

−



Hence,



∂R

∂t

∂S

∂t



= −

(div u

+idiv v

2σ

+iv

)

−

V +if(t) (15.9)

where f(t) is a real function, depending only on t, i.e., grad f (t)=0.Onthe

other hand,

∇

Ψ = Ψ



+iv

)

(div u

+idivv

)



. (15.10)

Comparing (15.8), (15.9)and(15.10), we see that the following equality holds:

∂Ψ

∂t

∇

Ψ − i



VΨ+if (t)Ψ. (15.11)

Adding to S an appropriate constant depending on t, one can obtain the

equality f =0.So(15.11) turns into the Schr¨odinger equation (15.7) with

potential V . 

So, Ψ(t, x) is a Schr¨odinger wave function corresponding to the above

mechanical system. Conversely, let Ψ satisfy the Schr¨odinger equation (15.7)

with potential V . Consider the functions R = Re log Ψ and S = Im log Ψ

and the vector ﬁelds u(t, x)=σ

grad R, v(t, x)=σ

grad S and a(t, x)=

v(t, x)+u(t, x).

360 15 The Newton-Nelson Equation

Theorem 15.5 Solutions ξ(t) of the Itˆo equation

dξ(t)=a(t, ξ(t))dt + σdw(t), (15.12)

where a(t, x) and σ are as introduced above, satisfy equation (15.6) with ¯α =

−grad V .

We leave the proof to the reader as a simple exercise. Note that for a

solution ξ(t)of(15.12)wehavev

(t, x)=v(t, x)andu

(t, x)=u(t, x), and

it should be emphasized that in this case the assumption that v

(t, x)isa

gradient is fulﬁlled automatically. Note also that strong solutions of (15.12)

are Markov diﬀusion processes (see Deﬁnition 6.17). Thus we can formulate:

Proposition 15.6 If a stochastic-mechanical trajectory ζ(t) with the force

¯α = −grad V is an unique strong solution of (15.12) corresponding to a

Schr¨odinger wave-function Ψ as described above, then ζ(t) is a Markov dif-

fusion process.

Indeed, ρ

(t, x)=|Ψ(t, x)|

= ρ

(t, x), where ρ

(t, x) is the probability

density of the diﬀusion process ξ(t), the solution of (15.12) constructed from

Ψ(t, x), which corresponds to ζ(t). The above diﬀusion process (trajectory)

can be shown to exist when V belongs to the very broad so-called Rellich

class (see [38]).

15.1.2 Existence theorems

In this Section we prove the existence of the trajectory assuming neither the

Schr¨odinger equation to be well-deﬁned nor its solution to exist. In this case

the trajectory is not a diﬀusion process but an Itˆo process of type (8.15).

First we consider deterministic initial data for the solution which leads to a

singularity at t = 0 resembling the Big Bang (see Remark 15.10 below). Then

we obtain another version of the construction that yields the existence of a

solution, for random initial data, such that its distribution is nowhere zero.

In the latter case no singularity at time zero arises.

Everywhere below in this Section we consider the vector force ﬁeld ¯α

of the form ¯α(t, x, X)=¯α

(t, x)+¯α

(t, x)X where ¯α

(t, x) is a vector

ﬁeld on R

depending on t ∈ [0,l]and¯α

(t, x) is a linear operator in

depending on the parameters t ∈ [0,l]andx ∈ R

, i.e., ¯α

(t, x)isa

(1.1)-tensor ﬁeld on R

. Consider the derivative of ¯α

(t, x), i.e., the ﬁeld

of bilinear operators ¯α



(t, x))(·, ·):R

× R

→ R

, and the vector ﬁeld

tr¯α



(t, x)(¯α

)ds =



i=1

¯α



(t, x)(¯α

(t, x)e

), where e

,...,e

is an arbi-

trary orthonormal frame in R

. We assume the following condition to be

fulﬁlled:

15.1 Stochastic Mechanics in R

361

Condition 15.7 The vector ﬁeld ¯α

(t, x) and the tensor ﬁeld ¯α

(t, x) are

Borel measurable jointly in t and x, tr¯α



(s, W (s))(¯α

) exists and is also Borel

measurable jointly in t, x and there exists a constant C>0 such that



(¯α

(t, x(t))

+ ¯α

(t, x(t))

+ tr¯α



(t, x(t))(¯α

)

)dt<C

for any continuous curve x(t) in R

, t ∈ [0,l],where¯α

(t, x) is the operator

norm.

