REPRESENTATION AND GAUSSIAN BOUNDS FOR THE DENSITY OF BROWNIAN MOTION WITH RANDOM DRIFT

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1 Communicaions on Sochasic Analysis Vol. 1, No. 2 (216) Serials Publicaions REPRESENTATION AND GAUSSIAN BOUNDS FOR THE DENSITY OF BROWNIAN MOTION WITH RANDOM DRIFT AZMI MAKHLOUF Absrac. We suy he ensiy, wih respec o Lebesgue measure, of he law of muliimensional Brownian moion wih ranom rif. We o no assume any Markovianiy, an he rif may be unboune. We sae a represenaion of his ensiy in erms of he Gaussian one, from which we erive explici upper an local lower bouns of Gaussian ype. 1. Inroucion In his paper, we are inerese in muliimensional Brownian moion wih ranom rif b, efine by X := W + b s s, (1.1) where (W ) is a -imensional Brownian moion an (b ) a sochasic process, aape o he naural filraion of W. We aim a suying he exisence an bouns of he ensiy of X (wih respec o Lebesgue measure). Several works have eal wih his opic in he paricular (Markovian) framework of SDE (sochasic ifferenial equaions) X = W + b( Xs )s, (1.2) where b(.) is a eerminisic measurable funcion. By working on he funamenal soluion of he PDE associae o (1.2), Aronson [1] obaine absolue coninuiy an Gaussian upper an lower bouns when b(.) is boune, allowing also for he presence of a uniformly ellipic iffusion coefficien σ(.) (in he W erm). There is a vas lieraure concerning he sochasic mehos use o erive such bouns. For regular rif b(.) an iffusion σ(.), we refer o [8], [2] an [5], where Malliavin calculus is use. For irregular rif, some resuls are obaine by sochasic mehos. Sill in he Markovian seing, sharp Gaussian bouns have been obaine for he ensiy of (1.2) by [11] (for boune rif b(.)) an [12] (for b(.) of linear growh). Receive ; Communicae by eiors. 21 Mahemaics Subjec Classificaion. Primary 6J65; Seconary 6H1; 6G99. Key wors an phrases. Brownian moion, ranom rif, sochasic equaions, ensiy, Gaussian bouns. 151

2 152 AZMI MAKHLOUF In [3], he auhors prove he exisence of a ensiy for one-imensional SDE (1.2), wih possibly pah-epenen coefficiens. They use a simple approach consising in freezing he coefficiens a ime ε, hen compuing he characerisic funcion (law Fourier ransform) ˆµ of he approximaing variable X ε, which is Gaussian coniionally on he pas up o ime ε. The absolue coninuiy of X follows by choosing ε such ha ˆµ can be shown o be in L 2 (R). However, his argumen canno be generalize o higher imensions, an oes no lea o any poinwise boun on he ensiy. The approach of [3] has been exene by [7] an [4] o suy he exisence an regulariy of he ensiy of muliimensional SDE (1.2) wih boune irregular rif b(.). In [6], he auhors have use a more general approach ha enables o sae an inegraion-by-pars formula for funcionals of SDE s, from which hey have erive Gaussian bouns on he ensiy. In he laer hree references, he auhors perform a change of measure by Girsanov heorem o ge ri of he irregular rif, an expan i as a sum of muliple sochasic inegrals, using Iô-Taylor expansion up o a cerain orer. In each inegral, a coniional expecaion is aken in orer o regularize he rif (relying on he Markovianiy of he SDE), so ha Malliavin calculus can be finally applie. Recenly, in [1], he auhors have combine he echniques of [7] an [6] wih a Clark-Ocone represenaion of he covariance, o sae Gaussian esimaes for oneimensional aiive funcionals of SDE s wih irregular rif. Besies, hey have applie he menione covariance represenaion o he ensiy esimaion of soluions of some one-imensional sochasic equaions, uner regulariy assumpions on he coefficiens (incluing he rif). In his paper, we consier (X ), he muliimensional Brownian moion wih ranom rif (b ) efine by (1.1), an we o no assume any Markovianiy nor bouneness on he rif. In such a general framework, we are no aware of any aemp o esablish exisence an Gaussian bouns for he ensiy of X. Our iea is simpler an more irec han hose cie above. We express some isance beween boh laws of he Brownian moion wih rif, X, an of he (rifless) Brownian moion W, by plugging he process X in a es funcion for he law of W. More precisely, we apply Iô s lemma o ϕ(s, X s ) for s, where ϕ(s, x) := E[f(W ) W s = x] an f is a es funcion. Then we ake he expecaion of he expansion, an we esimae how boh laws «iverge» from each oher, from ime o ime. This esimaion enables us o have exisence, a represenaion, an Gaussian bouns for he ensiy of X. The paper is organize as follows. We firs inrouce, in Secion 2, he noaions an assumpions use hroughou he aricle. Then we sae an commen our main resuls in Secion 3. The proofs are given in Secion 4, an a summarizing conclusion in Secion Noaions an Assumpions For x = (x 1,..., x ) R, x enoes is ranspose an x is Eucliian norm: x = ( i=1 x2 i ) 1 2. For any boune funcion f, f enoes is supremum norm.

