Lie symmetries of fundamental solutions of one (2+1)-dimensional ultra-parabolic Fokker--Planck--Kolmogorov equation
aa r X i v : . [ m a t h - ph ] A ug Lie symmetries of fundamental solutions of one(2+1)-dimensional ultra-parabolicFokker–Planck–Kolmogorov equation
Sergii Kovalenko, Valeriy Stogniy and Maksym Tertychnyi Department of Physics, Faculty of Oil, Gas and Nature Engineering,Poltava National Technical University, Poltava, Ukraine (e-mail: [email protected]) Department of Mathematical Physics, Faculty of Physics and Mathematics,National Technical University of Ukraine ”Kyiv Polytechnic Institute”, Kyiv, Ukraine (e-mail: valeriy [email protected]) Department of Mathematics and Statistics, Faculty of Science,University of Calgary, Calgary, Canada (e-mail: [email protected])
Abstract
A (2+1)-dimensional linear ultra-parabolic Fokker–Planck–Kolmogorov equation isinvestigated from the group-theoretical point of view. By using the Berest–Aksenov ap-proach, an algebra of invariance of fundamental solutions of the equation is found. Afundamental solution of the equation under study is computed in an explicit form as aweak invariant solution.
Keywords:
Fokker–Planck–Kolmogorov equation, fundamental solution, Lie symmetries.
Introduction
The idea that a solution of any well-posed boundary problem for a particular linear partialdifferential equation (PDE) can be reduced to the construction of some solution of a specialtype (known as a fundamental solution) is one of the most powerful approaches in a classicalmathematical physics. The method of integral transformations is a powerful and well-developedtool for the construction of the solutions of such a type. Unfortunately, this method is efficientonly for linear PDE’s with constant(or, at least, analytic) coefficients. Moreover, in some cases itis very difficult to investigate qualitative properties of fundamental solutions, because they cannot be presented in an explicit form, but only in the form of inverse integral transformations. Asa result, the development of more direct methods for the construction of fundamental solutionsis an important problem of a modern mathematical physics. Especially, it is urgent in the caseof PDE’s with alternating coefficients. One of such modern methods is a classical Lie methodfor investigation of symmetrical properties of PDE’s.Group analysis of differential equations is a mathematical theory with an area of interest inthe symmetry properties of differential equations. Basics of group analysis are in fundamentalworks of S. Lie and his scholars (see, for instance, [1]). Namely, Lie developed and was thefirst to use the tools of symmetry reduction. The main idea of this method is to search forthe solution of the equation under consideration in the form of a special substitution (ansatz),that reduces the given equation to the differential equation with less number of independentvariables.Further development of group-theoretical methods of differential equations is mainly con-nected with works of the following mathematicians: G. Bikhoff [2], L. I. Sedov [3], A. J. A. Mo-rgan [4], L. V. Ovsiannikov [5], N. H. Ibragimov [6–8], P. J. Olver [9,10], W. I. Fushchich [11–13]and others. At present, symmetry properties of many well-known equations of mechanics, gasdynamics, quantum physics, etc were investigated. Detailed analysis of results of research ofsymmetry properties of a wide range of linear and non-linear differential equations can be foundin the monographs [5, 6, 11–16].It is well-known, that as a rule fundamental solutions of linear PDE’s are invariant withrespect to the transformations, admitted by the prescribed equation [7, 17–25]. In particular,fundamental solutions of classical equations of mathematical physics such as the Laplace equa-2ion, the Heat equation, the Wave equation, etc have this property (see, for example, [18]). Itmeans, that if a linear PDE (especially in case of an equation with alternating coefficients) hasnon-trivial symmetry properties, then we can use group-theoretical methods to construct itsfundamental solution.The object of our investigation is the linear (2+1)-dimensional ultra-parabolic Fokker–Planck–Kolmogorov equation u t − u xx + xu y = 0 , ( x, y, t ) ∈ R , (1)where u = u ( t, x, y ), u t = ∂u∂t , u y = ∂u∂y , u xx = ∂ u∂x .Eq. (1) is the simplest and lowest dimensional version of the following (2 n +1)-dimensionallinear ultra-parabolic equation u t − n X j, l =1 ( k j l ( t, x, y ) u ) x j x l + n X j =1 ( f j ( t, x, y ) u ) x j + n X j =1 x j u y j = 0 , ( x, y, t ) ∈ R n +1 . (2)A. N. Kolmogorov introduced (2) in 1934 to describe the probability density of a system with2 n degrees of freedom [26]. Here, the 2 n -dimensional space is the phase space, x = ( x , . . . , x n )is the velocity and y = ( y , . . . , y n ) is the position of the system. It should be also stressed thatEq. (1) arises in mathematical finance in some generalization of the Black–Scholes model (see,for instance, [27, 28]).Research of symmetry properties of Eq. (1) was commenced in work [29], where severalpoint transformations of symmetries were found. But complete group analysis for this equationwas not performed.Maximal invariance algebra of Eq. (1) in Lie sense was calculated in work [30]. In addition,in this article an optimal system of sub-algebras was found for the calculated invariance alge-bra. Using two-dimensional sub-algebras, it was also performed the symmetry reduction andconstructed several explicit invariant solutions of Eq. (1).In this article, we continue research of symmetry properties of Eq. (1). The goal of ourwork is to find an invariance algebra of fundamental solutions of this equation and to constructin an explicit form the fundamental solution of Eq. (1) using already found symmetry algebra.3 Symmetries of fundamental solutions of linear PDE’s
Consider linear homogeneous PDE of p -th order with m independent variables Lu ≡ p X | α | =0 A α ( x ) D α u = 0 , x ∈ R m . (3)In (3) we use standard notations: x = ( x , . . . , x m ), α = ( α , . . . , α m ) is a multi-index withinteger non-negative components, | α | = α + . . . + α m ; D α ≡ (cid:18) ∂∂x (cid:19) α . . . (cid:18) ∂∂x m (cid:19) α m ; A α ( x ) are some smooth functions of the variable x . Fundamental solution of Eq. (3) is a function u ( x, x ) (namely, generalized), that yields thefollowing equation Lu = δ ( x − x ) , (4)where δ ( x − x ) is the Dirac delta function.Standard methods to find fundamental solutions of linear PDE’s are the method of integraltransformations (especially, in case of the equations with constant coefficients), the Green’sfunctions method, etc. [31, 32]. Here we consider an algorithm to find fundamental solutions oflinear homogeneous PDE’s by using symmetry groups of this equation.Remind, that a non-degenerate local substitution of the variables x, u ¯ x i = f i ( x, u, a ) , ¯ u = g ( x, u, a ) , i = 1 , . . . , m, (5)depending on a continuous parameter a is called a symmetry transformation of Eq. (3), if thisequation does not change its form with respect to the new variables ¯ x and ¯ u . A set G of allsuch transformations forms a Lie group (local, more precisely), that is called a symmetry group (or an acceptable group) of Eq. (3).According to the Lie theory, the construction of the symmetry group G of Eq. (3) isequivalent to finding its infinitesimal transformations:¯ x i ≈ x i + a · ξ i ( x, u ) , ¯ u ≈ u + a · η ( x, u ) , i = 1 , . . . , m, (6)where x i ( x, u ) and η ( x, u ) are some smooth functions.4inear differential operator of the first order X = m X i =1 ξ i ( x, u ) ∂∂x i + η ( x, u ) ∂∂u (7)is called an infinitesimal operator of the group G . The operator X is also called a symmetryoperator of Eq. (3).The group transformations (5) corresponding to the infinitesimal transformations (6) withthe operator (7) are found using so called Lie equations: d ¯ x i da = ξ i (¯ x, ¯ u ) , d ¯ uda = η (¯ x, ¯ u ) , i = 1 , . . . , m with the initial conditions ¯ x i | a =0 = x i , ¯ u | a =0 = u. So called infinitesimal criterion of invariance plays a fundamental role in the symmetryanalysis of differential equation, that is in case of Eq. (3) can be formulated in the followingform.
Theorem 1.
The infinitesimal operator (7) is a symmetry operator of Eq. (3) , if and only ifthere exists such a function λ = λ ( x ) , that yields the following identities: X p ( Lu ) ≡ λ ( x ) · Lu (8) for any function u = u ( x ) from the domain of Eq. (3) . In Eq. (8), X p is the prolongation of p -th order of the infinitesimal operator (7), that iscalculated by the well-known formula [5, 9, 13, 16]: X p = X + p X k =1 m X i ,...,i k =1 ϕ i ··· i k ∂∂u i ··· i k , where ϕ i ··· i k = D i · · · D i k η − m X i =1 ξ i u i ! + m X i =1 ξ i u i ··· i k i ,i , . . . , i k = 1 , . . . , m, k = 1 , . . . , p. Here we denote by D i the operator of total differentiation with respect to the variable x i : D i = ∂∂x i + u i ∂∂u + ∞ X j =1 m X i ,...,i j =1 u i ··· i j i ∂∂u i ··· i j , i = 1 , . . . , m.