Condition 15.7 is fulﬁlled, for example, if ¯α

(t, x) is continuous and uni-

formly bounded, ¯α

(t, x) belongs to the functional space C

(R×R

×R

, R

)

and has a bounded norm in that space.

For the sake of simplicity, without loss of generality we may assume σ =1.

We shall deal with stochastic processes starting at a non-random point in R

and for the sake of simplicity we assume this point to be equal to the origin.

Consider the probability space (

Ω,

F,ν)where

Ω = C

([0,l], R

F is the

σ-algebra generated by the cylindrical sets, and ν is a Wiener measure (see

Section 6.2.1). Denote by B

the σ-algebra generated by the cylindrical sets

with the bases over [0,t]; all the B

are completed by the sets of ν-measure

zero.

Recall that the coordinate process W (t, x(·)) = x(t), x(·) ∈

Ω,on(

Ω,

F,ν)

is a Wiener process in R

Let t

∈ (0,l)andfort ∈ [0,l] denote by t

(t) the function

(t)=

⎧

⎨

⎩

if t<t

if t ≥ t

(15.13)

Recall that we are looking for a solution as a process of type (8.15) that under

the above assumptions takes the form ξ(t)=



a(s)ds+ w(t)wherea(t)isto

be found. Pick a deterministic initial condition a

∈ R

for a(t) and consider

in R

the equation

a(t)=a



¯α

(s, W (·))ds +



¯α

(s, W (s))dW (s)+



tr¯α



(¯α

)ds

−



(s)a(s)ds +



(s)dW (s) −

(t)W (t) (15.14)

Equation (15.14) has a unique strong solution for t ∈ [0,l]. Indeed, the coef-

ﬁcients of (15.14) are either Lipschitz continuous and have a linear growth

with respect to a or do not depend on a. Since the solution is strong, it exists

for any Wiener process and is non-anticipative with respect to P

= B

.In

what follows we will consider a(t) for the realization of W (t) as the coordinate

process on (

Ω,

F,ν).

From Condition 15.7 it follows that a(t) satisﬁes (8.32).

362 15 The Newton-Nelson Equation

Below in this Section we will use the notation introduced in Section 8.3.

Consider the function θ(l)on

Ω determined for the process a(t) by formula

(8.33). From the above condition it easily follows that θ(l) is a probability

density on (

Ω,

F,ν). Denote by μ

the corresponding measure on (

Ω,

F), i.e.,

dμ

= θ dν (note that μ

and ν are equivalent), and by ξ(t) the coordinate

process on (

Ω,

F,μ

). Then (see Section 8.3) the process ξ(t) is expressed in

the required form

ξ(t)=



a(s)ds + w(t) (15.15)

where w(t) is a Wiener process on (

Ω,

F,μ

) adapted to P

= P

= B

Since ξ(t)andW (t) coincide as coordinate processes on (

Ω,

F), (15.14)

clearly turns into

a(t)=a



¯α

(s, ξ(s))ds +



¯α

(s, ξ(s))a(s)ds +



¯α

(s, ξ(s))dw(s)



tr¯α



(s, ξ(s))(¯α

)ds +



(s))dw(s) −

(t)ξ(t). (15.16)

Lemma 15.8

(i) D

∗

¯α

(s, ξ(s))dw(s)=−tr¯α



(t, ξ(t))(¯α

)+¯α

(t, ξ(t))

ξ(t)

−¯α

(t, ξ(t))E

(κ(t)).

