Fractional Fokker-Planck-Kolmogorov equations associated with SDES on a bounded domain

1 Introduction

The Fokker-Planck equation, including fractional versions, play an important role in the modeling of various random processes. Its applications can be found in statistical physics [6, 13, 35], quantum mechanics [6, 29], biology [7, 26], finance [23, 35], just to mention a few. The most of these applications assume an idealistic model in which a random quantity may take values in the whole d-dimensional space ℝ^d. Fractional generalizations of the Fokker-Planck equation in this case are obtained in works [9, 12, 14, 15, 24, 37]; see also references therein. The connection of a wide class of fractional Fokker-Planck equations with their associated stochastic differential equation driven by time-changed Lévy processes was studied in paper [14].

What concerns stochastic processes in a bounded domain, they are also spread out broadly, an example of which is a diffusion in a bounded region. The essential difference of the stochastic process in a bounded domain from the case of stochastic processes in ℝ^d is the influence of the boundary or near boundary processes to the whole picture. There is a rich literature on the stochastic processes and the associated (non fractional) Fokker-Planck equations on a bounded domain; see recent monograph by Taira [30].

In the present paper we will discuss fractional generalizations of Fokker-Planck-Kolmogorov (FPK for short) equations associated with nonlinear stochastic differential equations on a bounded domain with a smooth boundary. The fractional diffusion and associated FPK equation on a bounded domain was studied in papers [1, 8, 10, 21, 32] in various particular cases. Our goal in this paper is to study general initial-boundary value problems describing stochastic processes undergoing inside a bounded region, as well as on its boundary. We also discuss initial-boundary value problems for fractional FPK equations associated with stochastic differential equations driven by fractional Brownian motion on a bounded domain. Note that fractional FPK equation in the whole space ℝ^d associated with stochastic processes driven by fractional Brownian motion was considered in papers [2, 5, 15, 16].

For reader’s convenience we start the discussion with a well known particular cases of the influence of the boundary and related boundary conditions. Consider a nonlinear stochastic differential equation

where x ∈ ℝ^d is a random vector independent of m-dimensional Brownian motion B_t, and the mappings

satisfy Lipschitz and linear growth conditions. Namely, there exist positive constants C₁ and C₂ such that for all x, y ∈ ℝ^d the inequalities

and

hold. The stochastic differential equation (1.1) is understood as

with the second integral in the sense of Itô.

The backward Kolmogorov equation associated with SDE (1.1) in terms of the density function u(t, x) of Xtx = (X_t|X₀ = x) is

where t-dependent differential operator A(t) is defined in the form

with (d × d)-matrix-function {a_j,k(t, x), j, k = 1 … d}, coinciding with the matrix-function 12σ(t, x) σ^T(t, x). Here σ^T(t, x) is the matrix transposed to the matrix-function σ(t, x). The corresponding Fokker-Planck equation, or forward Kolmogorov equation, has the form

where A^* is the formally adjoint operator to A(t). Both equations (1.5) and (1.7) are accompanied with the initial condition

where u₀(x) is the density function of the initial vector X₀. Unifying the terminology used in these two equations we call the pair of equations (1.5) and (1.7)the FPK equation, the term used throughout the current paper.

If the solution process X_t of the stochastic differential equation in (1.1) is allowed to change only in a bounded region Ω ⊂ ℝ^d with a smooth boundary ∂Ω, then the probability ℙ(X_t ∈ ℝ^d \ Ω) of X_t being out of the region Ω is zero. The associated FPK equation in this case needs to be supplemented by boundary conditions. In order to see in what form the boundary conditions emerge, we need to introduce the notion of probability current, a d-dimensional vector field Φ(t, x), components of which are defined by

Using the probability current, one can write forward FPK equation (1.7) in the form of a conservation law:

or, the same,

where ∇=(∂∂x1,…,∂∂xd) is the gradient operator and the symbol “⋅” means the dot product of two vector objects.

Let S be a (d−1)-dimensional hyper-surface in Ω, and n_x, x ∈ S, be an outward normal to S at the point x ∈ S. Then the total flow of probabilities through the hyper-surface S can be calculated by the surface integral

Two boundary conditions are common in the study of stochastic processes in a bounded region:

1. reflecting boundary condition, and

2. absorbing boundary condition.

