On fractional heat equation

1 Introduction

Let {X_t, t ≥ 0; P_x, x ∈ E} be a strong Markov process in the phase space ℝ^d. Denote by T_t its transition semigroup (in an appropriate Banach space) and by L the generator of this semigroup. Let S_t, t ≥ 0 be a subordinator (i.e., a non-decreasing real-valued Lévy process) with S₀ = 0 and the Laplace exponent Φ:

We assume that S_t is independent of X_t.

Denote by E_t, t > 0 the inverse subordinator, and introduce the time-changed process Yt=XEt. Our goal is to analyze the properties of Y_t depending on the given Markov process X_t and the particular choice of subordinator S_t, see Theorem 3.1 and Theorem 4.1 below. There is a lot of interest on this kind of problem in diverse disciplines. In addition to purely stochastic motivations, the transform of the Markov process X_t in the non-Markov one Y_t implies the presence of effects in the corresponding dynamics. This feature is important in a number of physical models. In particular, progress in the understanding of this process may lead to the realization of useful models of biological time in the evolution of species and ecological systems. Currently, there exist rather complete studies of such problems in the case of so-called θ-stable subordinators (0 < θ < 1) [6, 15] and in special examples of initial processes X_t (see, e.g., [17], [12], [13]).

As a basic characteristic of the new process Y_t, we may study the time evolution

for a given initial data f.

As it was pointed out in several works, see e.g. [20], [7] and references therein, u(t, x) is the unique strong solution (in some appropriate sense) to the following Cauchy problem

Here we have a generalized fractional derivative (GFD for short)

with a kernel k uniquely defined by Φ.

Let u₀(t, x) be the solution to a similar Cauchy problem but with ordinary time derivative

In stochastic terminology, it is the solution to the forward Kolmogorov equation corresponding to the process X_t. Under quite general assumptions there is a convenient and essentially obvious relation between these evolutions that is known as the subordination principle:

where G_t(τ) is the density of E_t.

A similar relation holds for fundamental solutions (or heat kernels in another terminology) v(x, t) and v^E(x, t) of equations (1.2) and (1.1), respectively. For certain classes of a priori bounds for fundamental solutions v(x, t), the properties of the subordinated kernels were studied in [8]. The main technical tool used in this work involves a scaling property assumed for Φ [8] that is a global condition on the Lévy characteristic Φ(λ). It is nevertheless difficult to give an interpretation of this scaling assumption in terms of the subordinator.

The aim of the present work is to extend the class of random times for which one may obtain information about the time asymptotic of v^E(x, t). We consider the following three classes of admissible kernels k∈Lloc1(ℝ+), characterized in terms of the Laplace transforms K(λ) as λ → 0 (i.e., as local conditions):

We would like to emphasize that these classes of kernels leads to differential-convolution operators, in particular, the Caputo-Djrbashian fractional derivative (C1) and distributed order derivatives (C2),(C3). For each kernel of this type, we establish the asymptotic properties of the subordinated heat kernels from different classes of a priory bounds. It is important to stress that in working with much more general random times (i.e., without the scaling property), a price must be paid for such an extension, namely the replacement of v^E(x, t) by its Cesaro mean. This is the key technical observation that underlies the analysis of several different model situations.

2 Preliminaries

Let S = {S(t), t ≥ 0} be a subordinator, that is a process with stationary and independent non-negative increments starting from 0. They form a special class of Lévy processes taking values in [0, ∞) and their sample paths are non-decreasing. In addition we assume that S has no drift (see [3] for more details). The infinite divisibility of the law of S implies that its Laplace transform can be expressed in the form

where Φ:[0, ∞) → [0, ∞), called the Laplace exponent (or cumulant), is a Bernstein function. The associated Lévy measure σ has support in [0, ∞) and fulfills

such that the Laplace exponent Φ can be expressed as

which is known as the Lévy-Khintchine formula for the subordinator S. In addition we assume that the Lévy measure σ satisfy

For the given Lévy measure σ, we define the function k by

and denote its Laplace transform by K; i.e., for any λ ≥ 0 one has

We note that by the Fubini theorem, the function K is given in terms of the Laplace exponent. Specifically,

i.e.,

Given the inverse process E of the subordinator S, namely

the marginal density of E(t) will be denoted by G_t(τ), t, τ ≥ 0, more explicitly

An important characteristic of the density G_t(τ) is given by its Laplace transform. More precisely, does the τ-Laplace (or t-Laplace) transform of G_t(τ) are known for an arbitrary subordinator? Thus, we would like to compute the following integrals

The answer for the t-Laplace transform is affirmative and the result is given in (2.14) below. On the other hand, for the τ-Laplace transform a partial answer has been given for the class of θ-stable processes; namely

3 Subordinated heat kernel

In this section, we investigate the long-time behavior of the fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion by the inverse process E_t, t ≥ 0. We consider three classes of time change, namely those corresponding to the θ-stable subordinator, 0 < θ < 1, the distributed order derivative, and the class of Stieltjes functions. Henceforth L will always denotes a slowly varying function at infinity (SVF), that is,

see for instance [4] and [19]) for more details, while C, C′ are constants whose values are unimportant, and which may change from line to line.

Let v(x, t) be the fundamental solution of the heat equation

where Δ denotes the Laplacian in ℝ^d. It is well known that the solution v(x, t) of (3.1), called heat kernel (also known as Green function), is given by

and the associated stochastic process is the classical Brownian motion in ℝ^d. Notice that the heat kernel v(x, t) has the following long-time behavior

We are interested in studying the long-time behavior of the subordination of the solution v(x, t) by the density G_t(τ), that is,

Then v^E(x, t) is the fundamental solution of the general fractional time differential equation, that is,

Here Dt(k) are differential-convolution operators defined, for any nonnegative kernel k∈Lloc1(ℝ+), by

(See [11] for more details and examples.)

In order to study the time evolution of v^E(x, t), one possibility is to define its Cesaro mean

which may be written as

The long-time behavior of the Cesaro mean Mt(vE(x,t)) was investigated in [10, Sec. 3] for the three classes of admissible kernels and d ≥ 3. The method was based on the ratio Tauberian theorem from [12]. More precisely, the following theorem was shown.

In the next section we use an alternative method to find the long-time behavior of the Cesaro mean Mt(vE(x,t)).

4 Alternative method for subordinated heat kernel

The Laplace transform method is based on the result of Lemma 2.1 wherein the t-Laplace transform of the subordination v^E(x, t) is explicitly given by

The integral in (4.1) is computed according to the formula,

(see for instance [9, page 146, eqs. (27), (29)]), where a=|x|22, b=λK(λ), and K_ν(z) is the modified Bessel function of the second kind [2, Sec. 9.6]. The asymptotic of the Bessel function K_ν(z) as z → 0 is well known (e.g., see [2, Eqs. (9.6.8) and (9.6.9)]) and is given by

With these explicit formulas, we study each class (C1), (C2), and (C3) separately which constitutes the main contribution of this paper.