3 BROWNIAN MOTION WITH RANDOM DRIFT 153 For a space-iffereniable funcion f(, x) : [, ) R R, f(, x) sans for is graien wih respec o x, i.e. f(, x) = ( f x 1,..., f x )(, x). The funcion g(.,.) enoes he Gaussian hea kernel: for >, x R, g(, x) := (2π) x 2 2 ). For any ranom variable Z, an any real number p 1, Z p := (E[ Z p ]) 1 p. Le W be a -imensional Brownian moion, efine on a filere probabiliy space (Ω, F, (F ), P), where (F ) is he naural filraion of W, augmene wih he P-null ses. Le E enoe he expecaion uner P. Le (b ) be a R -value, (F )-aape sochasic process such ha b s s < (P-a.s.). We efine he Brownian moion wih ranom rif b by X := W + b s s. Throughou he paper, we use he following assumpion. Se E b,α () := E[exp(α L b,q () := sup E[ b s q ] 1 q s [,] b s 2 s)]; = sup b s q. s [,] Assumpion 2.1. There exis real consans α (, ) an q (1, ) such ha, for >, E b,α () + L b,q () <. (A b,α,q ) Noice ha no bouneness is assume for he rif process (b ), an ha he coniion on E b,α () is even weaker han he sanar Novikov coniion when α is smaller han 1 2. I will be hough sufficien o esimae he change of measure coming from he applicaion of Girsanov heorem (see he proof of Lemma 4.1, Secion 4.4). Le us menion here ha Novikov s coniion ensures (by Girsanov heorem) he exisence of a weak soluion o SDE (1.2), an ha a similar coniion has been recenly shown in [9] o be sufficien, in some cases, o have a srong soluion. One may hen apply our resuls o his paricular seing. 3. Main Resuls Theorem 3.1 (Exisence an represenaion). Le >. Assume ha (A b,α,q ) hols for some α (, ) an q (1, ), an se p := (1 + 1 q 1 )( α 2α ). If (i) eiher he imension = 1, (ii) or, > 1 an p < 1, hen X amis a ensiy γ(,.) wih respec o Lebesgue measure, an for x R, γ(, x) = g(, x) + E[ g( s, x X s )b s ]s. (3.1)

