The probabilistic point of view on the generalized fractional partial differential equations

2.1 Preliminaries: Standard fractional derivatives

Let I_af be the integration operator defined on the set of continuous curves f ∈ C([a, b]) as I_af(x) = ∫axf(t) dt. Integration by parts yields

Similarly by induction one gets the following formula for the iterated Riemann integral

This formula motivates the definition of the (left) fractional or Riemann-Liouville (RL) integral of order β > 0:

Noting that the derivation is the inverse operation to usual integration, the definition (2.2) of the fractional integral suggests two notions of fractional derivative, the so-called RL (left) derivatives of order β ∈ (n, n + 1), n a nonnegative integer (where x > a):

and the (left) Caputo-Dzherbashyan (CD) derivatives of order β ∈ (n, n + 1):

In this paper we shall be mostly concerned with the case β ∈ (0, 1) and its extensions, and occasionally look at the derivatives of higher order as the compositions of the derivatives of order β ∈ (0, 1). For β ∈ (0, 1) the definitions of the RL and CD derivatives turn to

and respectively

As is seen by direct calculations (see e.g. Appendix in [41]), for smooth enough f,

implying

In particular, it follows that for smooth bounded integrable functions, the RL and CD derivatives coincide for a = −∞, and one defines the fractional derivative in generator form as their common value:

Another useful rewriting of (2.7) and (2.8) (used in Section 3) is obtained by the change of the variable of integration:

When β ∈ (0, 1) and x < a, the corresponding right derivatives can be introduced by the formulas

implying

The right RL and CD derivatives coincide for a = ∞, and one defines the fractional derivative in generator form as their common value:

2.2 Generalized fractional differential operators

The fractional derivative d^βf/dx^β, β ∈ (0, 1), from (2.10) was suggested as a substitute to the usual derivative df/dx, which can model some kind of memory by taking into account the past values of f. An obvious extension widely used in the literature (see e.g. [18], [62], [81], [26] and references therein) represent various mixtures of such derivatives, both discrete and continuous,

To take this idea further, one can observe that d^βf/dx^β represents a weighted sum of the increments of f, f(x − y) − f(x), from various past values of f to the ‘present value’ at x. From this point of view, the natural class of generalized mixed fractional derivative represent the causal integral operators

with some positive measure ν on {y : y > 0} satisfying the one-sided Lévy condition:

which is just the condition ensuring that L_ν is well-defined at least on the set of bounded infinitely smooth functions on {y : y ≥ 0}. The sign − is introduced to comply with the standard notation of the fractional derivatives, so that, for instance,

with ν(y) = −1/[Γ (−β)y^1+β] (note that Γ(−β) < 0).

The dual operators to L_ν are given by the anticipating integral operators (weighted sums of the increments from the ‘present’ to any point ‘in future’):

As one of the most studied examples of D+(ν), let us mention the so-called tempered fractional derivatives given by the measure

with constants C, λ > 0 (see [6] and [58]). Section 3.4 yields another point of view on this derivative.

Looking at (2.10) with ’probabilistic eyes’ one recognises in the operators −d^βf/dx^β, β ∈ (0, 1) the generators of stable Lévy subordinators with the inverted direction (see e.g. [58], Ch.3 and [40], Ch.1 and 8), the operators −d^βf/d(−x)^β representing the generators of these subordinatots with the standard direction. Thus the probabilistic point of view suggests to take as fully mixed extensions of these operators the general Lévy subordinators, which leads again to the same operators (2.18) and (2.20), the condition (2.19) being the well known condition of the theory of Lévy processes defining the one-sided Lévy measures.

