 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 Email: fumi2003@mail.ru, amirmath20@yahoo.com 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,,,,,,tNkk kiuxt uxtauxht uxhtxtRR,  (1) supplemented with the initial data  0,0, .uxux xR0 (2) Here , for any , u0 is a given function and NN,kkaRh21, ,kN2x  is self-adjoint Laplas operator in the space 2LR with domain 22WRu. 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 ,,,,tuxtuxtg 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 ). 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 , 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 ). In work  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. Copyright © 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 21220, ,0, ,uCWR CLR  which Sa- tisfies the equation (1) and condition (2). Function 20, ,uC LR0ku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 , 0kN02) for each there is a strong solution of the Cauchy problem with initial condition , ukku0k3) the sequence of functions converges to fun- ukuctions in space . u20, ,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,,,,tNkk kiF uxtFu xtFa 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 21,,,expi,exp itNkkiUstsUstaUs tshUs tsh, and the initial condition (2) transforms into the equation:  00,0 ,UsUsFuss R . Then by using of the equations  21,,exp iexpi,tNkk kiUstsUstashshUs and the initial condition we obtain that  201,exp2cosNkkiUstUssasht. (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,UstTheorem 1. The formula (4) defines a strongly continuous semigroup , 0,Ut t transformation of the space 2LR. In fact, according to the unitarity of the the Fourier transform in space 2HLR 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 2LR 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, ,ssR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 ,supsRfsos,kk where 21i2cNfssa shsR . Therefore 12Nkka. 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0In fact, according to the proposition 1, if the solution of the Cauchy problem (1), (2) exists, then it re- presentable in the form u0121,*exp2 cos.Nkkiuxtu xFsasht (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 02Us LR satisfy the condition 202sUs LR then the function t,Us (see (4)): belongs to space 1220, ,0, ,CLRCL R; satisfies the inclusion 2,0,,2sUstCL R and the equation 21,2cosNkkiUstsbsh Ustt ,. 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 202uWRCopyright © 2012 SciRes. AJCM V. SAKBAEV, A. YAAKBARIEH Copyright © 2012 SciRes. AJCM 2970function under the action of the semigroup 202uWR,ttU0,0suplim nnHTtnut u FU operators According to Theorem 1 the semigroup has a bounded exponential growth tU, then the semigroup supposes continuation by tU2WRsatisfies for all . 0TLet us assume that 1N. For given in the Equation (1) parameters 1aa and 1Hh:RB we consider the operator-valued function , defined on H,Fcontinuity from space onto the space 22LR. Hence for any initial condition the function 02uLR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,kkNkhhka LSS ,02220,1exp 42πexp expd44ah tu utxxyttxyh xyhatuy ytt     F where hS 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 0uH linear manifold R0DC. 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. tFollowing the approach offered in , we define the operator-valued function equivalent in Chernoff sense to the semigroup of operators Chernoff’s theorem (see ) 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 1,0BHtt F,t for some 0 and, more- Lemma 1. If 0uD 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=020002dd.ah ttutuxau xhu xhxF ,0ttU. Then for any uH and any the 0Tequality  0,0suplim nnHtT tutn u UF . By following the definition in  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 22,,0, uutx txRtx . Since .uH the equality      222022222πexpexp expd4442π2π1exp expexp44 442πxyxyh xyhuatu y yttttttxy xyxyhxyhtt tttxat                   2222022expexp d4444yh xyhxyh xyhuyytttt      and V. SAKBAEV, A. YAAKBARIEH 298  22 22222222 20211exp exp24442π1exp exp24 441expexp d24 