4 154 AZMI MAKHLOUF Noice ha, uner Assumpion (A b,α,q ), he represenaion (3.1) always hols rue in he one-imensional case. For > 1, i sill hols if α an q are sufficienly large for coniion (ii) in Theorem 3.1 o be fulfille: his is he case if, for example (bu no necessarily), b is uniformly boune. Ieniy (3.1) may be seen as a firs-orer expansion of he ensiy γ(,.) of X, whose leaing erm is he ensiy g(,.) of Brownian moion W. One may hink abou Brownian moion wih eerminisic rif as a reference process: hen we erive he following generalizaion of Theorem 3.1. Corollary 3.2. Le > an β(.) : [, ] R any eerminisic funcion such ha (A b β,α,q ) hols for some α (, ) an q (1, ). Se I β () := β(s)s. Then uner coniions (i) an (ii) of Theorem 3.1, we have for x R, γ(, x) = g(, x I β ()) [ ( ) ] + E g s, (x I β ()) (X s I β (s)) (b s β(s)) s. Proof. Seing Y := W + (b s β(s))s, we have X = Y + I β (). Since β(.) is eerminisic, γ(, x) = γ Y (, x I β ()), where γ Y (,.) is he ensiy of Y. Applying Theorem 3.1 o Y proves Corollary 3.2. From Theorem 3.1 (an Lemma 4.1), we ge he following Gaussian bouns on he ensiy γ(, x). The upper boun (Theorem 3.3 an Corollary 3.4) is explicily given for any > an x R. The lower boun (Theorem 3.5) is only vali for small ime-space (, x). Theorem 3.3 (Upper boun). Uner he assumpions of Theorem 3.1, we have for > an x R, γ(, x) g(, x) + C ( ) p, L b,q ()E b,α () 1 2r + x g(p, x), wih C p, := ( 2 + 2p ) 1 C p,, an p, r an C p, efine as in Lemma 4.1. As wih Corollary 3.2, we immeiaely erive a boun which is no only opimal in he class of eerminisic rifs, bu also sharper han ha of Theorem 3.3. For insance, by choosing β(s) := E[b s ], he boun involves cenere momens L b β,q () (he sanar eviaion, if q = 2) of he rif b, insea of he crue momens L b,q (). Corollary 3.4. Uner he assumpions of Corollary 3.2, we have for > an x R, γ(, x) g(, x I β ()) + C ( ) p, L b β,q ()E b β,α () 1 2r + x Iβ () g(p, x I β ()).

5 BROWNIAN MOTION WITH RANDOM DRIFT 155 Theorem 3.5 (Lower boun). Assume ha he coniions of Theorem 3.1 hol. For > an R >, se O,R := {(, x) (, ) R, s.. < an x R}. Then for every R >, here exiss > (epening on R, α, q an ) such ha, for (, x) O,R, γ(, x) 1 g(, x). 2 Remark 3.6. I is no clear (a leas from our work) how an explici lower boun, vali for any ime, coul be. I canno be of he same ype as he bouns given by Theorems 3.3 an 3.5, in he sense ha oher powers of (smaller han 2 ) may appear. Here is an example: X := W + B s s, where B is a -imensional Brownian moion, inepenen from W. Then he ranom variable X is Gaussian wih ensiy γ(, x) = (2π + 2π 3 3 ) x 2 2( ) ), which is equivalen, for large, o ( 2π 3 3 ) 2 (hus negligible compare o 2 ). Fining a lower boun, vali for any ime, remains as a challenging problem. 4. Proofs The following lemma gives a crucial esimae for our proofs. For he convenience of he reaer, is proof is pospone o Secion 4.4. Lemma 4.1. Le < s <. Assume ha (A b,α,q ) hols for some α (, ) an q (1, ), an se p := (1 + 1 q 1 )( α 2α ) an r := (1 1 p 1 q ) 1. Then for x R, E[ g( s, x X s )b s ] ( ) C p, L b,q ()E b,α () 1 s 2 + ( ) 2p 2r ( s) 1 x 2 + g(p, x), where C p, is a consan epening only on p an, an explicily given by (4.6)- (4.5) Proof of Theorem 3.1 (Exisence an represenaion). Roughly speaking, he iea is he following: since we wan o compare he law of X o ha of W, we apply a es funcion characerizing he law of W, here E[f(W ) W s =.] ( s ), o he process X. We hen expan i using Iô s lemma; we only nee o be careful wih he possible blow up of he erivaive when s is close o. Bu his is conrolle by Lemma 4.1.