One can weight differently the points in past or future depending on the present position, and one can also add a local part to complete the picture, leading to the operators

with a non-negative b(x) and transition kernel ν(x, .), ∫ min(1, y)ν(x, dy) < ∞, which capture in full the idea of ’weighting the past’ and which can be called the one-sided, namely left or causal, weighted mixed fractional derivatives of order at most one. Symmetrically, one can define the right or anticipatingweighted mixed fractional derivatives of order at most one as

From the probabilistic point of view the extension to (2.22) and (2.23) represents the transition from time homogeneous monotonic processes to arbitrary decreasing or increasing Feller processes; operators (2.22) and (2.23) are known to represent the general form of the generators of such processes (Courrège theorem, see e.g. [40]).

To link with the theory of pseudo-differential operators, it is worth noting that the operators D+(ν)  and  D−(ν) are ΨDOs with the symbols −ψ_ν(−p) and −ψ_ν(p), where

is the symbol of the operator L_ν.

If ν is finite, the operators D+(ν) are bounded, which is not the case for usual derivatives. Thus the proper extensions of the derivatives represent only the operator D+(ν) arising from infinite measures ν satisfying (2.19). The operators arising from finite ν can be better described as the mixtures of finite differences approximating the derivatives.

The operators D±(ν) represent the extensions of the fractional derivatives D−∞+β  and  D∞−β in generator form. Looking for the corresponding extensions of the operators Da±β  and  Da±∗β with a finite a we note that, by (2.8) and (2.14), Da+∗β (resp. Da−∗β) is obtained from D−∞+β (resp. D∞−β) by the restriction of its action to the space C_{const(−∞,a]}(R) (resp. C_const[a,∞)(R). Therefore, the analogs of the CD derivatives should be defined as

By (2.9) and (2.13), the operators Da+β  or  Da−β, the analogs of the Riemann-Liouville derivatives, are obtained by further restricting the actions of D+β  and  D−β to the spaces C_{kill(−∞,a]}(R) and C_kill[a,∞)(R):

Similarly, the derivatives with a finite a based on the general one-sided operators (2.23) are obtained by the same reduction of these operators, that is, say for the left derivatives, they are given by

From the probabilistic point of view it is then seen that the operators (2.8) are the generators of the modifications of the stable subordinators obtained by forbidding them (interrupting on an attempt) to cross the boundary x = a with an a ∈ R, that is, all jumps aimed to jump over the chosen barrier-point a are forced to land exactly at a. On the other hand, the operators (2.7) are the generators of the modifications of the stable subordinators obtained by killing them on an attempt to cross the boundary x = a. Applying the same procedure to operators (2.23) and (2.22) lead to the operators (2.24) and (2.26), which represent thus the generators of Markov processes interrupted or killed on an attempt to cross the boundary point a.

Finally one can further extend these mixed derivatives by including additional killing mechanisms or by mixing killing and stopping (say, by working with the linear combinations of of CD and RL derivatives). Such extension leads to the following mixed derivative operators:

where S(a, x) and K(a, x) are the rates of stopping and killing respectively. Some formulas simplifies (this is especially relevant for the extensions to manifolds) if one counts jumps by their finite points (rather than relative to x) leading to the following modified version of (2.27):

2.3 Generalized fractional integrals: shift invariant case

Let us see what should be the proper analog of the fractional integral exploiting approaches from probability, semigroup theory and the generalized functions/ΨDEs theory. Let us consider first the case of operators (2.24) arising from the Lévy subordinators. We shall talk about the left derivatives for definiteness.

The operator d^β/dx^β ∘ I−∞β acts as the identical operator on functions with a compact support. Hence, in the language of the semigroups of operators, the operator I−∞β is the potential operator of the strongly continuous semigroup of linear operators in C_∞(R) generated by −d^β/dx^β, that is the limit of the resolvent operator R_λ = (λ + d^β/dx^β)⁻¹, as λ → 0. Notice that Iaβ is just the reduction of I−∞β to the space C_{kill(−∞,a]}(R). As is worth noting I−∞β is an unbounded operator in C_∞(R), but is bounded when reduced to C_{kill(−∞,a]}((−∞, b]) for any b > a. Thus we can define the generalized fractional integralIaν as the potential operator of the semigroup generated by Lν′ reduced to the space C_{kill(−∞,a]}(R).

