On a class of stochastic fractional kinetic equation with fractional noise

In this article we study a class of stochastic fractional kinetic equations with fractional noise which are spatially homogeneous and are fractional in time with H>1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H>1/2$\end{document}. The diffusion operator involved in the equation is the composition of the Bessel and Riesz potentials with any fractional parameters. We prove the existence of the solution under some mild conditions which generalized some results obtained by Dalang (Electron. J. Probab. 4(6):1–29, 1999) and Balan and Tudor (Stoch. Process. Appl. 120:2468–2494 , 2010). We study also its Hölder continuity with respect to space and time variables with b=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b=0$\end{document}. Moreover, we prove the existence for the density of the solution and establish the Gaussian-type lower and upper bounds for the density by the techniques of Malliavin calculus.


Introduction
This paper is concerned with the following stochastic fractional kinetic equation (SFKE for short) with zero initial condition (see, for example, Angulo et al. [1,3], Angulo et al. [2], and Márquez-Carreras [16]): with T > 0, d ≥ 1, α ≥ 0, γ > 0, b(·) is a measurable function andẆ denotes a fractional noise. We will specify later the required conditions on the noiseẆ . In the SFKE (1), I and are the identity and Laplacian operators, respectively, and the operators (I -) physical phenomena, such as diffusion in porous media with fractal geometry, kinematics in viscoelastic media, and propagation of seismic waves. In [2], the authors mainly studied the SFKE (1) with additive Gaussian space-time white noise in bounded and unbounded spatial domains. They connected it with the Eulerian theory of turbulence dispersion by means of the advection-diffusion equation. They also gave a very interesting connection with the Lagrangian theory. Nowadays we can find a lot of applications of these equations in turbulence, ecology, hydrology, geophysics, image processing, neurophysiology, economics and finance, etc. (see Angulo et al. [2], Anh et al. [4,5], Márquez-Carreras [16] and the references therein for more details). The composition of the Bessel and Riesz potentials plays an important role in describing the behavior of the process at the spatial macro and microscales. Apart from the classical context of heat conduction, an equation of form the SFKE (1) with α = 0 and γ = 2 also arises in neurophysiology; see [21] for example. Diffusion operators in the SFKE (1) with α = 0 and γ > 0 correspond to the generalized heat equation which have been used to define hyperviscosity and to study its effect on the inertial-range scaling of fully developed turbulence [13]. The presence of the Bessel operator is essential for a study of stationary solutions of the SFKE (1). One can also see [4] and [5] for related models.
After the nice work of Angulo et al. [2], several authors have also studied this kind of the SFKE (1) and other similar equations from a mathematical point of view. For example, Angulo et al. [1] considered a more generalized type of space-time fractional kinetic equation with Gaussian white noise or infinitely divisible noise as follows: with β n > β n-1 > · · · > β 1 > β 0 ≥ 0, A i > 0, i = 0, . . . , n, and the fractional-in-time derivative is defined in the Caputo-Djrbashian sense, i.e., where (·) is the gamma function. The solutions to the equation are proved in both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second-and higher-order properties. In particular, they also proved that the long-range dependence may arise in the temporal solution under certain conditions on the spatial operators. In [6], the authors provided a detailed review of the related literature. They considered a more general class of fractional (both in time and space) evolution equation defined on Dirichlet regular bounded open domains. They derived the sufficient conditions for the definition of a weak-sense Gaussian solution. The Hölder regularity of the solution with respect to the time and space variables is also derived. Meanwhile Márquez-Carreras [15] dealt with the SFKE (1) driven by a Gaussian noise which is white in time and correlated in space. They proved the existence and uniqueness of solution by means of a weak formulation and studied the Hölder continuity of this solution. Moreover, they also proved the existence of a smooth density associated to the solution process and studied the asymptotic behavior of this density. Later on Márquez-Carreras [16] studied the following kind of stochastic partial differential equations: with α > 0 and the processẆ is a Gaussian noise, white in time and correlated in space.
The existence and uniqueness of solution and the Hölder continuity of this solution was proved. Moreover, they proved the existence of the density of the solution and that its density was smooth.
In this paper, regarding the structure of the SFKE (1), we prove the existence and uniqueness of the solution and the Hölder continuity of this solution. Moreover, we show that the equation of the solution is absolutely continuous with respect to Lebesgue's measure on R d (with d < α + γ ) and establish the lower and upper bounds for its density by means of Malliavin calculus.
We would like to list some differences between this study and all the papers mentioned above. Firstly, the SFKE (1) we considered in this paper is driven by a more general Gaussian noise (fractional in time and correlated in space) which extended the former noises in Angulo et al. [2], Angulo et al. [1], and Márquez-Carreras [16]. Secondly, thanks to the fractional noise, the properties of the solution are checked for any α > 0 and γ > 0 and not for a more restricted region. Moreover, these properties do not depend on the dimension of x. Finally, we generalize some results of Balan and Tudor [7,8] to the fractional operator setting. We study some new properties of the mild solution to the SFKE (1). Here, we deal widely with the Hölder continuity in time and in space. We also study some density properties of the solution by using the techniques of Malliavin calculus; see, for example Nualart and Quer-Sardanyons [19,20], and Liu and Yan [14].
This article is organized as follows. In Sect. 2 we recall some preliminaries including the fractional noise and Malliavin calculus. Section 3 is devoted to describe what we understand by a solution of the SFKE (1) and prove the existence and uniqueness of this solution. We show that the solution of the SFKE (1) exists if (12) holds. In Sect. 4 we check that spatially the solution of the SFKE (1) with b = 0 is a Gaussian field with zero mean, stationary increments, and a continuous covariance function. We find its index (see Definition 4.1). We also show that the solution is not stationary in time. Finally in Sect. 5 we study the density properties of the solution of the SFKE (1), such as the existence of the density and related Gaussian-type lower and upper bounds for the density.

