American Journal of Computational Mathematics, 2012, 2, 295-301
http://dx.doi.org/10.4236/ajcm.2012.24040 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
Feynman Formulas Representation of Semigroups
Generated by Parabolic Difference-Differential Equations
V. Sakbaev, A. Yaakbarieh
Moscow Institute of Physics and Technology, People’s Friendship University of Russia, Moscow, Russia
Received July 10, 2012; revised October 27, 2012; accepted November 1, 2012
ABSTRACT
We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators
is the generator of unitary group in the Hilbert space of square integrable functions. The representation of semigroup of
Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.
Keywords: Difference-Differential Equations; Semigroup; Feynman Formula; Chernoff Theorem
1. Introduction
In this paper we investigate the questions of correct re-
solvability of Cauchy problem for modeling parabolic
differential-difference the equations of the form
 
  
=1
,,
,,,,
t
N
kk k
i
uxt uxt
auxht uxhtxtRR,


 
 (1)
supplemented with the initial data
 
0
,0, .uxux xR
0
(2)
Here , for any , u0
is a given function and
NN,
kk
aRh
2
1, ,kN
2
x
 

is self-adjoint Laplas
operator in the space

2
LR
with domain


2
2
WR
u
. The
equations of this form arise at the description of the
phenomena of diffusion or heat conductivity with the
sources, nonlocally dependent on the state . Physically
the state u means the density distribution of the con-
centration or the temperature. In particular, the equations
of a kind (1) arise at research of problems of manage-
ment by the phenomena of a heat transfer in which dy-
namics of a state is given by the differential equation u
 
,,,,
t
uxtuxtg txu ,
with management g. We obtain the Equation (1) in case
when management g is given by the action on a state
function u of a deviation argument operators in a com-
position with operators of differentiation and multipli-
cation by the function (see [1-3]).
In this work we obtain the representation of semigroup
solutions of the Cauchy problem for the functional-
differential equation through the Feynman formula (see
[4]). It means that although the representation of the
evolutionary operator of the Cauchy problem (1) can be
defined only in terms of the spectral decomposition (in
the simplest situation in terms of the Fourier transform of
the solution), nevertheless we obtain an approximation of
the evolutionary operator by sequence of n-fold com-
positions of integrated operators which kernels are ele-
mentary functions.
In the terms of the monography [1], differential-
difference Equation (1) concerns to type mixed diffe-
rential-difference equation without a deviation on time.
Nonlinear parabolic differential-difference the equations
arise in the investigation of nonlinear optical systems
with a feedback (see [3]). In work [5] the mixed problem
for nonlinear parabolic differential-difference equations
had been formulated. Also it was established what pro-
perties distinguish the specified problem from the mixed
problem for parabolic differential equations. The linear
Cauchy problem (1), (2) can be considered as a linea-
rization of specified nonlinear mixed problems.
Firstly we prove the correctness of Cauchy problem
(1), (2) by using of Fourier transformation and obtain the
representation of Fourier image of its solution. After that
we construct the approximation of solution by Feynman
formulas. We extend the obtained result onto the ope-
rators with distributed deviation of the space argument.
The obtained results gives not only the expression of
the decision of Cauchy problem with the help of con-
structive algorithms, but also the investigation of pro-
babilistic structure of the phenomena reject the argu-
ments in the heat equation. The above Feynman formulas
define the approximation of a Markov random process,
such that the mean value of some functional on this
process is the solutions of the Cauchy problem.
C
opyright © 2012 SciRes. AJCM

V. SAKBAEV, A. YAAKBARIEH 296
2. Correct Resolvability of the Cauchy
Problem and Generation of Semigroup
Operators
Let us determine the solution of the Equation (1),
satisfying the initial condition (2).
Definition. A strong solution of the Cauchy problem
(1), (2) call the function






21
22
0, ,0, ,uCWR CLR  which Sa-
tisfies the equation (1) and condition (2).
Function