On the other hand, the usual fractional integral (2.2) solves the boundary value problem D^βg = f with g(x) = 0 for x ≤ a. Recall that for a Feller process X_t(x) and a function f from the domain of its generator L the process f(X_t(x)) − ∫0t Lf (X_s(x)) ds is a martingale, called Dynkin’s martingale (see e.g. [40]). Applying to this martingale Doob’s optional sampling theorem shows that

where g = Lf and τ is the exit time of X_t(x) from a domain (at least if Eτ < ∞). This implies that

where Xtβ is the stable subordinator and τ_x is the time when Xtβ reaches the point x − a (and thus x − Xtβ reaches a). Therefore, from the probabilistic point of view, the generalized fractional integral, representing the analog of Iaβ for the case of the generalized derivative Da+(ν), is the path integral

where Xtν is the Lévy subordinator generated by operator (2.20) and τ_x the time for this process to reach a.

Finally, as is known (see e.g. [42]), the fundamental solution (vanishing on the negative half-line) to the fractional derivative operator d^β/dx^β is U^β(x) = x+β−1/Γ (β), so that the usual fractional integral Iaβf(x) represents the integral operator with the kernel being the fundamental solution of d^β/dx^β or, in other words, the convolution with this fundamental solution, restricted to the space C_{kill(−∞,a]}(R). Thus, from the point of view of the theory of PDEs and ΨDEs, the generalized fractional integral, representing the analog of Iaβ for the case of the generalized derivative Da+(ν), is the integral operator

where U^(ν)(dz) is the fundamental solution to the operator Lν′ (we denoted it U^(ν)(dz), as it turns out to be a measure, see below).

These three facets of the generalized fractional integral were given by analogy. Let us see now that they are all well defined and in fact represent the same objects. It is well known that the operators L_ν and Lν′ generate Feller processes on R (called Lévy subordinators) and the corresponding strongly continuous semigroups in spaces C_∞(R) and C_uc(R). The latter space is much less used as the former, but is handy for us as it includes the spaces of functions that are constants on the halflines. Recall that the potential measure is defined as the integral kernel of the potential operator. The potential measure for any Lévy subordinator is known to exist and to equal the vague limit

of the measures ∫0KG_(ν)(t, .) dt, K → ∞, such that U^(ν)(M) is finite for any compact M whenever ν does not vanish (see e.g. [72] or [44]). More precisely, for any λ > 0,

Here G_(ν)(t, dy) is the Green function of the Cauchy problem for the operator Lν′. Thus the definition of the Ia(ν) from the point of view of the semigroup theory is correct.

Turning to the probabilistic definition we note that the process obtained from a subordinator by killing on the attempt to cross the boundary is a well defined Feller sub-Markov process, so that formula (2.52) arising from a Dynkin’s martingale is well defined and represents a unique solution to the corresponding boundary problem. From this uniqueness it follows that the definitions of the generalized integral from the semigroup and probabilistic points of view coincide.

Finally, from the definitions of the potential measure it follows that the potential measure represents a fundamental solution vanishing on the negative half-line. Therefore to fix the definition arising from the PDEs theory it is only needed to show the uniqueness of such fundamental solution. This is the content of the following assertion from [43] (see also [44]), which also includes the λ-potential measures defined as

These measures are known to be bounded (see e.g. [72] or [44]):

The following result summarizes the properties of generalized fractional integrals.

In particular, applying (2.37) to Lν′ = −d^β/dx^β and comparing with the classical solution of the fractional linear equation via the Mittag-Leffler function yields the integral representation

which is equivalent to Zolotarev’s formula

thus yielding a proof of this formula from the semigroup theory by-passing subtle analytic manipulations of Zolotarev’s initial derivation (see [77] and [78]).