The reflecting boundary condition means that there is no probability flow across the boundary, leading to the condition

In this case X_t will stay in the region Ω forever and be reflected when X_t reaches the boundary ∂Ω. The absorbing boundary means that X_t will be absorbed by boundary as soon as X_t reaches the boundary, leading to the condition

Let a boundary operator 𝓑 be defined in the form

where the functions μ(x) and ν(x) are continuous on the boundary ∂Ω, satisfying some physical conditions discussed below, and n_x is the outward normal at point x ∈ ∂Ω. In order to define operators that will be used in the FPK equations associated with the SDEs in a bounded domain we introduce the following spaces:

with the boundary operator 𝓑 defined in (1.11). Introduce the operator A_𝓑(t) on the space CB2(Ω) :

In other words the operator A_𝓑(t) formally is the same as the operator A(t) defined in (1.6), but with the domain Dom(A_𝓑(t)) = CB2(Ω) for each fixed t > 0. The operator A_𝓑 is linear and maps the space CB2(Ω) to C(Ω).

Both boundary conditions for stochastic processes in a domain with absorbing or reflecting boundaries discussed above can be reduced to the boundary condition

Therefore, the forward and backward FPK equations in the case of bounded domain in terms of the density function u(t, x) of Xtx = (X_t|X₀ = x) can be formulated with the help of the operator A_𝓑. Namely, the backward FPK equation is given by

and the forward FPK equation is given by

where AB∗ is the formally adjoint operator to A_𝓑.

Thus, the explicit form of the FPK equation associated with a SDE with drift and diffusion coefficients independent of the time variable t is given by the initial-boundary value problem

where b_j(x), j = 1, …, d, are drift coefficients and a_i,j(x), i, j = 1, …, d, are diffusion coefficients; the functions μ(x′) and ν(x′) are continuous functions defined on the boundary ∂Ω. It is well known that the stochastic process solving SDE corresponding to the FPK equation in (1.12)-(1.14) is continuous. Such a stochastic process represents a diffusion process of a Markovian particle.

The paper is organized as follows. Section 2 provides necessary preliminary information on general (integer order) FPK equations through the Waldenfels operator and Wentcel’s boundary conditions. The corresponding stochastic processes represent solutions of associated stochastic differential equations on a bounded domain with combined continuous diffusion, jump, and other relevant phenomena. The main results of the paper is in Section 3. In this section we derive initial-boundary value problems for fractional order FPK equations. The associated stochastic processes represent solutions of stochastic differential equations in a bounded domain with an appropriate time-changed driving processes. In this section we also prove the existence of a unique solution to the obtained initial-boundary value problems.

2 Preliminaries and auxiliaries

FPK equation (1.12)-(1.14) does not take into account jumping events and some specific phenomena (diffusion on the boundary, jumps on the boundary or into the domain, viscosity, etc.) which may occur on the boundary. The general case with jump and viscosity effects involve pseudo-differential operators in the FPK equation and in the boundary condition. The theory of pseudo-differential operators was first developed by Kohn and Nirenberg [20] and Hörmander [17]. An operator A is called pseudo-differential with a symbol σ_A(x, ξ), if

where f~ (ξ) is the Fourier transform of f:

The symbols of pseudo-differential operators in [20, 17] are smooth and satisfy some growth conditions as |ξ| → ∞. However, symbols of pseudo-differential operators considered in the present paper are not smooth. We refer the reader to works [3, 18, 33] where appropriate classes of symbols are studied.

