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Exact asymptotic formulas for the heat kernels of space and time-fractional equations

  • Chang-Song Deng EMAIL logo and René L. Schilling
Published/Copyright: October 23, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 4

Abstract

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space

∂β∂tβu(t,x)=−(−Δx)γu(t,x),β,γ∈(0,1).
MSC 2010: Primary 60J35; Secondary 60G51; 60K99; 35R11; 35K08
Key Words and Phrases: heat kernel; asymptotic formula; space-fractional equation; time-fractional equation; subordinator; inverse subordinator

Appendix

We will need a moment formula for stable subordinators which can be found in Sato [31, Eq. (25.5), p. 162] (without proof but references to the literature). The following short and straightforward derivation seems to be new.

Lemma 4.1

The moments of orderκ ∈ (–∞, α) of aα-stable subordinator (St)t≥0exist and are given by

EStκ=Γ1−κβΓ(1−κ)tκ/β,t>0.

Proof

Since St has the same probability distribution as t1/αS1, it is enough to consider t = 1. Recall that the Laplace transform of S1 is 𝔼 e–tS1 = e–tα, t > 0. Substituting λ = S1 in the well-known formula [32, p. vii]

λ−r=1Γ(r)∫0∞e−λxxr−1dx,λ>0,r>0,

and taking expectations yields, because of Tonelli’s theorem,

ES1−r=1Γ(r)∫0∞Ee−xS1xr−1dx=1Γ(r)∫0∞e−xβxrdxx.

Now we change variables according to y = xα, and get

ES1−r=1Γ(r)1β∫0∞e−yyrβdyy=1rΓ(r)⋅rβΓrβ=Γ1+rβΓ(1+r).

Setting κ = –r proves the assertion for κ ∈ (–∞, 0). Note that this formula extends (analytically) to –r = κ < α. Alternatively, use the very same calculation and the formula [32, p. vii]

λr=rΓ(1−r)∫0∞1−e−λxx−r−1dx,λ>0,r∈(0,1),(4.1)

to get the assertion for κ ∈ (0, α).□

The following theorem is known in the literature in dimension n = 1, see [31, p. 163]. The multivariate setting and the short proof via subordination are new.

Lemma 4.2

Let (Xt)t≥0be a rotationally symmetricα-stable Lévy process on ℝnwith 0 < α < 2. For anyκ ∈ (–n, α),

E|Xt|κ=2κΓn+κ2Γ1−καΓn2Γ1−κ2tκ/α,t>0.

Ifκ ≤ –n orκ ≥ αthe moments are infinite.

Proof

Let (Bt)t≥0 be a Brownian motion on ℝn (starting from zero) with transition probability density given by (1.2), and (St)t≥0 be an independent α/2-stable subordinator, that is an increasing Lévy process. From Bochner’s subordination is well known that the time-changed process BSt, t ≥ 0, is a rotationally symmetric α-stable Lévy process on ℝn.

For any κ > –n and t > 0, we have

E|Bt|κ=1(4πt)n/2∫Rn|x|κexp−|x|24tdx=21−nt−n/2Γn2∫0∞rn+κ−1exp−r24tdr=2κΓn+κ2Γn2tκ/2.

Let 𝔼B and 𝔼S denote the expectations w.r.t. (Bt)t≥0 and (St)t≥0, respectively. Using Lemma 4.1, we obtain that for any κ ∈ (–n, α) and t > 0,

E|BSt|κ=ESEB|BSt|κ=2κΓn+κ2Γn2EStκ/2=2κΓn+κ2Γ1−καΓn2Γ1−κ2tκ/α.

If κ ≤ –n or κ ≥ α, we have 𝔼|Xt|κ = ∞, see [8, Theorem 3.1.e) and Remark 3.2.d)].□

Remark 4.1

We want to sketch another, slightly more general proof of Lemma 4.2 which avoids the subordination argument. Combining the well-known formulas

|x|κ=κ2κ−1Γn+κ2πn/2Γ1−κ2∫Rn∖{0}(1−cos⁡(x⋅ξ))|ξ|−κ−ndξ,κ∈(0,2),|x|κ=2κΓn+κ2πn/2Γ−κ2∫Rn∖{0}|ξ|−n−κe−ix⋅ξdξ,κ∈(−n,0)

(the second formula is to be understood in the sense of L. Schwartz distributions) with an Abel-type convergence factor argument and Fubini’s theorem, also yields the moment formula of Lemma 4.2.

Lemma 4.3

Letψ : [1, ∞) → ℝ be a bounded function such that lims→∞ψ(s) = 0 andω : [0, ∞) → (0, ∞) a non-increasing function satisfying∫1∞s–1ω(s) ds < ∞. For anyc > 0 andδ > 0 one has

limA→∞1log⁡A∫cA−δ∞s−1ω(s)ds=δω(0+)

and

limA→∞1log⁡A∫cA−δ∞s−1ω(s)ψc−1/δAs1/δds=0.

Proof

The first claim follows easily from l’Hospital’s rule. For n ∈ ℕ, we can use the first part of the lemma and get

1log⁡A∫cA−δ∞s−1ω(s)ψc−1/δAs1/δds≤1log⁡A∫cA−δ∞s−1ω(s)ψc−1/δAs1/δds≤∥ψ∥∞1log⁡A∫cA−δ∞−∫ncA−δ∞s−1ω(s)ds+∥ψ1[n1/δ,∞)∥∞1log⁡A∫ncA−δ∞s−1ω(s)ds→A→∞∥ψ∥∞(δω(0+)−δω(0+))+∥ψ1[n1/δ,∞)∥∞δω(0+)=∥ψ1[n1/δ,∞)∥∞δω(0+)→n→∞0,

and this completes the proof.□

The following asymptotic formula for integrals can be proved by the Laplace method, see e.g. de Bruijn [10, Section 4.2, pp. 63–65] for r0 = 0.

