INTEGRAL EQUATION FOR THE TRANSITION DENSITY OF THE MULTIDIMENSIONAL MARKOV RANDOM FLIGHT

Transcription

1 Theory of Sochasic Processes Vol. 2 (36), no. 2, 215, pp ALEXANDER D. KOLESNIK INTEGRAL EQUATION FOR THE TRANSITION DENSITY OF THE MULTIDIMENSIONAL MARKOV RANDOM FLIGHT We consider he Markov random fligh X() in he Euclidean space R m, m 2, saring from he origin R m ha, a Poisson-paced imes, changes is direcion a random according o arbirary disribuion on he uni (m 1)-dimensional sphere S m (, 1) having absoluely coninuous densiy. For any ime insan >, he convoluion-ype recurren relaions for he join and condiional densiies of he process X() and of he number of changes of direcion, are obained. Using hese relaions, we derive an inegral equaion for he ransiion densiy of X() whose soluion is given in he form of a uniformly convergen series composed of he muliple double convoluions of he singular componen of he densiy wih iself. Two imporan paricular cases of he uniform disribuion on S m (, 1) and of he circular Gaussian law on he uni circle S 2 (, 1) are considered separaely. 1. Inroducion Coninuous-ime random walks are an imporan field of sochasic processes (see, for insance, 1], 15, 16] and bibliography herein). An imporan case of such processes, random moions a finie speed in he mulidimensional Euclidean spaces R m, m 2, also called random flighs, became he subjec of inense research in las decades. The majoriy of published works deal wih he case of isoropic Markov random flighs when he moions are conrolled by a homogeneous Poisson process and heir direcions are aken uniformly on he uni (m 1)-dimensional sphere 3 8], 14], 19, 2]. The limiing behaviour of a Markov random fligh wih a finie number of fixed direcions in R m was examined in 2]. In recen years he non-markovian mulidimensional random walks wih Erlang- and Dirichle-disribued displacemens were sudied in a series of works 1 13], 17, 18]. Such random moions a finie velociies are of a grea ineres due o heir major heoreical imporance and numerous fruiful applicaions in physics, chemisry, biology and oher fields. When sudying such a moion, is explici disribuion is, undoubedly, he mos aracive aim of he research. However, despie many effors, he closed-form expressions for he disribuions of Markov random flighs were obained only in a few cases. In he spaces of low even dimensions such disribuions were obained in explici forms by differen mehods (see 2], 14], 6], 8] for he Euclidean plane R 2, 7] for he space R 4 and 3] for he space R 6 ). Moreover, in he spaces R 2 and R 4 such disribuions are surprisingly expressed in erms of elemenary funcions, while in he space R 6 he disribuion has he form of a series composed of some polynomials. As far as he random flighs in he odd-dimensional Euclidean spaces are concerned, heir analysis is much more complicaed in comparison wih he even-dimensional cases. A formula for he ransiion densiy of he symmeric Markov random fligh wih uni speed in he space R 3 was given in 19], however i has a very complicaed form of an inegral wih variable limis, involving he inverse hyperbolic angen funcion of he inegraion 21 Mahemaics Subjec Classificaion. 6K35, 6K99, 6J6, 6J65, 82C41, 82C7. Key words and phrases. Random fligh, coninuous-ime random walk, join densiy, condiional densiy, convoluion, ransiion densiy, inegral equaion, Fourier ransform, characerisic funcion, uniform disribuion on sphere, circular Gaussian law. 42