For simplicity we assume that Ω ⊂ ℝ^d is a bounded domain with a smooth boundary ∂Ω ⊂ ℝ^d−1. In the general case the (backward) FPK equation can be formulated as follows:

where 𝓐(x, D) is a second order Waldenfels operator acting on the space of twice differentiable functions defined on Ω, that is

and 𝓦(x′, D) is a boundary pseudo-differential operator defined on the space of twice differentiable functions defined on ∂Ω through the local coordinates x′ = (x₁, …, x_d−1) ∈ ∂Ω (see [30]) :

with (pseudo)-differential operators

and

Here τ_j(x′, y′), j = 1, 2, are local unit functions and ν_j(x′, dy′), j = 1, 2, are Lévy kernels satisfying some conditions indicated below, and u₀ is the density function of the initial state X₀. If the initial state X₀ = 0, then u₀(x) = δ₀(x). The boundary condition (2.3) is called a second order Wentcel’s boundary condition, to credit Wentcel’s contribution [36] to the theory of diffusion processes.

Obviously, initial-boundary value problem (2.2)-(2.4) recovers (1.12)-(1.14) if ν(x, ⋅) = 0, Q(x′, D) φ(x′)= γ(x′) φ(x′), δ(x′) = 0, and ν_j(x′, ⋅) = 0, j = 1, 2. Moreover, one can work with coefficients a_ij(x), i.j = 1.…, n, b_j(x), j = 1, …, d, c₀(x), and the symbol σ_A(x, ξ) of the operator A(x, D) in (2.5) satisfying mild conditions similar to Lipschitz (1.2) and linear growth condition (1.3). However, in this paper, for simplicity, we assume the following conditions:

1. a_ij(x) ∈ C^∞(Ω) ∩ C(Ω) and a_ij(x) = a_ij(x) for all i, j = 1, …, d, and x ∈ Ω, and there exists a constant a₀ > 0 such that

   ∑i,j=1daij(x)ξiξj≥a0|ξ|2,x∈Ω¯,ξ∈Rd;

2. b_j(x) ∈ C^∞(Ω) ∩ C(Ω), j = 1, …, d;

3. c₀(x) ∈ C^∞(Ω) ∩ C(Ω), and c₀(x) ≤ 0 in Ω;

4. the local unity function ϕ(x, y) and the kernel function ν(x, dy) are such that the symbol σ_A(x, ξ) of the operator 𝓐(x, D) belongs to the class of symbols S(Ω × ℝ^d) = C(Ω × ℝ^d) ∩ C^∞(Ω × (ℝ^d \ G)), where G ∈ ℝ^d is a set of d-dimensional measure zero; see [33] for the theory of pseudo-differential operators with such a nonregular symbols.

Equation (2.2) describes a diffusion process accompanied by jumps in Ω with the drift vector (b₁(x), …, b_n(x)) and diffusion coefficient defined by the matrix-function a_i,j(x), and jumps governed by the Lévy measure ν(x, ⋅). We assume additionally that the condition

1. A(x,D)[1(x)]=c0(x)+∫Ω[1−ϕ(x,y)]ν(x,dy)≤0,x∈Ω,

is fulfilled to ensure that the jump phenomenon from x ∈ Ω to the the outside of a neighborhood of x is “dominated” by the absorption phenomenon at x (see [30] for details).

The coefficients α_ij(x′), i, j = 1.…, d, β_j(x′), j = 1, …, d, and γ(x′) of the operator Q(x, D) in (2.5) satisfy the following conditions:

1. α_ij(x′) ∈ C^∞(∂Ω) and α_ij(x′) = α_ij(x′) for all i, j = 1, …, d−1, and x′ ∈ ∂Ω, and there exists a constant α₀ > 0 such that

   ∑i,j=1d−1αij(x′)ξiξj≥α0|ξ|2,x′∈∂Ω,ξ∈Rd−1;

2. β_j(x′) ∈ C^∞(∂Ω), j = 1, …, d;

3. γ(x′) ∈ C^∞(∂Ω), and γ(x′) ≤ 0 in ∂Ω.

   The symbols σ_Γ1(x′,ξ) and σ_Γ2(x′,ξ) of the boundary pseudo-differential operators Γ₁(x′, D) and Γ₂(x′, D) in equations (2.8) and (2.9) satisfy the following condition:

4. the local unity functions ϕ_k(x, y), k = 1, 2, and the kernel function ν_k(x, dy), k = 1, 2, are such that the symbols σ_Γk(x′,ξ) of operators Γ_k(x′, D), k = 1, 2, belong to the class of symbols S(∂Ω, ℝ^d) = C^∞(∂Ω × (ℝ^dG₀)), where G₀ ∈ ℝ^d is a set of d-dimensional measure zero.