Lemma 4.4

Assume that –∞ ≤ v < w ≤ ∞, h ∈ C2(v, w), and∫vwe−h(r)dr<∞.Letr0 ∈ (v, w). Ifh(r0) ≥ 0, h″(r0) > 0, andhis strictly decreasing on (v, r0] and strictly increasing on [r0, w), then

∫vwe−Ch(r)dr∼e−Ch(r0)2πCh″(r0)as  C→∞.

Lemma 4.5

Letϕ : (0, ∞) → (–1, ∞) be a continuous function such thatϕ(0+) = 0 and lim sups→∞s–θ(1 + ϕ(s)) < ∞ for someθ ∈ ℝ. For all constants a ∈ ℝ and b, c, d > 0 the following asymptotics holds

∫0∞sae−Bsb−cs−d(1+ϕ(s))ds∼I(B)as  B→∞

where the valueI(B) is given by the following expression

2πb+d(bB)−2(a+1)+d2(b+d)(cd)2(a+1)−b2(b+d)exp−(b+d)b−1cbb+dd−1Bdb+d.

Proof

First, we prove that

∫0∞sae−Bsb−cs−dds∼I(B)as  B→∞.(4.2)

If a ≠ –1, changing variables according to r = (c–1B)(a+1)/(b+d)sa+1 gives

∫0∞sae−Bsb−cs−dds=cB−1a+1b+d|a+1|∫0∞exp−cbb+dBdb+drba+1+r−da+1dr.

Lemma 4.4 with h(r) = rb/(a+1) + r–d/(a+1), r0 = (b–1d)(a+1)/(b+d), v = 0, w = ∞ and C = cb/(b+d)Bd/(b+d) yields (4.2).

If a = –1, we change variables according to r = log s + (b+d)–1 log(c–1B), and use Lemma 4.4 with v = –∞, w = ∞, h(r) = ebr + e–dr, r0 = (b + d)–1 log(b–1d) and C = cb/(b+d)Bd/(b+d) to obtain (4.2).

We still have to check that the following limit is zero:

limB→∞B2(a+1)+d2(b+d)exp(b+d)b−1cbb+dd−1Bdb+d∫0∞sae−Bsb−cs−dϕ(s)ds.

To this end, we fix n ∈ ℕ and observe that

I1(n,B):=∫01/nsae−Bsb−cs−d|ϕ(s)|ds≤ϕ1(0,1/n)∞∫01/nsae−Bsb−cs−dds≤ϕ1(0,1/n)∞∫0∞sae−Bsb−cs−dds.

Moreover, set

I2(n,B):=∫1/n∞sae−Bsb−cs−d|ϕ(s)|ds.

By our assumption, there exists a constant C(n) > 0 depending on n such that 1 + ϕ(s) ≤ C(n)sθ for all s ≥ 1/n. Thus,

|ϕ(s)|≤1+(1+ϕ(s))≤(ns)θ∨0+C(n)sθ≤C(n,θ)sθ∨0,s≥1/n,

where C(n, θ) := nθ∨0 + C(n)n(–θ)∨0. Using the dominated convergence theorem we deduce

B2(a+1)+d2(b+d)exp(b+d)b−1cbb+dd−1Bdb+dI2(n,B)≤C(n,θ)∫1/n∞B2(a+1)+d2(b+d)sa+(θ∨0)×exp(b+d)b−1cbb+dd−1Bdb+d−Bsb−cs−dds→B→∞C(n,θ)⋅0=0.

Combining these calculations gives

B2(a+1)+d2(b+d)exp(b+d)b−1cbb+dd−1Bdb+d⋅∫0∞sae−Bsb−cs−dϕ(s)ds≤B2(a+1)+d2(b+d)exp(b+d)b−1cbb+dd−1Bdb+d⋅(I1(n,B)+I2(n,B))≤B2(a+1)+d2(b+d)exp(b+d)b−1cbb+dd−1Bdb+d×ϕ1(0,1/n)∞∫0∞sae−Bsb−cs−dds+I2(n,B)→B→∞ϕ1(0,1/n)∞2πb+db−2(a+1)+d2(b+d)n(cd)2(a+1)−b2(b+d)+0→n→∞0.

This completes the proof.□

Acknowledgements

Chang-Song Deng gratefully acknowledges financial support through the National Natural Science Foundation of China (11401442, 11831015).

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Received: 2019-02-10
Published Online: 2019-10-23
Published in Print: 2019-08-27

© 2019 Diogenes Co., Sofia

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  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–4–2019)
  4. Research Paper
  5. Fractional equations via convergence of forms
  6. About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
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  9. Research Paper
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  11. Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
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https://doi.org/10.1515/fca-2019-0052
Keywords for this article
heat kernel; asymptotic formula; space-fractional equation; time-fractional equation; subordinator; inverse subordinator

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–4–2019)
  4. Research Paper
  5. Fractional equations via convergence of forms
  6. About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
  7. Survey Paper
  8. The vertical slice transform on the unit sphere
  9. Research Paper
  10. Well-posedness of time-fractional advection-diffusion-reaction equations
  11. Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
  12. Exact asymptotic formulas for the heat kernels of space and time-fractional equations
  13. Estimates of damped fractional wave equations
  14. A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis
  15. On the fractional diffusion-advection-reaction equation in ℝ
  16. Controllability of nonlinear stochastic fractional higher order dynamical systems
  17. Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2
  18. Analysis of fractional order error models in adaptive systems: Mixed order cases
  19. Fractional calculus of variations: a novel way to look at it
  20. Pricing of perpetual American put option with sub-mixed fractional Brownian motion
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