2 INTEGRAL EQUATION FOR THE TRANSITION DENSITY variable (see 19, formulas (1.3) and (4.21)]). Moreover, he densiy presened in his work evokes some quesions since is absoluely coninuous (inegral) par is disconinuous a he origin R 3 and his fac seems fairly srange. The characerisic funcions of he mulidimensional random flighs are much more convenien objecs o analyse han heir densiies. This is due o he following fac: if he densiies are funcions wih compac suppor hen heir characerisic funcions (Fourier ransforms) are analyical real funcions defined everywhere in R m. Tha is why hese characerisic funcions were he subjec of vas research, whose resuls were published in 4] and 9]. In paricular, in 4] he ime-convoluional recurren relaions for he join and condiional characerisic funcions of a Markov random fligh in he Euclidean space R m of arbirary dimension m 2 were obained. By using hese recurren relaions, he Volerra inegral equaion of second kind wih coninuous kernel for he uncondiional characerisic funcion was derived and a closed-form expression for is Laplace ransform was given. Such convoluional srucure of he characerisic funcions suggess a similar one for he respecive densiies. The discovery of convoluional relaions for he densiies of Markov random flighs in R m, m 2, is he main subjec of his aricle. The paper is organized as follows. In Secion 2 we inroduce general Markov random flighs in he Euclidean spaces R m, m 2, wih arbirary dissipaion funcion and describe he srucure of heir disribuion. Some basic properies of he join, condiional and uncondiional characerisic funcions of he process are also given. In Secion 3 we derive he recurren relaions for he join and condiional densiies of he process and of he number of changes of direcion in a form of double convoluions wih respec o he space and ime variables. Based on hese recurren relaions, an inegral equaion for he ransiion densiy of he process is obained in Secion 4, whose soluion is given in a form of uniformly converging series composed of he muliple double convoluions of he singular componen of he densiy wih iself. This soluion is unique in he class of funcions wih compac suppor in R m. Two imporan paricular cases of he uniform disribuion on S m (, 1) and of he circular Gaussian law on he uni circle S 2 (, 1) are considered in Secion Descripion of he process and is basic properies Consider he following sochasic moion. A paricle sars from he origin = (,..., ) of he Euclidean space R m, m 2, a he iniial ime insan = and moves wih some consan speed c (noe ha c is reaed as he consan norm of he velociy). The iniial direcion is a random m-dimensional vecor wih arbirary disribuion (also called he dissipaion funcion) on he uni sphere S m (, 1) = { x = (x 1,..., x m ) R m : x 2 = x x 2 m = 1 } having an absoluely coninuous bounded densiy χ(x), x S m (, 1). I should be emphasized ha here and hereafer he upper index m means he dimension of he space, in which he sphere S m (, 1) is considered, bu no he sphere s own dimension, which, clearly, is m 1. The moion is conrolled by a homogeneous Poisson process N() of rae λ > as follows. A each Poissonian insan he paricle insananeously akes on a new random direcion in S m (, 1) disribued wih he same densiy χ(x), x S m (, 1), independenly of is previous moion, and keeps moving wih he same speed c unil he nex Poisson even occurs. Then i akes on a new random direcion again and so on. Le X() = (X 1 (),..., X m ()) be he paricle s posiion a ime > which is referred o as he m-dimensional Markov random fligh. A arbirary ime insan > he paricle, wih probabiliy 1, is locaed in he closed m-dimensional ball of radius c cenred a he origin : B m (, c) = { x = (x 1,..., x m ) R m : x 2 = x x 2 m c 2 2}.

3 44 ALEXANDER D. KOLESNIK Consider he probabiliy disribuion funcion Φ(x, ) = Pr {X() dx}, x B m (, c),, of he process X(), where dx is an infiniesimal elemen in he space R m wih Lebesgue measure µ(dx) = dx 1... dx m. For arbirary fixed >, he disribuion Φ(x, ) consiss of wo componens. The singular componen corresponds o he case when no Poisson evens occur in he ime inerval (, ) and i is concenraed on he sphere S m (, c) = B m (, c) = { x = (x 1,..., x m ) R m : x 2 = x x 2 m = c 2 2}. In his case, a ime insan, he paricle is locaed on he sphere S m (, c) and he probabiliy of his even is Pr {X() S m (, c)} = e λ. If a leas one Poisson even occurs before a ime insan, hen he paricle is locaed sricly inside he ball B m (, c) and he probabiliy of his even is Pr {X() in B m (, c)} = 1 e λ. The par of he