(ii) D

∗



(s)dw(s)=

ξ(t)

− E



κ(t)



for t>t

where κ(t) is as in Lemma 8.35.

In order to prove (i) one should replace the Itˆo integral by a backward Itˆo

integral according to formula (6.7) and then apply formula (8.37) with the

same arguments as in the proof of Lemma 8.26(ii). Assertion (ii) follows from

formula (8.37), the deﬁnition of the Itˆo integral, and the deﬁnition of t

(t)

given by formula (15.13).

Theorem 15.9 For t ∈ (t

,l) the process ξ(t) satisﬁes (15.6), i.e., it is a

trajectory of the stochastic mechanical system with force ¯α(t, m, X).

Proof. Recall that for t ∈ (t

,l)wehavet

(t)=

.From(15.15) and Theorem

8.7 it follows that Dξ(t)=E

(a(t)). By (8.36) we obtain D

∗

ξ(t)=E

(a(t))+

ξ(t)

− E

(κ(t)). In particular

(t)=D

ξ(t)=E

(a(t)) +

ξ(t)

−

(κ(t)) (15.17)

and

(t)=D

ξ(t)=−

ξ(t)

(κ(t)). (15.18)

15.1 Stochastic Mechanics in R

363

Then for t>t

, using formula (8.38) and equation (15.16), as well as Lem-

mas 8.36, 8.37 and 8.38, one can obtain

∗

ξ(t)=¯α

(t, ξ(t)) + ¯α

(t, ξ(t))E

(a(t)) +

tr¯α



(s, ξ(s))(¯α

)



a(t)



−

ξ(t)

−



a(t)



a(t)

Analogously, using in addition formula (8.39) and Lemma 15.8, one can cal-

culate

∗

Dξ(t)=¯α

(t, ξ(t)) + ¯α

(t, ξ(t))E

(a(t)) + ¯α

(t, ξ(t))



x(t)



−¯α

(t, ξ(t)E

(κ(t)) −

tr¯α



(t, ξ(t))(¯α

ξ(t)

−



κ(t)



−



a(t)





κ(t)



(DD

∗

+ D

∗

D)ξ(t)=¯α

(t, ξ(t)) (15.19)

+¯α

(t, ξ(t))



(a(t)+

ξ(t)

−

κ(t))



From formula (15.17) it follows that (15.19) coincides with the ﬁrst equation

of (15.6).

Since ξ(t) is a process of the form (8.15) (see above), by Theorem 8.12

ξ = I (recall that we set σ = 1) and so system (15.6) is fulﬁlled. 

Remark 15.10. Generally speaking, ξ(t)fort ∈ (0,t

) does not satisfy

(15.6). Thus ξ(t) can be interpreted only as a trajectory of the stochastic

mechanical system beginning at the instant t

in the random conﬁguration

ξ(t

) with the initial forward derivative E

(a(t

)). It is clear that t

may be

chosen arbitrarily close to zero, and so we can bring the initial values of the

trajectory to the origin for the conﬁguration and, again as close as we want,

to a

for the forward derivative. But we cannot put t

= 0, since the integral



dw(s) does not exist (



ds diverges, see, e.g., [162] and Section 6.2.2),

i.e., when t

= 0 equations (15.14)and(15.16) are ill-posed. This behavior

of the solution is interpreted in Section 15.3 as a possible description of the

Big Bang (see Remark 15.51).

Remark 15.11. If ¯α = −grad V then there may exist a trajectory of the

stochastic mechanical system obtained by Carlen [38] (see above). Suppose

that both our and Carlen’s trajectories are connected with the same solu-

tion Ψ of the corresponding Schr¨odinger equation. Then those trajectories

364 15 The Newton-Nelson Equation

determine diﬀerent probability measures on the space of sample paths but

the densities ρ on R ×R

coincide and are equal to |Ψ|

Since the function t

(t) is piecewise smooth (see its deﬁnition by formula

(15.13)) we may consider its derivative t



(t) deﬁned by the formula



(t)=



0ift<t

−

if t ≥ t

(15.20)

Theorem 15.12 Equations (15.14) and (15.16) are equivalent to the follow-

ing equations of Stratonovich type

a(t)=a

−



(s)a(s)ds −





(s)W (s)ds



¯α

(s, W (·))ds +



¯α

(s, W (s)) ◦ dW (s) (15.21)

and

a(t)=a

−



(s)a(s)ds −





(s)ξ(s)ds +



¯α

(s, ξ(·))ds



¯α

(s, ξ(s))a(s)ds +



¯α

(s, ξ(s)) ◦ dw(s), (15.22)

respectively.

