Homogeneous Stochastic Differential Equations

WDS'1 Proceedings of Contributed Papers, Part I, 195 2, 21. ISBN 978-8-7378-139-2 MATFYZPRESS Homogeneous Stochastic Differential Equations J. Bártek Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. We study the relation between solutions to a certain type of nonlinear partial differential equation and solutions to its stochastic analogy driven by a fractional Brownian motion with Hurst index H > 1/2. The formula obtained is used to study properties of solutions to some stochastic differential equations, such as the stochastic porous medium equation. Introduction If we work with stochastic differential equations we are often not able to express the solution to our equation by a closed form formula and thus we have to study its properties indirectly by studying properties of coefficients in the equation. Therefore, it is desirable to find a way to obtain such an explicit formula. When dealing with so-called homogeneous equations it has been done so for stochastic equations driven by Wiener process in [Lototsky, 27]. In this work we will study an analogous situation, but considering fractional Brownian motion instead of Wiener process and also with a different aproach to stochastic integration. There are a few articles studying similar situations, for example [Duncan, Maslowski, Pasik-Duncan, 25] and [Tindel, Tudor, Viens, 23], but they deal with the linear case and possibly space-dependent fractional Brownian motion. This work itself consists of several sections. First, we give the definition of a fractional Brownian motion, then we describe the situation we are dealing with, define the notions of solutions to differential equations and prove the main theorem. Finally, we apply the results we obtain to study properties of the stochastic porous medium equation. Definition 1. Let H,1 is a constant. A scalar fractional Brownian motion fbm with Hurst index H is a continuous centered stochastic Gaussian process B H t t with covariance function [ ] E B H tb H s = 1 2 t2h + s 2H t s 2H. Fractional Brownian motion is not a semimartingale and has an infinite variation so it was necessary to develop a suitable integration theory. Here we utilize theory described in [Mishura, 28] which uses fractional calculus and we define the stochastic integral pathwise. Homogeneous equation Definition 2. We say that a function F is homogeneous of degree m 1, if for all λ > Fλx,λy,λz,... = λ m Fx,y,z,..., x R,y R d,z R d d,... 1 We say that the equation 2 is homogeneous of degree m 1, if the function F is homogeneous of degree m. Consider an equation v t = Fv,Dv,D 2 v,..., t >, x R d, 2 with initial condition v,x = v x. The unknown in the equation 2 is the function v = vt,x, function F is given, v t = v/ t and D k v is k-th derivative of v with respect to x. Notation Fv,Dv,D 2 v,... means that F depends on v and on a finite number of derivatives Dv,D 2 v,...,d m v,m N. 195

Further, consider a stochastic version of the equation 2 for unknown random field u = ut,x,t >, x R d of the following form du = Fu,Du,D 2 u,...dt + uftd B H t + gtdt, 3 with the same initial condition as in 2, i.e. u,x = u x = v x, where f C α+ε [,T] for all T >, for some < ε < min{1 α, α, H + α 1}, α 1 H,1 a g L loc R +. The symbol d B H means that we understand the stochastic integral in the Stratonovich sense for details see e.g. [Mishura, 28] or [Biagini, Yaozhong, Øksendal, Zhang, 28]. We will now investigate the relationship between solutions to the equations 2 and 3. First, we have to define the notion of a solution rigorously. Definition 3. Function v : R + R d R is a classical solution to the equation 2, if the following is satisfied: 1. v is continuous, 2. all partial derivatives of function v with respect to x involved in 2 exist and are continuous in variable x and are γ-hölder continuous in variable t on [,T] for all T >, where γ > 1 H. 3. The following equality holds for all t,x R + R d. vt,x = v x + Fvs,x,Dvs,x,D 2 vs,x,...ds Definition 4. Let τ : Ω R + {+ } be a stopping time. Random field u = ut,x is a classical solution to the equation 3, if there exists a set Ω Ω, PΩ = 1, and on the set,τ]] = {t,x,ω : t < τω, ω Ω, x R d } the following conditions are satisfied: 1. u is continuous in variable x and γ-hölder continuous in variable t for every t,x,ω,τ]], for some γ > 1 H, 2. all partial derivatives of function u with respect to x involved in 3 exist and are continuous in variable x and are γ-hölder continuous in variable t on [,T] for all < T < τ, for some γ > 1 H, 3. the equality ut,x = u x + holds for every t,x,ω,τ]]. Fus,x,Dus,x,...ds + us,xfsd B H s + gsds Define functions ht = exp gsds + fsd B H s and Ht = h m 1 sds. 4 The process h is called a geometric fractional Brownian motion and plays the key role in the correspondence of solutions to the equations 2 and 3. It is obvious that h = 1 and by a straigthforward application of the Itô formula see for example [Mishura, 28], Chapter 2 we obtain the following equality for h ht = 1 + Now we can state the main theorem hsfsd B H s + hsgs ds. 196