disribuion Φ(x, ) corresponding o his case is concenraed in he inerior in B m (, c) = { x = (x 1,..., x m ) R m : x 2 = x x 2 m < c 2 2} of he ball B m (, c) and forms is absoluely coninuous componen. Le p(x, ) = p(x 1,..., x m, ), x B m (, c), >, be he densiy of disribuion Φ(x, ). I has he form p(x, ) = p (s) (x, ) + p (ac) (x, ), x B m (, c), >, (2.1) where p (s) (x, ) is he densiy (in he sense of generalized funcions) of he singular componen of Φ(x, ) concenraed on he sphere S m (, c) and p (ac) (x, ) is he densiy of he absoluely coninuous componen of Φ(x, ) concenraed in in B m (, c). The densiy χ(x), x S m (, 1), on he uni sphere S m (, 1) generaes he absoluely coninuous and bounded (in x for any fixed ) densiy ϱ(x, ), x S m (, c), on he sphere S m (, c) of radius c according o he formula ϱ(x, ) = χ( 1 c x), x Sm (, c), >. Therefore, he singular par of densiy (2.1) has he form: p (s) (x, ) = e λ ϱ(x, )δ(c 2 2 x 2 ), >, (2.2) where δ(x) is he Dirac dela-funcion. The absoluely coninuous par of densiy (2.1) has he form: p (ac) (x, ) = f (ac) (x, )Θ(c x ), >, (2.3) where f (ac) (x, ) is some posiive funcion absoluely coninuous in in B m (, c) and Θ(x) is he Heaviside sep funcion given by { 1, if x >, Θ(x) =, if x. (2.4) Le p n (x, ), n, be he condiional densiies of he process X() condiioned by he random evens {N() = n}, n, where, as before, N() is he number of he Poisson evens ha have occurred in he ime inerval (, ). Obviously p (x, ) = ϱ(x, )δ(c 2 2 x 2 ). The condiional characerisic funcions (Fourier ransforms) of he process X() are: { } G n (α, ) = E e i α,x() N() = n, n 1, (2.5)

4 INTEGRAL EQUATION FOR THE TRANSITION DENSITY where α = (α 1,..., α m ) R m is he real m-dimensional vecor and α, X() is he inner produc of he vecors α and X(). According o 4, formula (6.4)], he funcions (2.5) are given by he formula: G n (α, ) = n! n+1 n dτ 1 dτ 2... dτ n ψ(α, τ τ 1 τ n 1 j τ j 1 ), n 1, (2.6) j=1 where ψ(α, ) = F x ϱ(x, )δ(c 2 2 x 2 ) ] (α) = S m (,c) e i α,x ϱ(x, ) ν(dx) (2.7) is he characerisic funcion (Fourier ransform) of densiy ϱ(x, ) concenraed on he sphere S m (, c) of radius c and ν(dx) is he Lebesgue measure on S m (, c). Noe ha, here and hereafer, he symbol F x means he sandard Fourier ransform wih respec o he spaial variable x. Consider he inegral facor in (2.6) separaely: n+1 J n (α, ) = dτ 1 dτ 2... dτ n ψ(α, τ τ 1 τ n 1 j τ j 1 ), n 1. (2.8) j=1 This funcion has a quie definie probabilisic sense, namely G n (α, ) = F x pn (x, ) ] (α) = (λ)n e λ n! α R m, >, n 1, G n (α, ) = λ n e λ J n (α, ), (2.9) is he characerisic funcion (Fourier ransform) of he join probabiliy densiy p n (x, ) of he paricle s posiion a ime insan and of he number N() = n of he Poisson evens ha have occurred by his ime momen. I is known (see 4, Theorem 5]) ha, for arbirary n 1, funcions (2.8) are conneced wih each oher by he following recurren relaion: J n (α, ) = ψ(α, τ) J n 1 (α, τ) dτ = ψ(α, τ) J n 1 (α, τ) dτ, n 1, (2.1) where, by definiion, J (α, ) def = ψ(α, ). Formula (2.1) can also be represened in he following ime-convoluional form: J n (α, ) = ψ(α, ) J n 1 (α, ), n 1, (2.11) where he symbol means he convoluion operaion wih respec o ime variable. From (2.11) i follows ha J n (α, ) = ψ(α, )] (n+1), n 1, (2.12) where (n + 1) means he (n + 1)-muliple convoluion in. Applying Laplace ransformaion L (in ime variable ) o (2.12), we arrive a he formula L J n (α, )] (s) = ( L ψ(α, )] (s) ) n+1, n 1. (2.13) I is also known (see 4, Corollary 5.3]) ha condiional characerisic funcions (2.6) saisfy he following recurren relaion G n (α, ) = n n τ n 1 ψ(α, τ) G n 1 (α, τ) dτ, G (α, ) def = ψ(α, ), n 1. (2.14)