The boundary condition (2.3) with the operator 𝓦(x, D) defined in equations (2.6)-(2.9) describes a combination of continuous diffusion and jumping processes undergoing on the boundary, as well as jumps from the boundary into the region, and the viscosity phenomenon near the boundary. Namely, the term

governs the diffusion process on the boundary, the term γ(x′) u(t, x′) is responsible for the absorption phenomenon at the point x′ ∈ ∂Ω, the term μ(x′)∂∂nx′ expresses the reflexion phenomenon at x′ ∈ ∂Ω, the term δ(x′) A(x′, D) u(t, x′) expresses the viscosity near x′ ∈ ∂Ω, and the terms Γ₁ (x′, D) u(t, x′) and Γ₂ (x′, D) u(t, x′) govern jump processes on the boundary and jump processes from the boundary into the region, respectively. We assume that the condition

1. W(x′,D)[1(x)]=γ(x′)+∫∂Ω[1−τ1(x′,y′)]ν1(x′,dy′)+∫Ω[1−τ2(x′,y)]ν2(x′,dy)≤0,x′∈∂Ω,

is fulfilled to ensure that the jump phenomenon from x′ ∈ ∂Ω to the outside of a neighborhood of x′ on the boundary ∂Ω or inward to the region Ω is “dominated” by the absorption phenomenon at x′.

We also assume the following transversality condition of the boundary operator 𝓦:

1. ∫Ων2(x′,dy)=∞,if μ(x′)=δ(x′)=0.

Fractional order FPK equations use fractional derivatives, and in more general case, fractional distributed order operators. Below we briefly recall the definitions and some properties of these operators referring the reader to the sources [19, 22, 33, 34] for details. By definition, the Caputo-Djrbashian derivative of order 0 < β < 1 is defined by

where τ ≥ 0 is an initial point and Γ(z), z ∈ ℂ \ {…, −2, −1, 0}, is Euler’s gamma function. We assume that the function g(t), t ≥ 0, is a differentiable function. We write D∗β when τ = 0. Making use of the fractional integration operator of order α > 0

we can write τD∗β also in the operator form

In our further analysis, without loss of generality, we assume that τ = 0.

Let μ be a Borel measure defined on the Borel sets of the interval (0, 1). The distributed order differential operator with the mixing measure μ, by definition, is

Two functions 𝓚_μ(t) and Φ_μ(s) defined with the help of the measure μ respectively by

play an important role in our further analysis. We denote by g~(s) the Laplace transform of a function g, that is

One can verify using the Laplace transform formula for fractional derivatives,

that the formula

holds. Moreover, using the formula [t−β]~(s)=Γ(1−β)sβ−1 (see [33]), one can also verify that the relation

holds.

The widest class of processes for which the Itô calculus can be extended is the class of semimartingales [27]. A semimartingale is a càdlàg process (the processes with paths which are right continuous and have left limits) which has the representation (see details in [27]) Z_t = Z₀ + V_t + M_t, where Z₀ is a d-dimensional random vector, V_t is an adapted finite variation process, and M_t is a local martingale. An example of a semimartingale is a Lévy process defined as follows. A stochastic process L_t ∈ ℝ^d, t ≥ 0, is called a Lévy process if the following three conditions are verified: (1) L₀ = 0; (2) L_t has independent stationary increments; (3) for all ϵ, t > 0, lim_s→t ℙ(|L_t − L_s | > ϵ) = 0. Lévy processes have càdlàg modifications. Lévy processes are fully described by three parameters: a vector b ∈ ℝ^d, a nonnegative definite matrix Σ, and a measure ν defined on ℝ^d \ {0} such that ∫ min (1, |x|²)dν < ∞, called its Lévy measure. The Lévy-Khintchine formula provides a characterization of Lévy process through its characteristic function Φ_t(ξ) = e^tΨ(ξ), with the Lévy symbol (see, e.g. [28])