6 156 AZMI MAKHLOUF Proof of Theorem 3.1. Applying he usual characerizaion of he probabiliy ensiy funcion of a ranom variable, we have o show ha, for every boune an coninuous es funcion f : R R, E[f(X )] = f(x)g(, x)x + f(x) E[ g( s, x X s )b s ]sx. (4.1) R R For fixe >, an for all s [, ] an y R, we se ϕ(s, y; ) := E[f(W ) W s = y] = E[f(y + W W s )]. Noe ha ϕ(, ; ) = E[f(W )] an ϕ(, y; ) = f(y), an ha ϕ(.,.; ) is coninuous on [, ] R. When s <, we have ϕ(s, y; ) = f(x)g( s, x y)x, R an R (y x) ϕ(s, y; ) = f(x) g( s, x y)x = f(x) g( s, x y)x. R s Moreover, he following esimaes hol: an (by he change of variable z = x y s ) ϕ(s, y; ) f, (4.2) ϕ(s, y; ) f. (4.3) s Clearly, for every ε (, ), he funcion ϕ(.,.; ) is of class C 1,2 on [, ε] R. By Iô s lemma (an he fac ha ϕ(.,.; ) is soluion o he backwar hea equaion), ϕ( ε, X ε ; ) = ϕ(, ; ) + ε ϕ(s, X s ; )W s + ε ϕ(s, X s ; )b s s. Esimaes (4.2) an (4.3) allow o ake expecaions in he above equaliy o ge E[ϕ( ε, X ε ; )] = E[f(W )] + ε = f(x)g(, x)x + R We have E[ ϕ(s, X s ; )b s ]s ε E[ f(x) g( s, x X s )b s x]s. R E[ f(x) g( s, x X s )b s ] f E[ g( s, x X s )b s ]. Now, Lemma 4.1 clearly shows ha, if 2 + 2p 1 2 > 1 (which is equivalen o eiher coniion (i) or (ii) of Theorem 3.1), hen E[ g( s, x X s )b s ] is (s x)-inegrable on [, ] R. Therefore, we apply Fubini s heorem in he las ieniy, an he ominae convergence heorem when ε, o obain (4.1) an prove Theorem 3.1.

7 BROWNIAN MOTION WITH RANDOM DRIFT 157 Le us poin ou here ha our meho woul no immeiaely apply if we ae o our framework a ranom iffusion coefficien in he Brownian erm of (1.1), since non-inegrable secon-orer erms (involving he secon erivaive of he Gaussian ensiy g) woul hen appear Proof of Theorem 3.3 (Upper boun). I is a irec consequence of Theorem 3.1 an Lemma 4.1. Proof of Theorem 3.3. Coniions (i) an (ii) of Theorem 3.1 ensure ha 2 + 2p 1 2 > 1, an hus he inegrabiliy of he boun given by Lemma 4.1 (wih respec o s). Then from hese wo resuls, γ(, x) = g(, x) + E[ g( s, x X s )b s ]s g(, x) + C p, L b,q ()E b,α () 1 2r ( s g(, x) + ( 2 + 2p ) 1 C p, L b,q ()E b,α () 1 2r ) 2 + ( 2p ( s) 1 x 2 + ( + x ) g(p, x). ) s.g(p, x) 4.3. Proof of Theorem 3.5 (Lower boun). We rely on he fac ha he inegral erm in he represenaion (3.1) of he ensiy becomes small for small an x (Noice he muliplicaive erm ( + x ) ha appears in he boun of Theorem 3.3. I is acually he same reason ha makes our lower boun only vali aroun he ime-space origin). Proof of Theorem 3.5. From Theorem 3.1, we have (using he riangle inequaliy) γ(, x) g(, x) E[ g( s, x X s )b s ]s. From Lemma 4.1 (see also he proof of Theorem 3.3), we ge γ(, x) g(, x) C ( ) p, L b,q ()E b,α () 1 2r + x g(p, x). Le R > an (, x) such ha x R. Then γ(, x) (2π) x 2 2 ) C p, L b,q ()E b,α () 1 2r (1 + R)(2πp) x 2 2p ) (2π) x 2 ( ) 2 ) 1 ϵ(), where ϵ() := C p, L b,q ()E b,α () 1 2r (1 + R)p 2 exp( R 2 2 R2 2p ).