5 46 ALEXANDER D. KOLESNIK The uncondiional characerisic funcion G(α, ) = E { e i α,x() } (2.15) of process X() saisfies he Volerra inegral equaion of second kind (see 4, Theorem 6]): G(α, ) = e λ ψ(α, ) + λ or in he convoluional form e λ( τ) ψ(α, τ) G(α, τ) dτ,, (2.16) G(α, ) = e λ ψ(α, ) + λ ( e λ ψ(α, ) ) G(α, ) ]. (2.17) This is he renewal equaion for he Markov random fligh X(). In he class of coninuous funcions inegral equaion (2.16) (or (2.17)) has he unique soluion given by he uniformly converging series G(α, ) = e λ n= λ n ψ(α, )] (n+1). (2.18) From (2.17) we obain he general formula for he Laplace ransform of characerisic funcion (2.15): L G(α, )] (s) = L ψ(α, )] (s + λ), Re s >. (2.19) 1 λ L ψ(α, )] (s + λ) These properies will be used in he nex secion for deriving he recurren relaions for he join and condiional densiies of Markov random fligh X(). 3. Recurren relaions Consider he join probabiliy densiies p n (x, ), n, x B m (, c), >, of he paricle s posiion X() a ime insan > and of he number of he Poisson evens {N() = n} ha have occurred before his insan. For n =, we have p (x, ) = p (s) (x, ) = e λ ϱ(x, )δ(c 2 2 x 2 ), >, (3.1) where, as before, p (s) (x, ) is he singular par of densiy (2.1) concenraed on he surface of he sphere S m (, c) = B m (, c) and i is given by (2.2). If n 1, hen, according o (2.3), join densiies p n (x, ) have he form: p n (x, ) = f n (x, )Θ(c x ), n 1, >, (3.2) where f n (x, ), n 1, are some posiive funcions absoluely coninuous in in B m (, c) and Θ(x) is he Heaviside sep funcion. The join densiy p n+1 (x, ) can be expressed hrough he previous one p n (x, ) by means of a recurren relaion. This resul is given by he following heorem. Theorem 1. The join densiies p n (x, ), n 1, are conneced wih each oher by he following recurren relaion: p n+1 (x, ) = λ p (x, τ) x p n (x, τ) ] dτ, n 1, x in B m (, c), >. (3.3) Proof. Applying Fourier ransformaion o he righ-hand side of (3.3), we have: λ F x p (x, τ) x p n (x, τ) ] ] dτ (α) (3.4) = λ ] F x p (x, τ) x p n (x, τ) (α) dτ

6 = λ = λ = λ INTEGRAL EQUATION FOR THE TRANSITION DENSITY F x p (x, τ) ] (α) F x pn (x, τ) ] (α) dτ e λ( τ) F x ϱ(x, τ)δ(c 2 ( τ) 2 x 2 ) ] (α) F x pn (x, τ) ] (α) dτ e λ( τ) ψ(α, τ) λ n e λτ J n (α, τ) dτ = λ n+1 e λ ψ(α, τ) J n (α, τ) dτ = λ n+1 e λ J n+1 (α, ) = F x pn+1 (x, ) ] (α), where we have used formulas (2.7), (2.9), (2.1). Thus, boh he funcions on he lef- and righ-hand sides of (3.3) have he same Fourier ransform and, herefore, hey coincide. The change of inegraion order in he firs sep of (3.4) is jusified because he convoluion p (x, τ) x p n (x, τ) of he singular par p (x, τ) of he densiy wih he absoluely coninuous one p n (x, τ), n 1, is an absoluely coninuous (and, herefore, uniformly bounded in x) funcion. From his fac i follows ha, for any n 1, he inegral in square brackes on he lef-hand side of (3.4) converges uniformly in x for any fixed. The heorem is proved. Remark 1. In view of (2.2) and (2.3), formula (3.3) can be represened in he following expanded form: e λ( τ) p n+1 (x, ) = λ { } ϱ(x ξ, τ)δ(c 2 ( τ) 2 x ξ 2 ) f n (ξ, τ)θ(cτ ξ ) ν(dξ) dτ, n 1, x in B m (, c), >, (3.5) where he funcion f n (ξ, τ) is absoluely coninuous in he variable ξ = (ξ 1,..., ξ m ) R m and ν(dξ) is he Lebesgue measure. The inegraion area in he inerior inegral on he righ-hand side of (3.5) is given by he sysem { x ξ 2 ξ R m = c 2 ( τ) 2, : ξ < cτ. The firs relaion of his sysem deermines a sphere S m (x, c( τ)) of radius c( τ) cenred a poin x, while he second one represens an open ball in B m (, cτ) of radius cτ cenred a he origin. Their inersecion M(x, τ) = S m (x, c( τ)) in B m (, cτ), (3.6) which is a par of (or he whole) surface of sphere S m (x, c( τ)) locaed inside he ball B m (, cτ), represens he inegraion area of dimension m 1 in he inerior inegral of (3.5). Noe ha he sum of he radii of S m (x, c( τ)) and in B m (, cτ) is c( τ)+cτ = c > x, ha is greaer han he disance x beween heir cenres and x. This fac, as well as some simple geomeric reasonings, show ha inersecion (3.6) depends on τ (, ) as follows. If τ (, 2 x 2c ], hen inersecion (3.6) is empy, ha is, M(x, τ) =. If τ ( 2 x ], hen inersecion M(x, τ) is no empy and represens 2c, 2 + x 2c some hypersurface of dimension m 1. If τ ( 2 + x 2c, ), hen Sm (x, c( τ)) in B m (, cτ) and, herefore, in his case M(x, τ) = S m (x, c( τ)).