Theorem 1. Assume that the function F in the equation 2 is continuous and homogeneous of degree m, and let function v = vt,x be a classical solution to the equation 2. Define function u as ut,x = htvht,x. 5 Then u is a classical solution to the equation 3. Proof. Let v be a classical solution to the equation 2. Note that H t = h m 1 t. We will use the Itô formula for stochastic integral w.r.t. fbm, see [Mishura, 28], Chapter 2. Our definition of classical solution ensures that the assumtions are satisfied. Thus, we have and dht = htftd B H t + gtdt dut,x = v t Ht,xh m tdt + vht,xhtftd B H t + gtdt. Using 2 and homogenity of F we obtain v t Ht,xh m tdt = h m tf vht,x,dvht,x,... = F htvht,x,htdvht,x,... which implies = Fut,x,Dut,x,D 2 ut,x,... dut,x = Fut,x,Dut,x,D 2 ut,xdt + ut,xftd B H t + gtdt and u is a classical solution to 3. Applications Now we can apply Theorem 1 and obtain some behavioral properties of solutions to some stochastic differential equations that have the form of equation 3. Example 1 Porous medium equation. The most important example is the porous medium equation. The equation describes various physical processes, such as heat conduction, gas flow in a porous medium, plasma radiation or population expansion. It has the following form v d t t,x = k=1 v m x k t,x, t >,x R d, 6 where d N and m 1. This equation has been studied extensively in many publications, for example [Vázques, 27]. Consider a function U [BT] defined as U [BT] t,x;b = 1 t α max, b m 1 1/m 1 2m β x 2 t 2β, t >, x R d, 7 where b > and β = 1 m 1d + 2, α = βd. The function U [BT] is called the Barenblatt s solution to 6 and it is an important solution which determines large-time behaviour for every other global solution to the equation 6. Similarly with some restrictions on functions H and h defined by 4 and on functions f and g it can be shown that the stochastic version of Barenblatt s solution constructed using Theorem 1 determines large-time bahaviour of many other solutions to the stochastic porous medium equation dut,x = u m t,xdt + ut,xftd B H t + gtdt, t >, x R d. 8 Now we state some properties of Barenblatt s solution, for more detailed description see for example [Aronson, 27], [Friedman, Kamin, 198] or [Vázques, 27]. 197

The total mass of the solution defined by M = R U [BT] t,x;bdx is independent of t and is d uniquely determined by the value of parameter b; namely m 1 d/2 Γ m M := M b = b 1/2βm 1 2πm β m 1 Γ m m 1 + d, where Γ is the Gamma function. U [BT] is a classical solution to the equation U t = U m on the set { } 2mb t,x : x m 1β tβ. Let U [BT] t,x;b be the Barenblatt s solution with mass M b and Ut,x is an arbitrary solution to 6 with R d U,x = M b. Then uniformly in x on sets of the form where C >. t βd Ut,x U [BT] t,x;b, t, 9 {t,x R + R d : x Ct β }, Now, using Theorem 1, we know that the random field u [BT] = u [BT] t,x;b = htu [BT] Ht,x;b 1 is a solution to 8. The mean value of mass M is E u [BT] t,x;bdx = Eht U [BT] Ht,x;bdx = M b Eht R d R d = M b E exp fsd B H s + gs ds { = M b exp gs ds + 1 } 2 H2H 1 fufv u v 2H 2 dudv. For a special choice of functions f and g we can study the large-time behavior of u [BT]. Assume that g and f in 8 are constants. The key role in limit properties of u [BT] have the processes h and H defined in 4. In this case they take the following form ht = exp gt + fb H t, 11 Ht = h m 1 sds = exp gsm 1 + fb H sm 1 ds. 12 First assume that g >. According to the law of iterated logarithm see e.g. [Arcones, 1995] there exists a constant K H > such that outside a P-null set for all ε > and for all δ > there exists time t t ω > such that for all t t the inequality B H t < K H + εt H+δ holds. This gives us estimates on h a H. For t t we have ht exp gt f K H + εt H+δ 198