8 158 AZMI MAKHLOUF Clearly, ϵ() (as C R ). Thus, here exiss > (epening on R, α, q an ) such ha, for, ϵ() 1 2. Therefore, γ(, x) 1 2 (2π) x 2 2 ) Proof of Lemma 4.1. The iea is o apply Girsanov heorem o make a change of measure uner which X is a Brownian moion. Since α may be smaller han 1 2, he sanar Novikov coniion is no fulfille. We procee in hree seps: firs we esablish an esimae for rifless Brownian moion, hen for Brownian moion wih boune rif b, an finally wih unboune b by approximaion. Proof of Lemma 4.1. Firs sep. Le us show ha, for any real number p (1, ), here exiss a consan C p, such ha, for x R, g( s, x W s ) p C p, ( s R ) 2 + 2p ( ( s) x ) g(p, x). (4.4) We have E[ g( s, x W s ) p ] = g( s, x y) p g(s, y)y R ( ) p y x = (2π( s)) p y x 2 p s 2( s) )(2πs) y 2 2s )y. By he change of variable z = y x s, E[ g( s, x W s ) p ] = (2π) p 2 ( s) p 2 p (2πs) 2 z p exp( p z 2 R 2 z s + x 2 )z. 2s By sraighforwar compuaion, we have, seing ξ := z s + x 2 Then p z 2 2 2s ( s + ps) = z + ξ 2 2s 2s z + ξ x 2. s s+ps x, p 2( s + ps) x 2 E[ g( s, x W s ) p ] (2π) p 2 ( s) p 2 p (2πs) x 2 2 ) z p 2 z + ξ exp( )z R 2s = (2π) p 2 ( s) p 2 p (2π) x 2 p 2 ) s ζ ξ exp( ζ 2 2 )ζ, R

9 BROWNIAN MOTION WITH RANDOM DRIFT 159 where we have mae he change of variable ζ := (z + ξ). From we ge s ζ ξ E[ g( s, x W s ) p ] p 2 p 1 ( s ζ p s + ξ p ) 2 p 1 ( ζ p + ( s) p 2 p x p ), 2 p 1 (2π) p ( s) p 2 p (2π) x 2 2 )µp p, + 2 p 1 (2π) p ( s) p x p p (2π) x 2 2 ) 2 p 1 (2π) p 2 + p 2 µ p p, ( s) (( s) p 2 + x p p) (2π) x 2 2 ), where µ p, := ( ζ p (2π) ζ 2 R 2 )ζ) 1 p = N (, I ) p. (4.5) Therefore (using ( a + b ) 1 p a 1 p + b 1 p ), (E[ g( s, x W s ) p ]) 1 p p (2π) 2 + 2p µp, ( s) 2 + 2p = C p, ( s) 2 + 2p where We have prove esimae (4.4). (( ) s) x 1 (2π) x 2 2p exp( 2p ) (( s) x 1 ) 2 2p (2πp) x 2 2p ), C p, := p p 2 µp,. (4.6) Secon sep. Suppose ha he rif process b is uniformly boune. By Girsanov heorem, (X s ) is a Brownian moion uner he equivalen probabiliy measure Q efine by Γ s := P F s s = exp( b Q uw u + 1 b u 2 u). 2 We enoe he expecaion uner Q by E Q. By Höler s inequaliy (noice ha p, q, r > 1 an p 1 + q 1 + r 1 = 1), E[ g( s, x X s )b s ] = E Q [ g( s, x X s )b s Γ 1 q s Γ 1 1 q s ] (E Q [ g( s, x X s ) p ]) 1 p (EQ [ b s q Γ s ]) 1 q (EQ [Γ r(1 1 q ) s ]) 1 r = (E[ g( s, x W s ) p ]) 1 p (E[ bs q ]) 1 q (EQ [Γ r p +1 s ]) 1 r.

10 16 AZMI MAKHLOUF The boun on he firs erm is given by esimae (4.4). The secon one is boune by L b,q (). For he las erm, seing r := r p, we have Now an E Q [Γ r +1 s ] = E[Γ r s ] = E[exp( = E[exp( 2r 2 + r r b uw u E[exp((2r 2 + r ) = E[exp((2r 2 + r ) = E[exp( r p (2r p + 1) b u 2 u) exp( r b u 2 u)] r b uw u b u 2 u)] 1 2 E[exp( 2r b uw u 1 2 b u 2 u)] 1 2 b u 2 u)] 1 2. r b u 2 u)] r p = p 1 (1 1 p 1 q ) 1 = (p(1 1 q ) 2α 1) 1 = α, 2r b u 2 u)] 1 2 r p (2r p + 1) = 2α 1 + 4α α 1 + 8α 1 + = α( α) α ( α) = α. 2 Therefore, (E Q [Γ r p +1 s ]) 1 r E[exp(α We have prove he lemma for uniformly boune b. b u 2 u)] 1 2r Eb,α () 1 2r. Thir sep. We now examine he general case of unboune b. We approximae b s by he sequence b n s := b s 1 bs n (n ), an we se X n s := W s + Clearly, b n s b s an Xs n X s, P-a.s. n n From he previous sep, we have E[ g( s, x X n s )b n s ] C p, b n s q E[exp(α b n u 2 u)] 1 2r For all r > an y R, we have ( s b n uu. g(r, y) = y r (2πr) y 2 2r ). ) 2 + ( ) 2p ( s) 1 x 2 + g(p, x). (4.7)