7 48 ALEXANDER D. KOLESNIK Thus, formula (3.5), as well as (3.3), can be rewrien in he expanded form: p n+1 (x, ) = λ + λ 2 + x 2c 2 x 2c 2 + x 2c e λ( τ) e λ( τ) M(x,τ) S m (x,c( τ)) ϱ(x ξ, τ) f n (ξ, τ) ν(dξ) dτ ϱ(x ξ, τ) f n (ξ, τ) ν(dξ) dτ (3.7) and he expressions in curly brackes of (3.7) represen surface inegrals over M(x, τ) and S m (x, c( τ)). Remark 2. By means of he double convoluion of wo arbirary generalized funcions g 1 (x, ), g 2 (x, ) S, x R m, >, g 1 (x, ) x g 2 (x, ) = R m g 1 (ξ, τ) g 2 (x ξ, τ) dξ dτ (3.8) formula (3.3) can be represened in he succinc convoluional form ] p n+1 (x, ) = λ p (x, ) x p n (x, ). (3.9) Taking ino accoun he well-known connecions beween he join and condiional densiies, we can exrac from Theorem 1 a convoluion-ype recurren relaion for he condiional probabiliy densiies p n (x, ), n 1. Corollary 1. The condiional densiies p n (x, ), n 1, are conneced wih each oher by he following recurren relaion: p n+1 (x, ) = n + 1 n+1 τ n p (x, τ) x p n (x, τ) ] dτ, n 1, x in B m (, c), >, (3.1) where p (x, ) = ϱ(x, )δ(c 2 2 x 2 ) is he condiional densiy corresponding o he case when no Poisson evens occur before ime insan. Proof. The proof immediaely follows from Theorem 1 and recurren formula (2.14). Remark 3. Formulas (3.3) and (3.1) are also valid for n =. In his case, for arbirary >, hey ake he form: p 1 (x, ) = λ p 1 (x, ) = 1 p (x, τ) x p (x, τ) ] dτ, (3.11) p (x, τ) x p (x, τ) ] dτ, (3.12) where, as before, he funcion p (x, ), defined by (3.1), is he singular par of he densiy concenraed on he surface of he sphere S m (, c). The derivaion of (3.11) is a simple recompilaion of he proof of Theorem 1 where one should ake ino accoun he boundedness of he densiy p (x, ) ha jusifies he change of inegraion order in (3.4). Formula (3.12) follows from (3.11).