2
0, ,uC LR

0k
u

u

R
call a weak solution
of the Cauchy problem (1), (2), if there is a sequence of
initial data such that
1) the sequence converges in space H to the
element ,
0k
N
0
2) for each there is a strong solution of
the Cauchy problem with initial condition ,
ukk
u
0k
3) the sequence of functions converges to fun-
u

k
u
 
ctions in space . u
 
2
0, ,CL
Note that both strong and weak solution satisfies the
Equation (1) and condition (2) in the sense of the integral
identity.
Suppose that the existence of solution
of Cauchy problem (1), (2) is ob-
tained. To find a representation of the solution of the
Cauchy problem (1), (2) through the initial condition, we
apply the Fourier transform F to the left and right part of
Equation (1):

,, 0,,uxt txR
 

 
 

=1
,,
,,
t
N
kk k
i
F uxtFu xt
Fa ux htux ht





,
k


t


 



(3)
Let the function be the
 
,,,0,Usts Rt
Fourier transform in the first variable. Then
Equation (3) takes the form:

,uxt
 
    
2
1
,,
,expi,exp i
t
N
kk
i
UstsUst
aUs tshUs tsh





,
and the initial condition (2) transforms into the equation:
   
00
,0 ,UsUsFuss R .
Then by using of the equations
 



2
1
,,
exp iexpi,
t
N
kk k
i
UstsUst
ashshUs




and the initial condition we obtain that
  
2
0
1
,exp2cos
N
kk
i
UstUssasht




 


. (4)
Thus, established the following statements
Proposition 1. If the Cauchy problem (1), (2) has a
solution , then the Fourier transform defined
by equality (4).
u

,Ust
Theorem 1. The formula (4) defines a strongly
continuous semigroup


, 0,Ut t transformation of
the space


2
LR.
In fact, according to the unitarity of the the Fourier
transform in space


2
H
LR it is sufficient to verify,
that the one-parametrical family of operators


,0tt

U, of multiplication on the function


,,Ustt ,R

 is strongly continuous semigroup ope-
rators in space


2
LR
with norm not greater than one.
Semigroup property

 
12 1
,, ,UstUstUst t
0t
2
fol-
lows from the properties of the exponential function. The
strong continuity in point of operator-function


, 0tt

U follows from the uniform on any interval of
real line convergence of function
to the unit function

,, ,UsttRs

,R


1, ,
s
sR0t
0t
at . Then the
strong continuity at any point follows from the
semigroup property. Moreovet the type of above semi-
groupe

is equal to the value

,
sup
sR
f
s


os,
kk
where

2
1 i
2c
N
f
ssa shsR


 
. Therefore
1
2
N
k
k
a 



. The theorem 1 is proved.
Theorem 2. The Cauchy problem (1), (2) has a unique
generalized solution


, 0,ut t

U which is defined as the action of the semigroup on the initial
condition .
,0tt
0
In fact, according to the proposition 1, if the solution
of the Cauchy problem (1), (2) exists, then it re-
presentable in the form
u





0
12
1
,
*exp2 cos.
N
kk
i
uxtu x
F
sasht














 






 (5)
Conversely, if function is de-
fined by the equality (4) then the function

,,, 0,Ust sRt






1
,uxtF Ust

,
satisfy the equality (5), the equa-
lity (1) and the condition (2). In fact, if the function




02
Us LR satisfy the condition



2
02
s
Us LR
then the function


t,Us (see (4)):
belongs to space





  
1
22
0, ,0, ,CLRCL 

R;
satisfies the inclusion
  

 
2
,0,,
2
s
UstCL R
and the equation



2
1
,2cos
N
kk
i
Ustsbsh Ust
t

 


,.
Then, by the unitarity of the Fourier transform, for
each initial function there is the function
(5) which is a strong solution of the Cauchy problem (1),
(2). Hence, the formula (5) defines the image of the