The preliminaries
This section is devoted to recalling some preliminaries about the fractional noise and related Malliavin calculus.

Fractional noise
Let us start by introducing some basic notions on Fourier transforms of functions: the space of real valued infinitely differentiable functions with compact support is denoted by D(R d ) and by S(R d ) the Schwartz space of rapidly decreasing C ∞ functions in R d . For ϕ ∈ L 1 (R d ), we let Fϕ be the Fourier transform of ϕ so that the inverse Fourier transform is given by Similarly to [8] or [11] for the general case, on a complete probability space ( , F, F t , P), for H > 1/2, we consider a zero-mean Gaussian process with α H = H(2H -1) and (·) : R d → R + is a non-negative definite function and its Fourier transform F = μ is a tempered measure. Moreover, we assume that there is an integer m ≥ 1 such that We callẆ the fractional noise; it has a spatial covariance (·) and has the covariance of a fractional Brownian motion with Hurst parameter H > 1/2 in time.
Let H be the completion of D([0, T] × R d ) endowed with the inner product where Fϕ refers to the Fourier transform with respect to the space variable only and the last equality in (2) Moreover, we can interchange the order of the integrals ds dt and μ(dξ ), since for the indicator functions φ and ϕ, the integrand is a product of a function of (t, s). Hence, we have The space H may contain distributions, but it contains the space |H| of measurable functions ϕ : [0, T] → R d such that We shall make a standard assumption on the spectral measure μ, which will prevail until the end of the paper (see Dalang [10] for some details about this hypothesis).

Hypothesis 1
The measure μ satisfies the following integrability condition: Remark 2.1 Since the spectral measure μ is a non-trivial positive tempered measure, we can ensure that there exist positive constants c 1 , c 2 and k such that The following estimate (see, for example, [18]) will be needed in the sequel: where C H > 0 is a constant depending only on Hurst parameter H.