8 INTEGRAL EQUATION FOR THE TRANSITION DENSITY Inegral equaion for ransiion densiy The ransiion probabiliy densiy p(x, ) of he mulidimensional Markov fligh X() is defined by he formula p(x, ) = p n (x, ), x B m (, c), >, (4.1) n= where he join densiies p n (x, ), n, are given by (3.1) and (3.2). The densiy (4.1) is defined everywhere in he ball B m (, c), while he funcion p ac (x, ) = p n (x, ) (4.2) n=1 forms is absoluely coninuous par concenraed in he inerior in B m (, c) of he ball. Therefore, he series (4.2) converges uniformly everywhere in he closed ball B m (, c ε) for arbirary small ε >. In he following heorem we presen an inegral equaion for he densiy (4.1). Theorem 2. The ransiion probabiliy densiy p(x, ) of he Markov random fligh X() saisfies he inegral equaion: p(x, ) = p (x, ) + λ p (x, τ) x p(x, τ) ] dτ, x B m (, c), >. (4.3) In he class of funcions wih compac suppor, inegral equaion (4.3) has he unique soluion given by he series p(x, ) = λ n p (x, )] x (n+1), (4.4) n= where he symbol x (n + 1) means he (n + 1)-muliple double convoluion wih respec o spaial and ime variables defined by (3.8), ha is, p (x, )] x (n+1) = p (x, ) x p (x, ) x... x p (x, ). }{{} (n+1) erms The series (4.4) is convergen everywhere in he open ball in B m (, c). For any small ε >, he series (4.4) converges uniformly (in x for any fixed > ) in he closed ball B m (, c ε) and, herefore, i deermines he densiy p(x, ) which is coninuous and bounded in his ball. Proof. Applying Theorem 1 and aking ino accoun he uniform convergence of he series (4.2) and of he inegral in he formula (3.3), we have: p(x, ) = p n (x, ) n= = p (x, ) + = p (x, ) + λ = p (x, ) + λ p n (x, ) n=1 n=1 p (x, τ) x p n 1 (x, τ) ] dτ p (x, τ) x p n 1 (x, τ) ] dτ n=1

9 5 ALEXANDER D. KOLESNIK = p (x, ) + λ = p (x, ) + λ = p (x, ) + λ p (x, τ) x p (x, τ) x { }] p n 1 (x, τ) dτ n=1 { }] p n (x, τ) dτ n= p (x, τ) x p(x, τ) ] dτ, proving (4.3). Anoher way of proving he heorem is o apply he Fourier ransformaion o boh sides of (4.3). Afer jusifying he change of he order of inegrals in he same way as i was done in (3.4), we arrive a Volerra inegral equaion (2.16) for Fourier ransforms. Using noaion (3.8), he equaion (4.3) can be represened in he convoluional form p(x, ) = p (x, ) + λ p (x, ) x p(x, ) ], x B m (, c), >. (4.5) Le us check ha he series (4.4) saisfies he equaion (4.5). Subsiuing (4.4) ino he righ-hand side of (4.5), we have: ( p (x, ) + λ p (x, ) x )] λ n p (x, )] x (n+1) = p (x, ) + n= = p (x, ) + = λ n+1 p (x, )] x (n+2) n= λ n p (x, )] x (n+1) n=1 λ n p (x, )] x (n+1) n= = p(x, ) and, herefore, he series (4.4) is he soluion o he equaion (4.5) indeed. Noe ha by applying Fourier ransformaion o (4.3) and (4.4) and aking ino accoun (2.2), we arrive a he known resuls (2.17) and (2.18), respecively. The uniqueness of soluion (4.4) in he class of funcions wih compac suppor follows from he uniqueness of he soluion of Volerra inegral equaion (2.16) for is Fourier ransform (2.18) (i.e. characerisic funcion) in he class of coninuous funcions. Since he ransiion densiy p(x, ) is absoluely coninuous in he open ball in B m (, c), hen, for any ε >, i is coninuous and uniformly bounded in he closed ball B m (, c ε). From his fac and aking ino accoun he uniqueness of he soluion of inegral equaion (4.3) in he class of funcions wih compac suppor, we can conclude ha series (4.4) converges uniformly in B m (, c ε) for any small ε >. This complees he proof. 