2
02
uWR

V. SAKBAEV, A. YAAKBARIEH 297
0
function under the action of the semigroup

2
02
uWR

,ttU

  
0,
0 sup
lim n
nH
Ttnut u
 FU
operators According to Theorem 1 the
semigroup has a bounded exponential growth

tU
,

then the semigroup supposes continuation by

tU

2
WR
satisfies for all .
0T
Let us assume that 1N

. For given in the Equation
(1) parameters 1 aa

and 1
H
h
:RB

we consider the
operator-valued function , defined on

H
,
F
continuity from space onto the space

2


2
LR .
Hence for any initial condition the function

02
uLRah


0,R

 and taking values in a Banach space
 
0,ut tuU is a generalized solution of the Cauchy


BH of bounded linear operators in Hilbert space H.
For each value of we define its value 0t


,ah tF by
the equality
problem (1), (2). Theorem 2 is proved.
Corollary 1. The generator of the semigroup

,0ttU is operator

1
,
kk
N
khh
k
a

 

LSS





 
,0
2
22
0
,
1exp 4
2π
exp expd
44
ah tu utx
xy
t
t
xyh xyh
atuy y
tt











 
 

 

 


 



F
where h
S
is the shift operator on value . hR
The obtained representation of the solution of the
Cauchy problem (1), (2) is not constructive. For appro-
ximation of the solution obtained by using of sequences
of multiple integrals of elementary functions we use the
approach from papers [4,6,7] based on the Feynman
formulas.
(6)
3. Chernoff’s Approximation of Solutions of
the Cauchy Problem which is satisfied for arbitrary function from the
dense in space
0
u
H
linear manifold


R0 DC
. The
proposed form of Chernoff’s approximation of the semi-
group related to the fact that the first term in formula (6)
corresponds to the dynamics, generated by the unper-
turbed heat equation, and the second and third terms for
small values of the variable presente the influence of
the displaced sources.
t
Following the approach offered in [4], we define the
operator-valued function equivalent in Chernoff sense to
the semigroup of operators Chernoff’s
theorem (see [8]) states that:

,0ttU.
, Let the operator-valued function with
values in Banach space

,0ttF


BH continuous in strong
operational topology, supposes the estimation We verify the conditions of Chernoff's theorem for
operator-valued function


,, . ah ttR 
F


1,0
BH
tt   F,t for some 0

 and, more- Lemma 1. If 0
uD

the equality holds
over, operator is closable and its closure is the
generat or o f a strongly co ntinuous se migroup

0

F

  
,0
=0
2
0
00
2
d
d
.
ah t
tu
t
u
x
au xhu xh
x



F

,0ttU. Then for any uH

and any the 0T
equality

  
0,
0
sup
lim n
nH
tT tutn u
 
UF .
By following the definition in [4] the operator-valued
function will be called equivalent by
Chernoff semigroup if for any

, 0,ttFU

,0tt
To prove of lemma 1 we compute the function

2
2,,0,
uu
tx txR
tx






 . Since
.uH

the
equality
  

   
222
0
2222
2
πexpexp expd
444
2π2π
1exp expexp
44 4
4
2π
xyxyh xyh
uatu y y
tttt
tt
xy xyxyhxyh
tt t
t
t
x
at






 
 

 

 

   


   



  
 

   
  

  







2222
0
22
expexp d
44 44
yh xyhxyh xyhuyy
tt tt


 
 

 

 


 


and

V. SAKBAEV, A. YAAKBARIEH 298
 



22 2
2
22
2
2
22 2
0
2
11
exp exp
244 4
2π
1exp exp
24 4 4
1expexp d
24 4 4
xy xyxy
u
tt t xt
t
xyh xyhxyh
at tt t
t
xyhxyhxyh uyy
tt t t


 
 

 

 
 

 
 

 
 
 



 
 

 

 

 


then
   
22
2
0
2
1
,exp exp
44
2π
xyh xyh
uu
txaau y y
tt xt



 

 
 
 

 
d
t
.
Consequently

   
 

,0
=0
22
2
0
2
0
2
0
00
2
d
d
,exp exp
lim 44
2π
.
ah t
t
tu
t
xyh xyh
ua
txu y y
tt
xt
uxauxhuxh
x








  









F
d
Lemma 1 is proved.
For arbitrary and for given in Equation (1)
NN
coefficients ,, 1,
kk
ahkN we define the operator fun-
ction