44xy xyxyutt txttxyh xyhxyhat tt ttxyhxyhxyh uyytt tt               then   222021,exp exp442πxyh xyhuutxaau y yttxt     dt. Consequently    ,0=022202020002dd,exp explim 442π.ah tttutxyh xyhuatxu y yttxtuxauxhuxhx Fd Lemma 1 is proved. For arbitrary and for given in Equation (1) NNcoefficients ,, 1,kkahkN we define the operator fun- ction ,, 0,AH ttFDF assuming that for each function its image defined by equality 0u,AH tu0 ,22=11exp 42πexp exp.44AHNkkktuxxyttxyh xyhat tt 2kF (7) Hence, by lemma 1 on linear manifold D the operator ,0ddAH tttF coincides with the generator of the semi- group , 0.Ut tLemma 2. The operator-valued function ,,0AH ttF, is continuous in the strong operator topology on the semiaxis and supposes an esti- 0tmation on norm ,=112, 0NahkBH kttat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  ,221,, 2πexp exp44ah txy txyh xyhat tt . And ,0ahh htatt 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: 01BHt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2BH at. If NN then  ,0 01,0kkNAHkh hkttatt t.FF FSS Therefore the operator-valued function ,AH tF is continuous in the strong operator topology and satisfy the estimate ,112NAH kBH ktt 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Copyright © 2012 SciRes. AJCM V. SAKBAEV, A. YAAKBARIEH 299of 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U4. 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,, ,,tNkk kkRKuxt uxtauth uthKx yuyt yutxxtR 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 1LRˆK of the func- tion K is a bounded continuous real-valued function. Let us define ,,Ust uxt and ˆKK. According to our assumption ˆKLR. Then we obtaine the relation  2ˆ,exp iexpi2π,NtkkkikUstsashshKsUst  by applying Fourier transform F to the equality (8). Hence we obtain, that  20=10ˆ,exp2cos2πNkkiKUstsashttKsUstU s U (9) and therefore  12=1ˆ,*exp2cosexp2πNkkiuxtuxtsashtKs (10) Therefore the following analogue of Theorems 1 and 2: Proposition 2. If the kernel 1KLR, then equality (9) defines a one-parameter semigroup of ,0Ktt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KU on the initial condition . 0So that, to find the approximate solution of the Cauchy problem by Feynman formula, we define, generalizing the formula (6), operator-valued function u,, ,0AHKttF, such that (see (7))   ,,0 ,020=1expd d42πAHK AHRtu tuxzKzyzuyyttFF (11) Lemma 3. For any 00uCR the equality ,, 00=0ddAHK Kttu utFL holds. According to the formula (11) ,, 0=021ddˆ2cosAHKtNkkktu stsahsKsu Fs therefore  ,, 00=0d,dAHK Kttu xutFL hence lemma 3 is proved. Lemma 4. Operator-valued function ,, ,0AHKttF, is continuous in the strong operator topology on the and admits the estimate in the norm 0,t1,,112 ,NAHKk LBH kttaK0tF. Operator-valued function admits the repre- sentation ,,AHKF  ,, 00010dkkNAHKk hhkyRtttaStSttStK yy FF FFF Since 01BHtF then Copyright © 2012 SciRes. AJCM V. SAKBAEV, A. YAAKBARIEH 300 1,, 12AHKLBHttat 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FS is bounded and does not depend on the variable t. Therefore operator-valued function is continuous in the strong operator topology. t,,AHKAccording to lemmas 3 and 4 the function satisfies all conditions of Chernoff theorem. Hence the following statement is obtained. F,, ,0AHKttF,Theorem 4. Let 1KLR. Then operator-valued function ,, , 0,AHKttF is equivalent by Chernoff to the semigroup , 0Ktt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 mmmmahmmRR RUtu xFtt xyFttyyFtyyuyyy   1 where jjtNt. For any mN the -multiply inte- mgral 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,jjACR RtB jn (where jB is bounded set from algebra Rorel subsets of R, 2, 3,n of B 12 ) is ity andgiven by10nt tt equal  1221123221,nnBnnBnnBBLRAttt ttttt UPU UPU (Here B—characteristic function of set B, and BP —the projective operator of multiplication to characteri- stic function of set B). Then according to the work  the following Feynman-Kac formula 0,,0, ,BBCR Rutt utB R0d,  uniquely defines the solution of the Cauchy problem (1), (2).  0ut tuU6. 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