5. Some paricular cases In his secion we consider wo imporan paricular cases of general Markov random flighs described in Secion 2 when he dissipaion funcion has he uniform disribuion on he uni sphere S m (, 1) or he circular Gaussian law on he uni circle S 2 (, 1) Symmeric random flighs. Suppose ha he iniial and every new direcion are chosen according o he uniform disribuion on he uni sphere S m (, 1). Such processes in he Euclidean spaces R m of differen dimensions m 2, which are referred o as symmeric Markov random flighs, have become he subjec of a series of works 3 8], 14], 19, 2]. In his symmeric case he funcion ϱ(x, ) is he densiy of he uniform disribuion on he surface of he sphere S m (, c) and, herefore, i does no depend on spaial variable

10 INTEGRAL EQUATION FOR THE TRANSITION DENSITY x. Then, according o (2.2), he singular par of he ransiion densiy of process X() akes he form: p (s) (x, ) = e λ Γ ( ) m 2 2π m/2 (c) m 1 δ(c2 2 x 2 ), m 2, >. (5.1) Therefore, according o Theorem 1, for arbirary dimension m 2, he join probabiliy densiies f n (x, ), n 1, of a symmeric Markov random fligh are conneced wih each oher by he following recurren relaion: f n+1 (x, ) = λ Γ ( ) m 2 2π m/2 c m 1 e λ( τ) ( τ) m 1 { M(x,τ) x = (x 1,..., x m ) in B m (, c), m 2, n 1, >, } f n (ξ, τ) dξ dτ, (5.2) where he inegraion area M(x, τ) is given by (3.6). I is known (see 5, formula (7)]) ha, in arbirary dimension m 2, he join densiy of he symmeric Markov random fligh X() and of he single change of direcion is given by he formula f 1 (x, ) = λe λ 2 m 3 Γ ( ) m ( 2 m 1 π m/2 c m F, m m ; m 2 ; x 2 ) c 2 2, (5.3) where x = (x 1,..., x m ) in B m (, c), m 2, >, F (α, β; γ; z) = k= (α) k (β) k (γ) k is he Gauss hypergeomeric funcion. Then, by subsiuing (5.3) ino (5.2) (for n = 1), we obain he following formula for he join densiy of process X() and of wo changes of direcion: f 2 (x, ) = λ 2 e λ 2m 4 Γ ( )] m 2 2 π m c 2m 1 { ( m 1 M(x,τ) F 2 z k k!, m 2 + 2; m 2 ; ξ 2 ) } c 2 τ 2 dξ dτ (τ( τ)) m 1, x = (x 1,..., x m ) in B m (, c), m 2, >. (5.4) In he hree-dimensional Euclidean space R 3, he join densiy (5.3) was compued explicily by differen mehods and i has he form (see 6, formula (25)] or 19, he second erm of formulas (1.3) and (4.21)]): ( c + x f 1 (x, ) = λe λ 4πc 2 x ln x = (x 1, x 2, x 3 ) in B 3 (, c), x = ), (5.5) c x x x2 2 + x2 3, >. By subsiuing his join densiy ino (5.2) (for n = 1, m = 3), we arrive a he formula: f 2 (x, ) = λ2 e λ { ( ) } cτ + ξ dξ dτ 16π 2 c 4 ln cτ ξ ξ τ( τ) 2, (5.6) M(x,τ) x = (x 1, x 2, x 3 ) in B 3 (, c), >. Formula (5.6) can also be obained by seing m = 3 in (5.4).

11 52 ALEXANDER D. KOLESNIK According o Theorem 2 and (5.1), he ransiion densiy of he m-dimensional symmeric Markov random fligh solves he inegral equaion Γ ( ) m { 2 e λ p(x, ) = 2π m/2 c m 1 m 1 δ(c2 2 x 2 ) ( ) ] } (5.7) e λ( τ) + λ ( τ) m 1 δ(c2 ( τ) 2 x 2 x ) p(x, τ) dτ, x = (x 1,..., x m ) B m (, c), >. In he class of funcions wih compac suppor, he equaion (5.7) has he unique soluion given by he series ( ( Γ m ) ) n+1 ] p(x, ) = λ n 2 e λ x (n+1) 2π m/2 c m 1 m 1 δ(c2 2 x 2 ). (5.8) n= 5.2. The circular Gaussian law on a circle. Consider now he case of a non-symmeric planar random fligh when he iniial and each new direcion are chosen according o he disribuion on he uni circle S 2 (, 1) wih he wo-dimensional densiy χ k (x) = 1 2π I (k) exp x = (x 1, x 2 ) R 2, x = ( ) kx1 δ(1 x 2 ), (5.9) x x x2 2 k R, where I (z) is he modified Bessel funcion of order. Formula (5.9) deermines he oneparameric family of densiies {χ k (x), k R}, and for any fixed real k R he densiy χ k (x) is absoluely coninuous and uniformly bounded on S 2 (, 1). If k =, hen formula (5.9) yields he densiy of he uniform disribuion on he uni circle S 2 (, 1), while for k i produces non-uniform densiies. In he uni polar coordinaes x 1 = cos θ, x 2 = sin θ, he