Malliavin calculus
Gaussian, we might develop the Malliavin calculus (refer to Nualart [18] for more details) with respect to fractional noise introduced in Sect. 2.1 in order to study the density of the solution to the SFKE (1). We will also recall briefly the results in Nourdin and Viens [17] in order to establish the lower and upper bounds for the density. Recall the notation W (ϕ) = T 0 R d ϕ(t, x)W (dt, dx) for ϕ ∈ H, and let S be the class of smooth and cylindrical random variables of the form where f ∈ C ∞ b (R n ) (the set of all functions with bounded derivatives of all orders) and ϕ i ∈ H (i = 1, . . . , n and n ∈ N). For each F ∈ S, define the derivative D t,x F by Let D 1,2 be the completion of S under the norm Then D 1,2 is the domain of the closed operator D on L 2 ( ). We also denote by D h the closure of S under the norm with D h F = DF, h H . Let {h n , n ≥ 1} be an orthonormal basis of H. Then F ∈ D 1,2 if and only if F ∈ D h n for each n ∈ N and ∞ n=1 E|D h n F| 2 < ∞. On the other hand, the divergence operator δ is the adjoint of the derivative operator D characterized by with u ∈ L 2 ( ; H). Then Dom(δ), the domain of δ, is the set of all functions u ∈ L 2 ( ; H) such that for all F ∈ D 1,2 , where C u is some constant depending on u. Another important operator in the theory of Malliavin calculus is the generator of the Ornstein-Uhlenbeck semigroup, which is usually denoted by L (see, for example, Nualart [18]). It is related to the Malliavin derivative D and its adjoint δ through the formula δDF = -LF in the sense that F belongs to the domain of L if and only if it belongs to the domain of δD.
The authors in [17] considered a random variable F ∈ D 1,2 with mean zero and defined the following function on R: where L -1 denotes the pseudo-inverse of L. Then Nourdin and Viens [17] proved the following.

Proposition 2.3
Assume that there exists a positive constant C 1 , such that g F (F) ≥ C 1 > 0, a.s., then the law of F has a density p(·) whose support is R and satisfies, almost everywhere in R, An immediate consequence of the above proposition, is that, if one also has g F (F) ≤ C 2 a.s., then the density p(·) satisfies, for almost all z ∈ R, In order to deal with particular applications of this method, Proposition 3.7 in Nourdin and Viens [17] established an alternative formula for g F (F). That is, where, for any random variable F defined in ( , F, P), F denotes the shifted random variable in × , for some probability space , given by Notice that, indeed, F depends on the parameter ζ , but we have decided to drop its explicit dependence for the sake of simplification. In Eq. (7), E stands for the expectation with respect to .

Existence and uniqueness
In this section, we will study the Cauchy problem for the SFKE (1) driven by fractional noise. Following Walsh [22], let us recall the notation of a mild solution to the SFKE (1).
where G(t, x) is the fundamental solution (called also the Green function) of Moreover, according to [15], the Green function G(t, x) can be written as with i 2 = -1 and its Fourier transform FG(t, ·)(ξ ) is given by When H > 1/2, it turns out that under relatively mild assumptions on the fundamental solution G given by (9), the condition provides a necessary and sufficient condition for the stochastic integral with respect to Gaussian process W given by (11) is also the necessary and sufficient condition for the existence of the solution in the linear case, i.e.
Next we firstly give an integrability condition on the spectral measure μ.

Hypothesis 2
The measure μ satisfies the following integrability condition: Before we prove the equivalence between (11) and (12). Let us now recall some of the main examples of spatially covariances (·) (or the tempered measure μ), which will be our guiding examples in the remainder of the present paper.

Proposition 3.1 Assume that the condition (12) holds, then
Proof The proof of this proposition can be completed by using Proposition 3.2.