For general ν the classical interpretation of the solution Rλ′g(x) is subtle for g ∈ C([a, ∞)) not vanishing at a. However, Rλ′g(x) may well belong to the domain locally, outside the boundary point a. And in fact, the requirement for the solution to belong to the domain outside a boundary point is common for the classical problems of PDEs. The following assertion illustrates this point concretely.

Apart from generalized solutions arising from duality as considered above, one uses also the notions of generalized solution by approximation. Namely, for a measurable bounded function g(x) on [a, ∞), a continuous curve f(x), t ≥ a, is the generalized solution via approximation to the problem D+(ν)f = −λ f + g on C([a, b]), if there exists a sequence of curves gⁿ(.) ∈ C_{kill(−∞, a]}(R) such that gⁿ → g a.s., as n → ∞, and the corresponding classical (i.e. belonging to the domain) solutions fⁿ(x), given by (2.37) with gⁿ(x) instead of g(x), converge point-wise to f(t), as n → ∞.

The following assertion is a consequence of Proposition 2.2.

2.4 Generalized fractional integrals: weighted mixed derivatives

Let us turn to operators (2.22). It is known (see e.g. [40]) that the operator Lν,γl generates a conservative Feller semigroup T_t in C_∞(R) with invariant core C∞1(R) whenever the following conditions hold:

1. b ∈ C¹(R^d) and b ≥ 0;

2. ∇ ν (x, dy), the gradient of the Lévy kernel with respect to x, exists in the weak sense as a signed measure that depends weakly continuously on x, in the sense that ∫ f(y) ∇ ν(x, dy) is a continuous function for any f ∈ C(R^d) with a support separated from zero;

3. supx∫min(1,y)ν(x,dy)<∞,supx∫min(1,y)|∇ν(x,dy)|<∞,(2.43)

   and for any ϵ > 0 there exists a K > 0 such that

   supx∫K∞ν(x,dy)<ϵ,supx∫K∞|∇ν(x,dy)|<ϵ,supx∫01/Kyν(x,dy)<ϵ.(2.44)

The next result shows the existence of the potential measures describing the integral kernel of the resolvent and potential operators of this Feller semigroup:

with λ ≥ 0, where P_(ν,γ) denote the transition probabilities of the semigroup T_t and Π(ν,γ)λ(x, dy) the λ-potential measure.

Consequently, under the assumptions of Proposition 2.5, the potential operator Π_(ν,γ) is an integral operator, which is well defined for functions with a support bounded below. It is a bounded operator when reduced to the spaces of functions with a fixed compact support. The integral kernel of the potential operator, Π_(ν,γ)(x, dy), is such that, for any x, the measure Π_(ν,γ)(x, ⋅ .) is supported on the set (−∞, x]. Consequently, from the point of view of the semigroup theory, we can define the generalized mixed fractional (weighted) integralIa(ν,γ) as the potential operator Π_(ν,γ) reduced to the space C_{kill(−∞,a]}(R):

On the space C_{kill(−∞,a]}(R) the composition Da+(ν,γ)∘Ia(ν,γ) acts as identity, as it should be.

Since the process generated by L(ν,γ)l and stopped at the attempt to cross the given boundary-point a is a well defined Feller process with a regular boundary point, we can apply the theory of Dynkin’s martingale that implies that if a function g solves the boundary-value problem D^βg = f with g(x) = 0 for x ≤ a, then it can be expressed as the path integral yielding the probabilistic definition of the generalized fractional integral:

where Xtν(x) is the decreasing process generated by operator (2.23) started at x and τ_x the time for this process to reach a.

As the fundamental solutions are less in use for the operators that are not shift invariant, we shall not give detail on this interpretation here.

The results of the previous section extend now automatically to the case of weighted integrals. For instance, the following holds.

Similarly, Proposition 2.4 on the generalized solutions has the straightforward extension.

