Random Spherical Hyperbolic Diffusion

Abstract

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background (CMB) radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to CMB data are presented to illustrate the theoretical results.

Appendices

Appendix A: Diffusion Length of a Local Disturbance

Consider a density disturbance u of total mass Q originating at the origin. The well-known point source solution to linear diffusion in three space dimensions is given by

$$\begin{aligned} u=\frac{1}{8[\pi Dt]^{3/2}}e^{-r^2/4Dt}. \end{aligned}$$

The density level set at some low significance value u is at

$$\begin{aligned} r=2[Dt]^{1/2}\ln ^{1/2}\left( \frac{Q}{u[8\pi Dt]^{3/2}}\right) , \end{aligned}$$

so

$$\begin{aligned} \frac{dr}{dt}=(D/t)^{1/2}\frac{\ln \left( \frac{Q}{u[8\pi Dt]^{3/2}}\right) -3/2}{\ln ^{1/2}\left( \frac{Q}{u[8\pi Dt]^{3/2}}\right) }. \end{aligned}$$

The level set reaches its maximum extent when $\frac{dr}{dt}=0,$ implying $t=\frac{(Q/u)^{2/3}}{8e\pi D},$ so the diffusion length is

$$\begin{aligned} r_D=\frac{1}{2}\left( \frac{3}{\pi e}\right) ^{1/2} \left( \frac{Q}{u}\right) ^{1/3}\approx 0.296 \left( \frac{Q}{u}\right) ^{1/3}. \end{aligned}$$

For example, for a mass disturbance the size of a solar mass, and a neutron diffusion length of 0.3 light-year at the temperature of neutrino dissociation from weak nuclear interactions, estimated from [6] and [25], the marginal disturbance density u is around 1 solar mass per cubic light year. This is meant to have occurred at a time when the cosmological expansion factor a(t) was less than $10^{-3},$ so after expansion to the current level, the equivalent marginal density would be less than one nucleon mass per cubic metre, around the current mean density of the universe.

Appendix B: Proofs

Proof of Theorem 1

By substituting (19) into Eq. (15) and using (18), we obtain

$$\begin{aligned} \sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left[ \frac{1}{c^{2}}\frac{{ d} ^{2}b_{lm}(t)}{{ d}\, t^{2}}+\frac{1}{D}\frac{{d}\, b_{lm}(t)}{{ d}\, t}+l(l+1)k^{2}b_{lm}(t)\right] Y_{lm}({\mathbf {x}})=0. \end{aligned}$$

(31)

To find particular solutions of (31), we need to solve the ordinary differential equation

$$\begin{aligned} \frac{1}{c^{2}}\frac{{ d}^{2}b_{lm}(t)}{{ d}\, t^{2}}+\frac{1}{D} \frac{{d}\, b_{lm}(t)}{{d}\, t}+l(l+1)k^{2}b_{lm}(t)=0. \end{aligned}$$

(32)

The initial conditions for this equation can be determined from (20) and (16) and they are

$$\begin{aligned} b_{lm}(t)|_{t=0}= {\tilde{Y}}_{lm}^{*}({\mathbf {0}}),\quad \left. \frac{ { d}\, b_{lm}(t)}{{ d}\, t}\right| _{t=0}=0. \end{aligned}$$

(33)

The characteristic equation of (32) is $\frac{1}{c^{2}}z^{2}+\frac{1}{D}z+l(l+1)k^{2}=0,$ with the roots $z_{1,2}=-{c^{2}}/(2D)\pm K_l.$ Therefore, the general solution of Eq. (32) is given by the formula:

$$\begin{aligned} b_{lm}(t)=M_{1}e^{z_{1}t}+M_{2}e^{z_{2}t}, \end{aligned}$$

where $M_{1},M_{2}$ are some constants. From the initial conditions in (33) we obtain

$$\begin{aligned} M_{1} = \left( \frac{1}{2}+\frac{c^{2}}{4DK_l}\right) {\tilde{Y}}_{lm}^{*}({\mathbf {0}}) , \quad M_{2} = \left( \frac{1}{2}-\frac{c^{2}}{4DK_l}\right) {\tilde{Y}}_{lm}^{*}({\mathbf {0}}). \end{aligned}$$