Proposition 3.2 Denote
Then we have with two positive constants C 3.1 and C 3.2 given by Proof Recall that the Fourier transform FG(t, ·)(ξ ) of G(t, x) with respect to the spatial variable x is given by (10). Then we can rewrite N t (ξ ) defined by (13) as follows: Now we want to find the lower and upper bound for N t (ξ ). Firstly assuming that |ξ | < 1 and using the fact e -x ≤ 1 for any x > 0, then one obtains where we have used the fact that 1 < 2 1+|ξ | 2 when |ξ | < 1. Suppose now that |ξ | ≥ 1, by means of the change of variables, inequality (5) and the fact that 1e -x < 1 for all x > 0, we have where we have used the fact So combining the two estimates for N t (ξ ) with |ξ | < 1 and |ξ | ≥ 1, we have . Next let us proceed to prove the lower bound for N t (ξ ). Suppose firstly t|ξ Using e -x > 1x for any x > 0, we conclude that Hence one obtains that where for the last inequality we have used the fact that 1 ≥ 1 1+|ξ | 2 .
where we have used the fact that |ξ | α (1 + |ξ | 2 ) Thus we can conclude to the following lower bound for N t (ξ ): Thus the proof of this proposition is complete.
Now we can state the main result in this section. The proof of this theorem could be derived by the standard arguments with some estimates of the Green function G(t, x) and some properties of the stochastic integral in (8). However, we have preferred to give the complete proof. We shall also make the following hypothesis on the coefficient b.
(H.1): The function b satisfies the Lipschitz condition as follows: Theorem 3.1 Assume that (12) holds and the coefficient b satisfies (H.1), then there exists a unique solution u(t, x) to the SFKE (1) such that for any T > 0 and p ≥ 2.
Firstly let us give a useful estimate associated with the Green function G(t, x) given by (9).
where the notation f g means that there exist two constants c, C such that cg ≤ f ≤ Cg.
Proof Using the Plancherel theorem and equality (10), we can write where we have used the integration in polar coordinates in the last equation above and S d is a positive constant resulting from the integration over the angular spherical coordinates. Now using the fact r α (1 + r 2 ) γ /2 ≥ r α+γ with r > 0, we get, with the change of variable formula u = 2tr α+γ , where ( d α+γ ) is a Gamma function. On the other hand, using the fact r α (1 + r 2 ) γ /2 ≤ (1 + r 2 ) α+γ 2 with r > 0, we get with the change of variable formula u = 2t(1 + r 2 ) where the last integral is finite. Then we can conclude the proof of this lemma. Now let us prove the main result in this section.
Proof of Theorem 3.1 We use the Picard approximation to get a solution to (8). Define Firstly, we will prove that It follows from (17) where Note that, by the Hölder inequality and the fact that For the term B (n) p (t, x), since the stochastic integral dy) is Gaussian, according to Proposition 3.2 and Eq. (12), we have Combining (18), (19) and (20), we have Then that fact that (19) together with the Gronwall lemma ensures that and consequently {u (n) (t, x), n ≥ 1} is well defined. Moreover, by Lemma 15 in Dalang [10], one can obtain Secondly let us prove that {u (n) (t, x), n ≥ 1} converges in L p ( ). As for n ≥ 2, Then Gronwall's lemma yields Hence, {u (n) (t, x)} n≥0 is a Cauchy sequence in L p ( ). Let t, x).
Taking n → ∞ in L p ( ) in both sides of (17) shows The uniqueness can be checked by a standard argument.
Remark 3.2 1. Our result Theorem 3.1 here is an extension of the one in Márquez-Carreras [15] (when H = 1/2) to fractional noise. However, the noises considered in Márquez-Carreras [15] is multiplicative. 2. The cases α = 0 and γ = 2, then Eq. (1) reduces to the classical stochastic heat equation with fractional version considered in Balan and Tudor [8]. Then the condition (12) is coherent with the results found in [8] for the stochastic heat equation with fractional noise.

Index-β Gaussian random field
In this section we will prove that the solution to the SFKE (1) satisfies the following property defined by Definition 4.1; see Márquez-Carreras [15] for example. As a related problem, we also study the sample paths of the solution to the SFKE (1).
it is seen that B H is an index-H random field.
Moreover, from Angulo et al. [1], the following results hold with probability one. 1. dim H (graph(X)) = d + 1β, where dim H is the Hausdorff dimension which quantifies the irregularity of a set and graph(X) := {(t, X(t)), t ∈ R d }. 2. X is Hölder continuous of order ρ strictly less than β. However, for any ρ < β, X fails to satisfy any uniform Hölder condition of order ρ. In this section we essentially show that the solution to the SFKE (1) has similar properties to the solution studied in Angulo et al. [1], Márquez-Carreras [15]. However before we state our main result in this section, we will firstly give another condition on the tempered measure μ which is slightly stronger than Hypothesis 2 because of the appearance of η ∈ (0, 1).