,, 0, AH ttF
DF assuming that for each function its image defined by equality
0
u

,AH tu 0


 
,
2
2
=1
1exp 4
2π
exp exp.
44
AH
Nk
k
k
tux
xy
t
t
xyh xyh
at tt











 2
k















F


(7)
Hence, by lemma 1 on linear manifold D the operator

,
0
d
dAH t
t
t
F coincides with the generator of the semi-
group

, 0.Ut t
Lemma 2. The operator-valued function

,,0 AH ttF, is continuous in the strong operator
topology on the semiaxis and supposes an esti-
0t
mation on norm


,
=1
12, 0
N
ahk
BH k
ttat



F.
Firstly we prove this statement for the case 1N

.
The operator-valued function is the sum of
function

,ah tF


0tF with the integral kernel of the heat
equation and the function with the integral

,ah t
kernel

 
,
22
1
,, 2π
exp exp
44
ah txy t
xyh xyh
at tt







 

 
 













.
And






,0ahh h
tatt  SSF, where h

S
—shift
operators on the value of h

. It is well known that the
operator-valued function is continuous in
the strong operational topology and uniformly bounded

0,ttF0,
the norm topology:



01
BH
tF0.t The ope-
rators h

S
are bounded and have unit normed. Hence,
the operator-valued function is continuous in
the strong operator topology and satisfy the estimate

,ah t

2
BH at.
If N

N then
 

,0 0
1
,0
kk
N
AHkh h
k
ttatt t


.



FF FSS
Therefore the operator-valued function


,AH tF is
continuous in the strong operator topology and satisfy the
estimate


,
1
12
N
AH k
BH k
tt

  Fa for any . 0t
Thus lemma 2 is proved.
Lemmas 1 and 2 exibit that the function


,, AH tF
satisfies all conditions of Chernoff theorem.
Therefore the next theorem is proved as the main result
0t

V. SAKBAEV, A. YAAKBARIEH 299
of Feynman type representation of solution of Cauchy
problem for functional differential equation with devi-
ation of space argument.
Theorem 3. Let the above assumption on the para-
meters of equation (1) are satisfies. Then the operator-
valued function is equivalent on Chern-
off semigroup

,, 0, AH ttF

, 0.ttU
4. Some Generalization on the Case of
Distributed Deviation of the Space
Argument
At the end of our article we study the Cauchy problem
for the perturbed Equation (1), in which the deviation of
the argument presented by the convolution of unknown
function with some kernel. As such a perturbation we
consider the equation




  

 
1
,,
,d
,, ,,
t
N
kk k
k
R
K
uxt uxt
auth uth
Kx yuyt y
utxxtR R








L
(8)
where K is some even function of space and the
remaining terms are defined in the consideration of the
Equation (1). Hence Fourier transform

1
LR
ˆ
K
of the func-
tion K is a bounded continuous real-valued function.
Let us define



,,Ust uxt


and


ˆ
K
K

.
According to our assumption

ˆ
K
L

R. Then we
obtaine the relation
 

 
2ˆ
,exp iexpi2π,
N
tkkk
ik
UstsashshKsUst


 



by applying Fourier transform
F
to the equality (8). Hence we obtain, that
   



2
0
=1
0
ˆ
,exp2cos2π
N
kk
i
K
UstsashttKsUs
tU s


 






U

(9)
and therefore
   
12
=1
ˆ
,*exp2cosexp2π
N
kk
i
uxtuxtsashtKs


 




 


 



 (10)
Therefore the following analogue of Theorems 1 and 2:
Proposition 2. If the kernel

1
K
LR, then equality
(9) defines a one-parameter semigroup of


,0
KttU,
contractive transformations of the space H, and for any
0 of the Cauchy problem for Equation (8) has a
unique solution defined by equality (10), i.e. by the
action of the semigroup
uH


,0,tt
K
U on the initial
condition .
0
So that, to find the approximate solution of the Cauchy
problem by Feynman formula, we define, generalizing
the formula (6), operator-valued function
u