The mixed fractional integrals can be also defined when arising from more general fractional derivatives (2.28). Namely, they represent the potential operators of the semigroups and processes generated by (2.28) and killed at the boundary. Their probabilistic or path integral representation can be written as in (2.52):

where X_t is the decreasing sub-Markov process generated by the operator

Alternatively, using the Feynman-Kac theory, it can be rewritten as

where Y_t is the decreasing Markov process on [a, ∞) generated by the operator

2.5 Further extensions of mixed fractional operators

Extensions of the fractional operators and the fractional differential equations introduced above corresponds to monotone stochastic processes. They do not exhaust all interesting problems. Apart from the extensions with β ∈ (1, 2) (which we shall not touch here, see [41]) important additional situations include two-sided problems (very natural in the fractional calculus of variations) and multidimensional problems. Unlike the problems considered above, working with potential operators for these cases is usually much more complicated, because we cannot in general use the potential operator of the free problem (without a boundary) and just reduce it to problems with a one-sided support. On the other hand, the representation via Dynkin’s martingale works usually equally well in all cases. However, it gives only uniqueness of the solution and the integral representation for it, that is, a generalized solution in probabilistic sense. The difficulty lies in the identification of the analytic characteristics of thus obtained generalized solutions and of the conditions when they are classical.

The most general extension of mixed fractional derivatives within the probability theory arises from an arbitrary Markov process by interrupting it on the attempt to cross a boundary. We refer to [41] for detail and only mention here briefly three examples.

1. Two-sided problem mixed with a diffusion (see detail in [21]). The typical case is the problem

   (Da+∗β1+Db−∗β2+A)f=g,f(a)=fa,f(b)=fb,

   with a given function g and a diffusion operator A. Extending the derivatives Da+∗β1  and  Db−∗β2 to fully mixed derivatives − Lν′ and L_ν leads to the two-sided problem

   (L+A)f=g,f(a)=fa,f(b)=fb,,

   with

   Lf(x)=∫a−xb−x(f(x+y)−f(x))ν(x,dy)+ga(x)dfdx+(f(b)−f(x))∫b−x∞ν(x,dy)+(f(a)−f(x))∫−∞a−xν(x,dy).(2.56)

   The generalized solution to this problem (the two-sided analog of the fractional integral) can be given by Dynkin’s martingale (2.29) with τ the exit time from (a, b) of the process X_t(x) generated by L + A, or simply the process generated by

   ∫a−xb−x(f(x+y)−f(x))ν(x,dy)+ga(x)dfdx+Af(x),

   as these two processes coincide before exiting (a, b). Spectral problems for two-sided derivatives will be analyzed in Section 5.1.

2. One-sided multidimensional fractional equations. These are the equations in the orthant

   O={x=(x1,⋯,xk):xj≥aj}

   of the type

   (Da1+∗β1+⋯+Dak+∗βk)f(x1,⋯,xk)=g(x1,⋯,xk),

   where D_j acts on the variable x_j. When f is given on the boundary of the orthant O, the generalized solution is well defined by Dynkin’s martingale (2.29) with τ the exit time from the interior of O.

3. Fully multidimensional case. The analog of RL derivative arising from a process in R^d and a domain D ⊂ R^d is the generator of the process killed on leaving D. For CD version this is more subtle, as we have to specify a point where a jump crosses the boundary. The most natural model assumes that a trajectory of a jump follows the shortest path. Namely, assume that A is a generator of a Feller process X_t(x) in R^d with the generator of type

   Af(x)=(γ(x),∇)f(x)+∫Rd(f(x+y)−f(x))ν(x,dy)(2.57)

   with a kernel ν(x, .) on R^d ∖ {0} such that

   supx∫Rdmin(1,|y|)ν(x,dy)<∞,(2.58)

   that is, in the terminology of [39], [40], a generator or order at most one.

Let D be an open convex subset of R^d with boundary ∂D and closure D̄. For x ∈ R^d and a unit vector e let L_x,e = {x + λe, λ ≥ 0} be the ray drawn from x in the direction e. For x ∈ D, let