and thus assuming < δ < 1 H, Further, BÁRTEK: HOMOGENEOUS STOCHASTIC EQUATIONS κ > t 1 t 1 ω t > e gm 1 κ ht e gm 1+κ. Ht 1 e gsm 1+fm 1BH s ds + e gm 1+κ ds t 1 C 2 ω + e[gm 1+κ]t gm 1 + κ and Ht e gsm 1+fm 1BH s gsm 1 f m 1KH+εsH+δ ds + e ds t C 2 e[gm 1 κ]t ω + gm 1 κ. Hence it follows, for < α < 1, Analogously, H α t C 3 ω + H 2β t C 4 ω + eα[gm 1+κ]t gm 1 + κ α. e2β[gm 1 κ]t gm 1 κ 2β. If we use these estimates on stochastic Barenblatt s solution 1 we get u [BT] t,x e g κt C 3 + eα[gm 1+κ]t gm 1+κ α max,b m 1 2m β x 2 1 m 1. C 4 + e2β[gm 1 κ]t gm 1 κ 2β Further x R d t 2 t 2 ω > t t 2 m 1 2m β x 2 C 3 + e2β[gm 1 κ]t gm 1 κ 2β < b 2 hence t t 1 t 2 u [BT] t,x e g κt C 4 + eα[gm 1+κ]t [gm 1 + κ] α 1 1 b m 1 2 e g κt = C 5 C 6 ω + eα[gm 1+κ]t. 13 We can sum up these calculations to the following statement Proposition 1. Assume that g > and f are constants and u [BT] is a solution to 8 defined by 1. Then there exists a measurable set A, PA = 1, such that for all ω A, for all κ > and for all x R d there exist constants C >, K Kω > and t t ω,x such that for all t > t the equality e g κt u [BT] t,x C K + e α[gm 1+κ]t holds. In particular, outside a P-null set x R d lim inf u[bt] t,x = +. 199

Proof. By 13 lim inf u[bt] t,x lim C 5 e g κt C 6 + e α[gm 1+κ]t = C e g κt 5 lim e α[gm 1+κ]t = C 5 lim e {g κ α[gm 1+κ]}t. 14 It is sufficient to choose κ1 + α < κ < = 2g m 1d + 2 2g d [m 1d + 2] 1 + m 1d+2 2g[m 1d + 2] [m 1d + 2][md + 2] = 2g md + 2 and the coefficient {g κ α[gm 1 + κ]} in the exponent of 14 is positive. In the case of g < we obtain by an analogous argument as above the following statement. Proposition 2. Assume that g < and f are constants and u [BT] is a solution to 8 defined by 1. Then there exists a measurable set A, PA = 1, such that for all ω A, for all κ > and for all x R d there exist constants C >, K Kω > and t t ω,x such that for all t > t the equality holds. In particular, outside a P-null set e g+κt u [BT] t,x C K + e α[gm 1 κ]t x R d lim u [BT] t,x =. Acknowledgments. This work was supported by the GA UK grant no. 16751 and by the grant SVV 261315/21 and by the Czech Science Foundation grant no. P21/1/752. References Arcones, M. A., On the law of the iterated logarithm for gaussian processes, Journal of Theoretical Probability, 84, 87794, 1995. Aronson, D. G.: The Porous Medium Equation. In Nonlinear Diffusion Problems Montecatini Terme, 1985, Lecture Notes in Math. 1224 1 46, Springer, Berlin, 1986. Biagini F., Hu Y., Øksendal B., Zhang T.: Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, 28. Duncan T. E., Maslowski B., Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stoch. Process. Appl. 1158 1357 1383, 25. Friedman, A., Kamin S.: The Asymptotic Behavior of Gas in n-dimensional Porous Medium, Trans. Amer. Math. Soc., 2622, 551-563, 198. Lototsky, S. V.: A random change of variables and applications to the stochastic porous medium equation with multiplicative time noise, Communications on Stochastic Analysis 13 343 355, 27. Mishura Y. S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer Verlag, Berlin, 28. Tindel S., Tudor C. A., Viens F.: Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields 1272 186 24, 23. Vázques J. L.: The Porous Medium Equation Mathematical Theory, Oxford Mathematical Monographs, Oxford, 27. 2