,, ,0 AHKttF, such that (see (7))
 
 
,,0 ,0
2
0
=
1expd d
4
2π
AHK AH
R
tu tu
xz
K
zyzuyy
t
t


 



FF
(11)
Lemma 3. For any


00
uCR

 the equality

,, 00
=0
d
dAHK K
t
tu u
t




FL
holds.
According to the formula (11)



 
,, 0
=0
2
1
d
d
ˆ
2cos
AHK
t
N
kk
k
tu s
t
s
ahsKsu





 



F
s
therefore
 
,, 00
=0
d,
dAHK K
t
tu xu
t




FL
hence lemma
3 is proved.
Lemma 4. Operator-valued function


,, ,0 AHKttF,
is continuous in the strong operator topology on the
and admits the estimate in the norm 0,t


1
,,
1
12 ,
N
AHKk L
BH k
ttaK


0t





F.
Operator-valued function admits the repre-
sentation
,,AHK
F
 
 
,, 000
1
0d
kk
N
AHKk hh
k
y
R
tttaStSt
tStK yy



 




FF FF
F
Since



01
BH
t

F then

V. SAKBAEV, A. YAAKBARIEH 300

 1
,, 12 AHK
L
BH
ttat FK. Operator-valued fun-
ction is continuous in the strong operator
topology and uniformly bounded in the norm topology,
and the operators h

0, 0,ttF
S
is bounded and does not depend
on the variable t. Therefore operator-valued function
is continuous in the strong operator topology.

t
,,AHK
According to lemmas 3 and 4 the function
satisfies all conditions of Chernoff
theorem. Hence the following statement is obtained.
F

,, ,0 AHKttF,
Theorem 4. Let

1
K
LR. Then operator-valued
function


,, , 0, AHKttF is equivalent by Chernoff to
the semigroup


, 0
KttU.
5. Remarks on Feynman-Kac
Representation
Using the result of Theorem 3, we obtain approximate
solution of the Cauchy problem (1), (2) by sequence of
multiple integrals, which integrand expression contains
elementary functions and the initial condition. Therefore
we obtain the solution by passing to the limit




 
0
,11,1212,110000
,,,,, ,dd,
lim ahmmah mmmmahm
mRR R
Utu x
Ftt xyFttyyFtyyuyyy
 

 
 

1

where jj t
N
t. For any m

N the -multiply inte- m
gral under the limit operation defines the values of
measure Feynman-Kac on cylindrical sets and hence, on
the algebra of cylindrical sets in the space of continuous
maps of semiaxis into the coordinate space R (see
[4,9]).
R
Conversely we can obtaine the expression for the
Cauchy problem solution by Feynman-Kac formula. The
markovian measure

(see [10,11]) is defined by
means of semigroup


,0tt

,CRR

,

U on the algebra of
cylindrical sets in space of continuous maps
of semiaxis



0,

 into coordinate space R by the
following rule. The value of measure

on an arbitrary
cylindrical set




,: ,1,
jj
A
CR RtB jn 


(where
j
B is bounded set from algebra

Rorel
subsets of R, 2, 3,n of B

 12 ) is
ity
and
given by
1
0n t tt
equal

  


12
2
1123221
,
nn
BnnBnnBB
L
R
A
ttt ttttt




 UPU UPU
(Here
B

—characteristic function of set B, and
B
P
—the projective operator of multiplication to characteri-
stic function of set B).
Then according to the work [12] the following
Feynman-Kac formula
 


    

0
,
,
0, ,
BB
CR R
utt u
tB R
0d,




 


uniquely defines the solution of the
Cauchy problem (1), (2).
 
0
ut tuU
6. Conclusion
In this paper we obtain the approximation of solution of
Cauchy problem for parabolic difference-differential
equation with deviation of the space arguments by
Feynman formulas. This result gives the opportunity to
find the approximation of exact solution of Cauchy pro-
blems by using only integration with analitic fun-
ction. Also this result gives the approximation of diffu-
sion type random process with values in coordinate space
such that the mean value of functional depending on this
process is the solution of considered Cauchy problem.
N
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