Hypothesis 3
For some ψ ∈ (0, 1), the measure μ satisfies the following integrability condition: Let us denote which is the mild solution to the SFKE (1) with b = 0. Then we have
Next let us move to the case of U with respect to the time variable. The result is given as follows.

Theorem 4.2
Assume that the measure μ satisfies Hypothesis 3 for some η ∈ (0, 1). Then, for t ∈ R + , τ ∈ R such that t + τ ∈ R + and x, z ∈ R d , the spatial-temporal covariance function of U(t + τ , x) and U(t, z) is

Moreover, U(·, x) is asymptotically in time with an index-β Gaussian field with
Since the process U is not stationary in time but as t tends to infinity, it converges to a stationary process. That means the limiting-time process is stationary in time and space.

Analysis of the density
This section is devoted to a study of the density of the solution to the SFKE (1) at any fixed (t, x) ∈ [0, T] × R d . This will be done by using Malliavin calculus. The aim in this section is two-fold. Firstly we will prove that the solution to the SFKE (1) at any fixed (t, x) ∈ [0, T] × R d is a random variable whose equation admits a density. Secondly we apply the results obtained by Nourdin and Viens [17] to the SFKE (1) to obtain the upper and lower Gaussian-type estimates for the density (see recent work by Nualart and Quer-Sardanyons [19,20], and Liu and Yan [14]).

Existence of the density
The main result in this subsection is stated as follows. Before giving the proof of Theorem 5.1, we firstly give the following.
Proof Let u (n) (t, x)(n ≥ 1) be the solution of Eq. (17). Since b is Lipschitz, by a standard argument, one can see that the sequences u (n) converges to u in L p ( ) for any p ≥ 2 and (t, x) ∈ [0, T] × R d as n → ∞. Then a similar argument to that in Zhang and Zheng [23] shows that, for each n ∈ N and h ∈ H, Since u (n) (t, x) → u(t, x) as n → ∞ in the L p ( ) sense, there exists a random field u h (t, x) such that D h u (n) (t, x) → u h (t, x) as n → ∞ uniformly on (t, x) ∈ [0, T] × R d , and the latter satisfies Hence, from the closeness of the operator D h , it follows that u(t, x) ∈ D h , D h u(t, x) = u h (t, x) and Next we proceed to proving that u(t, x) ∈ D 1,2 . Recall the sequence {h n , n ≥ 1} introduced in Sect. 2. By (48), one gets with two positive constants C 5. Then, by (49), the Hölder inequality with p = q = 2 and estimates (16) for the Green function, we have Then the Gronwall lemma yields where C 5.1.7 and C 5.1.8 are independent of m. Let m → ∞ to get That means that u(t, x) ∈ D 1,2 . Since From (48), (51) and Fubini's theorem, it follows that Thus we can conclude the proof of this proposition.
We also need the following lemma concerning the estimates for the L 2 -norm of the Malliavin derivative Du(t, x).

Lemma 5.1
For ε ∈ (0, t) and d < α + γ , there exist two positive constants C 5.1.9 and C 5.1.10 such that and Proof We will only deal with the proof of (52), since (53) can be checked by using exactly the same arguments. For s ∈ [tε, t], set Then from the proof of Proposition 5.1, we get Let us invoke the linear equation (45) satisfied by the Malliavin derivative Du(s, y) for (s, y) With the estimate (16) associated with the Green function G(t, x), we have Then For the second term L ε,2 (s, y), we apply the Hölder inequality, the fact that b is bounded and Fubini's theorem, so that we end up with By (56) and Lemma 5.1, one gets Then, for each ε 0 > 0, according to (58), (59) and (60), Thus the proof of this theorem is complete.