11 BROWNIAN MOTION WITH RANDOM DRIFT 161 g(r,.) is coninuous, an g(r, y) = 1 r (2πr) 2 C r (2πr) 2, where C = sup z R {z exp( z2 2 )} = exp( 1 2 ). Then y r exp( y 2 2r ) g( s, x X n s )b n s exp( 1 2 )( s) 1 2 (2π( s)) 2 bs. Therefore, by he ominae convergence heorem, E[ g( s, x X n s )b n s ] n E[ g( s, x X s)b s ]; E[exp(α b n s q n b s q ; b n u 2 u)] n E[exp(α b u 2 u)]. Taking he limi when n in (4.7) proves Lemma Conclusion We represen he ensiy of Brownian moion wih ranom rif in erms of he Gaussian one. This represenaion enables us o measure he isance beween boh ensiies, leaing o explici Gaussian upper an (local) lower bouns. In paricular, his work exens o a non-markovian seing similar resuls known for sochasic ifferenial equaions wih irregular rif, using simpler an more general argumens. References 1. Aronson, D. G.: Bouns for he funamenal soluion of a parabolic equaion, Bull. Amer. Mah. Soc., 73 (1967) Bally, V.: Lower bouns for he ensiy of locally ellipic Iô processes, Annals of Probabiliy, 34 (26) Fournier, N. an Prinems, J.: Absolue coninuiy for some one-imensional processes, Bernoulli, 16(2) (21) Hayashi, M., Kohasu-Higa, A., an Yûki, G.: Höler coninuiy propery of he ensiies of SDEs wih singular rif coefficiens, Elecron. J. Probab., 19(77) (214) Kohasu-Higa, A.: Lower bouns for ensiies of uniformly ellipic non-homogeneous iffusions, Proceeings of he Sochasic Inequaliies Conference in Barcelona. Progress in Probabiliy, 56 (23) Kohasu-Higa, A. an Makhlouf, A.: Esimaes for he ensiy of funcionals of SDEs wih irregular rif, Sochasic Processes an heir Applicaions, 123(5) (213) Kohasu-Higa, A. an Tanaka, A.: A Malliavin Calculus meho o suy ensiies of aiive funcionals of SDE s wih irregular rifs, Ann. Ins. H. Poincarï Probab. Sais., 48(3) (212) Kusuoka, S. an Sroock, D.: Applicaions of Malliavin calculus, par III, J. Faculy Sci. Univ. Tokyo Sec. 1A Mah., 34 (1987) Lanconelli, A.: A comparison heorem for sochasic ifferenial equaions uner he Novikov coniion, Poenial Analysis, 41(4) (214)

12 162 AZMI MAKHLOUF 1. Nguyen, T. D., Privaul, N., an Torrisi, G. L.: Gaussian esimaes for soluions of some one-imensional sochasic equaions. To appear in Poenial Analysis. Preprin available a hp:// Qian, Z., Russo, F., an Zheng, W.: Comparison heorem an esimaes for ransiion probabiliy ensiies of iffusion processes, Probabiliy Theory an Relae Fiels, 127(3) (23) Qian, Z. an Zheng, W.: A represenaion formula for ransiion probabiliy ensiies of iffusions an applicaions, Sochasic Processes an heir Applicaions, 111(1) (24) Azmi Makhlouf: Naional Engineering School of Tunis (ENIT) an Laboraory of Mahemaical an Numerical Moelling in Engineering Sciences (LAMSIN), Universiy of Tunis El Manar, Tunis 12, Tunisia aress: azmi.makhlouf@gmail.com