Proof. Indeed, by the Itˆo formula (6.10)forf(t, x)=t

(t)x we have

d(t

(t)W (t)) = t



(t)W (t)dt + t

(t)dW (t)

and

d(t

(t)ξ(t)) = (t

(t)a(t)+t



(t)ξ(t))dt + t

(t)dw(t).

An application of formula (6.23) (which follows from (6.6)) to A =¯α

com-

pletes the proof. 

Remark 15.13. Equations (15.21)and(15.22) show that a(t) is a vector

(belongs to the tangent bundle), a fact which plays a signiﬁcant role in the

transition to manifolds. Note also that if ¯α

=0,a(t) becomes a process with

a.s. smooth sample paths.

Now we are in position to prove the existence of a solution of the Newton-

Nelson equation (15.6) with random initial data. Here we shall obtain a pro-

cess that is a solution for all positive times, i.e., without singularity at t =0.

However, the initial density must be nowhere equal to zero.

The main technical diﬀerence from the above case is that here we can

calculate some derivatives “beforehand” and then use them to construct a

solution of (15.6).

15.1 Stochastic Mechanics in R

365

As above, consider equation (15.6) with ¯α(t, x, X)=¯α

(t, x)+¯α

(t, x)X

and suppose that ¯α

and ¯α

satisfy Condition 15.7. Consider the initial data

for (15.6)oftheform

ξ(0) = x

,Dξ(0) = a

, (15.23)

where x

and a

are random elements with values in R

, x

has a smooth

distribution ρ

with respect to Lebesgue measure on R

with ρ(x) = 0 for all

x and a

is bounded.

All processes below in this section are considered on some ﬁnite time in-

terval [0,l].

Let W

(t) be the standard Wiener process on R

, i.e., the coordinate

process on the probability space (

Ω,

F,ν) introduced at the beginning of this

section. The ﬁltration P

is generated by cylinder sets with bases on [0,t].

Consider the process W (t)=x

+ W

(t). Its density, denoted by ρ

(t, x),

satisﬁes the diﬀusion equation

∂

∂t

(t, x)=

Δρ

(t, x)

with initial condition ρ

(0,x)=ρ

. Thus the density ρ

(t, x)canbe

considered as given aprioriand we use it in the further construction. By

Lemma 8.17 the osmotic velocity u

(t, W (t)) of W (t) can be found by the

formula

(t, x)=

grad log ρ

(t, x).

Hence u

(t, x) is uniquely constructed from ρ

(t, x) (and, like ρ

(t, x), will

also be used later in the construction).

Consider the Itˆo equation

a(t)=a



¯α

(s, W (s)) ds (15.24)



¯α

(s, W (s)) dW (s)+



tr ∇¯α

(¯α

(s, W (s))) ds



(s, W (s)) ds +



(a(s) ·∇)u

(s, W (s)) ds,

where D

is deﬁned in (8.7) and may be represented as D

∂

∂t

Δ (see

(8.22)and(8.24)). Thus D

(s, W (s)) is known and we can prove the

existence of a solution a(t) of equation (15.24) by imitating the method used

for equation (15.14). Since (15.24) is linear in a, under the above conditions

this equation has a unique strong solution that we shall again denote by a(t).

Let θ(l) be deﬁned by formula (8.33) with a(t)andW (t) as above. Clearly

θ(t) is a martingale with expectation 1. Introduce a new probability measure