A Lévy subordinator is a nonnegative and nondecreasing Lévy process. A Lévy subordinator with the Lévy symbol Ψ(s) = s^β, s > 0, is called a β-stable Lévy subordinator. The composition of two Lévy processes is again a Lévy process. In particular, a time-changed process L_Dt with a stable Lévy subordinator D_t is again a Lévy process. Transition probabilities of such a time-changed process satisfy FPK type equation with a pseudo-differential operator on the right hand side, whose symbol is continuous but not smooth. However, the composition L_Wt, where W_t = inf{τ ≥ 0: D_τ > t}, the inverse to the stable subordinator D_t is not a Lévy process, but still is a semimartingale. This drastically changes the associated FPK equation: now it is a time-fractional pseudo-differential equation [14, 15], implying non-Markovian behaviour of the process.

The properties of the density function of the inverse process W_t are listed below in the more general case when W_t represents the inverse of a process which belongs to the class 𝓢_μ of mixtures of an arbitrary number of independent stable subordinators with a mixing measure μ. Namely, a non-negative increasing Lévy stochastic process D_t belongs to the class 𝓢_μ if

It is known (see e.g. [28]) that mixtures of independent stable processes of different indices are no longer stable. For a stochastic process in 𝓢_μ we use the notation D_μ,t showing the dependence on the measure μ, and for their inverses the notation W_μ,t. The density function of W_μ,t is denoted by ftμ(τ) . Note that such mixtures model complex diffusions and other types of stochastic processes with several simultaneous diffusion modes. Their associated FPK type equations are distributed order differential equations (see e.g. [19, 22, 34]).

For the proofs of these lemmas we refer the reader to [33].

3 Main results: fractional FPK equations associated with SDEs on a bounded domain

In this section we derive fractional order FPK equations associated with SDEs on a bounded domain as initial-boundary value problems. We also prove existence of a unique solution to obtained initial-boundary value problems.

In the particular case of W_t being the inverse of a single Lévy’s subordinator D_t with the stability index β ∈ (0, 1) this theorem implies the following result:

An important question is the existence of a unique solution of the initial-boundary value problem in equations (3.13)-(3.15).

Now consider the following initial-boundary value problem (2.2)-(2.4) for t-dependent generalization of the FPK equation

where 𝓑(x, D) is a pseudo-differential operator whose order is strictly less than the order of the operator 𝓐(x, D). We assume that 𝓐(x, D) is an elliptic Waldenfels operator defined in (2.5) and 𝓦(x′, D) is a Wentcel’s boundary pseudo-differential operator defined in (2.6). The parameter γ runs in the interval (−1, 1). Equation (3.13) is a parabolic equation: if 0 < γ <1, then it is degenerate; if −1 < γ < 0, then it is singular. Obviously, the initial-boundary value problem in (3.13)-(3.15) recovers problem (2.2)-(2.4), if 𝓑(x, D) = 0 and γ = 0.

Here is the motivating example: if 𝓑(x, D) = 0, γ = 2H − 1 and 𝓐(x, D) = Δ, Laplace operator, then equation (3.13) reduces to

which represents the FPK equation associated with the fractional Brownian motion with the Hurst parameter H ∈ (0, 1) (see, e.g. [5, 16]). Therefore, initial-boundary value problem (3.13)-(3.15) can be considered as a FPK equation of a stochastic process driven by fractional Brownian motion. By definition, the fractional Brownian motion with the Hurst parameter H is a stochastic process BtH, Gaussian with mean zero for each fixed t ≥ 0, with continuous paths and the following covariance function:

Fractional Brownian motion is not Markovian and is not a semimartingale. Therefore, a stochastic integral and stochastic differential equation driven by a fractional Brownian motion can not be defined in the Itô sense. Nevertheless, one can define stochastic differential equations driven by fractional Brownian motion in a meaningful sense [4, 11, 25]. Thus, the class of initial-boundary value problems of the form (3.13)-(3.15) describes stochastic processes in a bounded region Ω driven by not only Lévy processes, but also fractional Brownian motion. Below we derive the fractional FPK equation associated with such a stochastic process with a time-changed driving process.