5t was shown in [33], that the linear homogeneous PDE (3) given additional conditions p ≥ m ≥ X = m X i =1 ξ i ( x ) ∂∂x i + ( α ( x ) u + β ( x )) ∂∂u . (9)In the class of infinitesimal operators of the form (9), the maximal invariance algebra of Eq.(3) as a vector space is a direct sum of two sub-algebras: the sub-algebra, that consists of theoperators of the form X = m X i =1 ξ i ( x ) ∂∂x i + α ( x ) u ∂∂u , (10)and the infinite-dimensional subalgebra, that is generated by the operators X = β ( x ) ∂∂u , (11)where β = β ( x ) is an arbitrary smooth solution of Eq. (3).It is clear, that the infinitesimal operators (11) are symmetry operators of Eq. (4). Hence,in the sequel we will consider only operators of the form given by (10).Constructive method to find symmetries of the form (10) of linear inhomogeneous PDE’swith δ -function in a right-hand side was proposed in works [17, 20]. In the same articles, it wasintroduced an algorithm for construction of invariant fundamental solutions of the equations ofthe form (3).Main result of these works is in the following statement. Theorem 2.
The Lie algebra of symmetry operators of the form (10) of Eq. (4) is a sub-algebra of the Lie algebra of symmetry operators of Eq. (3) , which is defined by the followingconditions: ξ i ( x ) = 0 , (12) λ ( x ) + m X i =1 ∂ξ i ( x ) ∂x i = 0 , (13) where i = 1 , . . . , m . Formulate the algorithm to find fundamental solutions of a linear PDE using properties ofits invariance algebra:1. find a general form of symmetry operator of Eq. (3) and a relevant function λ ( x ), thatyields identities (8); 6. using conditions (12) and (13), find a Lie algebra of symmetry operators of Eq. (4);3. construct invariant fundamental solutions of Eq. (3) using symmetry operators of Eq. (4). Remark 1.
Formulated algorithm for construction of fundamental solutions using symmetriesof linear PDE’s with δ -function in a right-hand side is especially efficient for multi-dimensionallinear equations and for the equations with alternating coefficients in case, if they allow ratherwide invariance algebras. Remark 2.
To find generalized invariant fundamental solutions, it is necessary to solve reducedequations (these equations are written in invariants of corresponding transformation groups) onthe set of generalized functions (see, for example, [19, 20, 34]). (1)
Apply the method described in the previous section to the object of our research, in otherwords to the linear Fokker–Planck–Kolmogorov equation (1).We define fundamental solution of Eq. (1) as a generalized function u = u ( t, x, y, t , x , y ),that depends on t , x , y as parameters and yields the equation u t − u xx + xu y = δ ( t − t , x − x , y − y ) , (14)given the additional condition u | t Write the general form of symmetry operator of the finite-dimensional part of invariancealgebra of Eq. (1) X = X i =1 a i X i , (15)or in a more detailed form X =(2 a t + a t + a ) ∂ t + ( a + a x + a ( tx + 3 y ) + 3 a t + 2 a t ) ∂ x ++ ( a t + 3 a y + 3 a ty + a t + a t + a ) ∂ y ++ ( − a − a (2 t + x ) + 3 a ( y − tx ) − a x + a ) u∂ u , where a i ( i = 1 , . . . , 8) are any real constants.Substituting the infinitesimal operator (15) into Eq. (8), where we put Lu ≡ u t − u xx + xu y , p = 2, we find the function λ = λ ( t, x, y ), that corresponds to this operator: λ ( t, x, y ) = − a − a (4 t + x ) + 3 a ( y − tx ) − a x + a . (16)Substituting (15) and (16) into Eqs. (12) and (13) (see, Th. 1), we obtain the followingequalities: 2 a t + a t + a = 0; a + a x + a ( t x + 3 y ) + 3 a t + 2 a t = 0; a t + 3 a y + 3 a t y + a t + a t + a = 0;2 a − a (2 t − x ) + 3 a ( y − t x ) − a x + a = 0 , a = − a x − a ( t x + 3 y ) − a t − a t ; a = − a t − a t ; a = a ( t x − y ) + a t x + 2 a t + a t ; a = − a + a (2 t − x ) − a ( y − t x ) + a x . Substituting in (15) the constants a , a , a , and a by the calculated expressions and hav-ing split with respect to the independent constants a , a , a , a , we obtain that the finite-dimensional part of invariance algebra of Eq. (14) is four-dimensional and generated by thefollowing operators: Y = 2( t − t ) ∂ t + ( x − x ) ∂ x − ( x ( t − t ) − y − y )) ∂ y − u∂ u ,Y = ( t − t ) ∂ t + (( tx + 3 y ) − ( t x + 3 y )) ∂ x + (3( y − y ) t − t x ( t − t )) ∂ y −− (2( t − t ) + x − x ) u∂ u ,Y = 3( t − t ) ∂ x + ( t − t t + 2 t ) ∂ y − tx − y − ( t x − y )) u∂ u ,Y = 2( t − t ) ∂ x + ( t − t ) ∂ y − ( x − x ) u∂ u . Invariance property of Eq. (14) with respect to the operators of the form β ( t, x, y ) ∂ u , where β ( t, x, y ) is an arbitrary smooth solution of Eq. (1) is straightforward.The proof is now completed.Show how the results of Th. 3 can be applied to the construction of invariant fundamentalsolutions of Eq. (1). Use for this, for example, the two-dimensional algebra of operators h Y , Y i . Classical invariants corresponding to these operators can be derived from the systemof equations Y I = 0 , Y I = 0 , where I = I ( t, x, y, u ), and the equations are solved in a classical sense. We obtain after relatedcalculations: I = ( t − t ) exp (cid:20) ( x − x ) t − t ) (cid:21) u, I = ( t − t )( x + x ) − y − y )( t − t ) / . Classical invariant solution is found from the equality I = ϕ ( I ), from which we obtain asubstitution (ansatz) u = 1( t − t ) exp (cid:20) − ( x − x ) t − t ) (cid:21) ϕ ( ω ) , ω = I , (17)9hat reduces the equation under study (1) to the following ordinary differential equation: d ϕdω + 32 ω dϕdω + 32 ϕ = 0 . It is well-known, that a particular solution of this equation is the following function [35, P. 216]: ϕ = C exp (cid:20) − ω (cid:21) . (18)Substituting (18) to (17), we derive a classical invariant solution of Eq. (1) corresponding tothe operators Y and Y : u = C ( t − t ) exp " − ( x − x ) t − t ) − t − t ) (cid:18) y − y − ( t − t ) x + x (cid:19) . (19)Substituting (19) to Eq. (14), it is easy to show that Lu ( t, x, y ) = 0. As a result, thesolution (19) is not a fundamental solution of the Fokker–Planck–Kolmogorov equation (1). Toconstruct its weak invariant solution we use Statement 1 from work [36], in other words wesearch for weak invariant solutions in the form u = h ( t, x, y )( t − t ) exp " − ( x − x ) t − t ) − t − t ) (cid:18) y − y − ( t − t ) x + x (cid:19) , where h = h ( t, x, y ) ∈ D ′ ( R ) (denote by D ′ = D ′ ( R ) the space of generalized functions). Theequations Y u = 0 and Y u = 0 give us correspondingly:2( t − t ) h t + ( x − x ) h x − ( x ( t − t ) − y − y )) h y = 0 , t − t ) h x + ( t − t ) h y = 0 . It is easy to see that the generalized function h ( t, x, y ) = C θ ( t − t ) + C is a solution of theseequations (here, θ ( t − t ) is the Heaviside step function). We obtain using this fact: u = C θ ( t − t ) + C ( t − t ) exp " − ( x − x ) t − t ) − t − t ) (cid:18) y − y − ( t − t ) x + x (cid:19) . (20)Substituting (20) to Eq. (14), we find the constant C = √ π .Hence, the fundamental solution of the Fokker–Planck–Kolmogorov equation (1) u = √ π θ ( t − t ) + C ( t − t ) exp " − ( x − x ) t − t ) − t − t ) (cid:18) y − y − ( t − t ) x + x (cid:19) (21)is found as a weak invariant solution with respect to a two-dimensional algebra h Y , Y i of pointsymmetries of Eq. (14). In formula (21) we can put C = 0, because a fundamental solution isdefined up to the addition of any solution of a homogeneous equation.10 emark 3. The fundamental solution (21) was found by A. N. Kolmogorov [37] without us-ing the methods of symmetry analysis of differential equations. Our calculations give group-theoretical background for this solution and justify once again an empiric observation, thatfundamental solutions of linear PDE’s should be searched among the invariant solutions. Remark 4. Using a similar scheme, as it was performed for Eq. (1) , we can investigate thesymmetry properties of fundamental solutions of other many-dimensional and more complicatedKolmogorov type equations (2) . In this article, we restricted our attention to Eq. (1) in ordernot to lose the main idea of Berest–Aksenov method of construction of fundamental solutions oflinear homogeneous PDE’s using its group properties under complicated technical calculations. In this article, using the Berest–Aksenov method [17, 20] we found an invariance algebra offundamental solutions of linear Fokker–Planck–Kolmogorov equation (1). Its operators wereused to construct invariant fundamental solutions of this equation. It was shown that the fun-damental solution (21) of Eq. (1), found by A. N. Kolmogorov is a weak invariant fundamentalsolution. References [1] S. Lie: Theory of Transformation Groups , vol. 1–3, B. G. Teubner, Leipzig 1888–93 (inGerman).[2] G. 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