wo-dimensional densiy (5.9) akes he form of he circular Gaussian law (also called he von Mises disribuion): χ k (θ) = cos θ ek, θ π, π), k R. (5.1) 2π I (k) For arbirary real k R, densiy (5.9) on he uni circle S 2 (, 1) generaes he densiy p (s) e λ ( ) (x, ) = 2πc I (k) exp kx1 δ(c 2 2 x 2 ), (5.11) x x = (x 1, x 2 ) R 2, x = x x2 2, >, k R, concenraed on he circle S 2 (, c) of radius c. Then, according o Theorem 1, he join densiies are conneced wih each oher by he recurren relaion λ f n+1 (x, ) = 2πcI (k) { ( ) } k(x 1 ξ 1 ) e λ( τ) exp f n (ξ 1, ξ 2, τ) dξ 1 dξ 2 dτ, (x1 ξ 1 ) 2 + (x 2 ξ 2 ) 2 τ M(x,τ) x = (x 1, x 2 ) in B 2 (, c), n 1, >, k R. (5.12) According o Theorem 2 and (5.11), he ransiion densiy of a planar Markov random fligh wih dissipaion funcion (5.9) saisfies he inegral equaion e λ ( ) p(x, ) = 2πc I (k) exp kx1 δ(c 2 2 x 2 ) x

12 + INTEGRAL EQUATION FOR THE TRANSITION DENSITY λ 2πc I (k) ( ( ) ) ] e λτ kx1 exp δ(c 2 τ 2 x 2 x ) p(x, τ) dτ, τ x x = (x 1, x 2 ) B 2 (, c), x = x x2 2, >, k R. (5.13) In he class of funcions wih compac suppor, equaion (5.13) has he unique soluion given by he series ( ) n+1 ( ) ] p(x, ) = λ n 1 e λ x (n+1) kx1 exp δ(c 2 2 x 2 ). (5.14) 2πc I (k) x n= Acknowledgemen. The auhor wishes o hank he referee for helpful commens and suggesions. References 1. P. Becker-Kern, M. M. Meerschaer and H.-P. Scheffler, Limi heorems for coupled coninuous ime random walks, Ann. Probab. 32 (24), A. Ghosh, R. Rasegar and A. Roiershein, On a direcionally reinforced random walk, Proc. Amer. Mah. Soc. 142 (214), A. D. Kolesnik, The explici probabiliy disribuion of a six-dimensional random fligh, Theory Soch. Process. 15(31) (29), A. D. Kolesnik, Random moions a finie speed in higher dimensions, J. Sais. Phys. 131 (28), A. D. Kolesnik, Asympoic relaion for he densiy of a mulidimensional random evoluion wih rare Poisson swichings, Ukrain. Mah. J. 6 (28), A. D. Kolesnik, A noe on planar random moion a finie speed, J. Appl. Probab. 44 (27), A. D. Kolesnik, A four-dimensional random moion a finie speed, J. Appl. Probab. 43 (26), A. D. Kolesnik and E. Orsingher, A planar random moion wih an infinie number of direcions conrolled by he damped wave equaion, J. Appl. Probab. 42 (25), A. D. Kolesnik and M. A. Pinsky, Random evoluions are driven by he hyperparabolic operaors, J. Sais. Phys. 142 (211), G. Le Caër, Two-sep Dirichle random walks Physica A 43 (215), G. Le Caër, A new family of solvable Pearson-Dirichle random walks, J. Sais. Phys. 144 (211), G. Le Caër, A Pearson random walk wih seps of uniform orienaion and Dirichle disribued lenghs, J. Sais. Phys. 14 (21), G. Leac and M. Piccioni, Dirichle random walks, J. Appl. Probab. 51 (214), J. Masoliver, J. M. Porrá and G. H. Weiss, Some wo and hree-dimensional persisen random walks, Physica A 193 (1993), M. M. Meerschaer and P. Sraka, Semi-Markov approach o coninuous-ime random walk limi processes, Ann. Probab. 42 (214), M. M. Meerschaer and H.-P. Scheffler, Limi heorems for coninuous ime random walks wih infinie mean waiing imes, J. Appl. Probab. 41 (24), A. A. Pogorui and R. M. Rodriguez-Dagnino, Random moion wih uniformly disribued direcions and random velociy, J. Sais. Phys. 147 (212), A. A. Pogorui and R. M. Rodriguez-Dagnino, Isoropic random moion a finie speed wih K-Erlang disribued direcion alernaions, J. Sais. Phys. 145 (211), W. Sadje, Exac probabiliy disribuions for non-correlaed random walk models, J. Sais. Phys. 56 (1989), W. Sadje, The exac probabiliy disribuion of a wo-dimensional random walk, J. Sais. Phys. 46 (1987), Insiue of Mahemaics and Compuer Science Academy Sree 5, Kishinev 228, Moldova address: kolesnik@mah.md