Lower and upper bounds for the density
Let us consider T > 0 and let u = {u(t, x), (t, x) ∈ [0, T] × R d } be the unique mild solution to Eq. (1). This section is devoted to proving the following result concerning with the Gaussian-type estimates for the density of u(t, x) at any fixed (t, x) ∈ [0, T] × R d .

Theorem 5.2
Fix t ∈ [0, T] and x ∈ R d . Suppose that Hypothesis 3 is satisfied for some η ∈ (0, 1). Moreover, the coefficient b(·) is of class C 1 (R d ) and has a bounded Lipschitz continuous derivative. Then the density of the random variable u(t, x) satisfies the following: for almost every z ∈ R where m = Eu(t, x) and C 5 2. There exists a positive constant k 2 such that, for any t ∈ [0, T], Remark 5.1 It is worth mentioning that the integrability condition (12) was sufficient for us to prove the existence of density for the solution u(t, x) at any fixed point (t, x) ∈ [0, T] × R d . However, as will be made clearer in Lemma 5.2, we will really need lower and upper bounds of the form (63) and (64) in order to obtain lower and upper bounds for the density of u(t, x) at any fixed (t, x) ∈ [0, T] × R d .
Remark 5.2 It is interesting to note that the lower and upper bounds obtained in this proposition did not include the parameter α and γ .
Theorem 5.2 will be a consequence of Theorem 3.1 in [17] and Proposition 5.1. We use the notation F = u(t, x) -Eu(t, x) and we recall that we will need to find almost sure lower and upper bounds for the random variable g F (F), which is given by where DF = (DF)(e -ζ ω + √ 1e -2ζ ω ).
In order to prove Proposition 5.1, we will also need the following lemma, whose proof is similar to that of Lemma 5.1, Lemma 4.6 in Nualart and Quer-Sardanyons [19] As a consequence, we have the following estimate: Let Then according to (64), we have proved that Then a suitable generalization of the Gronwall-type lemma (see, for example, Lemma 15 in Dalang [10]) allows us to conclude the proof. The estimation (68) can be checked using exactly the same arguments.
Proof of Proposition 5. 1 We first recall that the Malliavin derivative of u(t, x), (t, x) ∈ [0, T] × R satisfies D v,z u(s, y) ≥ 0, for all (v, z) ∈ [0, T] × R d , a.s. This is because the Malliavin derivative solves the linear equation (45). Let us deal with the proof of (66) in two steps. Our method used here is essentially due to Nualart and Quer-Sardanyons [19] and [20].
Let us finally estimate | 3 (t, x; δ)|. For this, we apply Fubini's theorem, the fact that b is bounded, the Cauchy-Schwartz inequality, and we finally invoke Lemma 5.3,
Step 2. The upper bound. The upper bound in (66) is almost an immediate consequence of the computations which we have just performed for the lower bound. More precisely, according to g F (F) and the considerations in the first part of the proof, we have the following: where we notice that we have substituted δ by 1 in i (t, x; δ), i = 0, 1, 2, 3. We have already seen that, for i = 1, 2, i (t, x; 1) ≤ C 5.2.14 t 2H+1-d 2(α+γ ) and 3 (t, x; 1) ≤ C 5.2.17 t 2H+1-d α+γ .
Therefore we conclude that with the positive constants C 5.2.2 depending on T. Therefore the proof of this proposition is complete.
Proof of Theorem 5.2 For any fixed (t, x) ∈ [0, T] × R, we know that the random variable F = u(t, x) -E(u(t, x)) is centered and belongs to D 1,2 and by (66); we have 0 < C 5.2.1 t 2H ≤ g F (F) for all t ∈ [0, T]. We then apply Theorem 3.1 and Corollary 3.3 in Nourdin and Viens [17], and find that the probability density ρ : R → R of the random variable F is given by dy , for almost every z ∈ R. Then the density p of the random variable u(t, x) satisfies p(z) = E|u(t, x) -E(u(t, x))| 2g F (z -E(u(t, x))) expz-E(u(t,x)) 0 y g F (y) dy .
In order to conclude the proof, we only need to apply the bounds obtained in Proposition 5.1 to (80).