Thus, the solution of the Cauchy problem (32)–(33) is given by

$$\begin{aligned} b_{lm}(t)= & {} \left( \frac{1}{2}+\frac{c^{2}}{4DK_l}\right) {\tilde{Y}}_{lm}^{*}({\mathbf {0}}) \exp \left[ -t\left( \frac{c^{2}}{2D}-K_l\right) \right] \\&\quad + \left( \frac{1}{2}-\frac{c^{2}}{4DK_l}\right) {\tilde{Y}}_{lm}^{*}({\mathbf {0}}) \exp \left[ -t\left( \frac{c^{2}}{2D}+K_l\right) \right] . \end{aligned}$$

Returning now to (19), we obtain the solution of the Cauchy problem (15)–(16) in the form

$$\begin{aligned} {\tilde{p}}({\mathbf {x}},t)= & {} \sum _{l=0}^{\infty }Q_{l}({\mathbf {x}}) \left( \left( \frac{1}{2}+\frac{c^{2}}{4DK_l}\right) \exp \left[ -t\left( \frac{c^{2}}{2D} - K_l\right) \right] \right. \nonumber \\&\quad + \left. \left( \frac{1}{2}-\frac{c^{2}}{4DK_l}\right) \exp \left[ -t\left( \frac{c^{2}}{2D} + K_l\right) \right] \right) . \end{aligned}$$

(34)

Note that the multiplier of $Q_{l}({\mathbf {x}})$ on the right-hand side of (34) equals

$$\begin{aligned} \exp \left( -\frac{c^{2}t}{2D}\right) \biggl \{ \cosh \left( tK_l\right) + \frac{c^{2}}{2DK_l} \;\sinh \left( tK_l\right) \biggr \}. \end{aligned}$$

By substituting this expression into (34), we get

$$\begin{aligned} {\tilde{p}}({\mathbf {x}},t) = \exp \left( -\frac{c^{2}t}{2D} \right) \sum _{l=0}^{\infty } Q_{l}({\mathbf {x}})\biggl \{ \cosh \left( tK_l\right) + \frac{c^{2}}{2DK_l} \;\sinh \left( t K_l\right) \biggr \} . \end{aligned}$$

Finally, using $K_l'$ and rewriting the Green function we obtain the statement of the theorem. $\square$

Proof of Theorem 2

The solution of the initial value problem (22)–(24) can be written as a spherical convolution of the Green function $p(\theta ,\varphi ,t)$ from Sect. 4 and the random field $T(\theta ,\varphi ),$ if the corresponding Laplace series converges in the Hilbert space $L_{2}(\varOmega \times {\mathbb {S}}^2,\sin \theta d\theta d\varphi ).$

Let the two functions $f_1(\cdot )$ and $f_2(\cdot )$ on the sphere ${\mathbb {S}}^2$ belong to the space $L_{2}({\mathbb {S}}^2,\sin \theta d\theta d\varphi )$ and have the Fourier–Laplace coefficients

$$\begin{aligned} a_{lm}^{(i)}=\int _{{\mathbb {S}}^2}f_{i}(\theta ,\varphi )Y_{lm}^{*}(\theta ,\varphi )\sin \theta d\theta d\varphi , \quad i=1,2. \end{aligned}$$

Recall (see, i.e., [15]) that their non-commutative spherical convolution is defined as the Laplace series

$$\begin{aligned}{}[f_{1}*\ f_{2}](\theta ,\varphi )=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}a_{lm}^{(*)}\ Y_{lm}(\theta ,\varphi ) \end{aligned}$$

(35)

with the Fourier–Laplace coefficients given by

$$\begin{aligned} a_{lm}^{(*)}= \sqrt{\frac{4\pi }{2l+1}}a_{lm}^{(1)}a_{l0}^{(2)}, \end{aligned}$$

provided that the series (35) converges in the corresponding Hilbert space.

Thus, the random solution $u(\theta ,\varphi ,t)$ of Eq. (22) with the initial values determined by (23) and (24) can be written as a spherical random field with the following Laplace series representation

$$\begin{aligned} u(\theta ,\varphi ,t)=[T*\ p_{t}](\theta ,\varphi )=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}a_{lm}^{(t)}Y_{lm}(\theta ,\varphi ), \end{aligned}$$

(36)

provided that this series is convergent in the Hilbert space $L_{2}(\varOmega \times {\mathbb {S}}^2,\sin \theta d\theta d\varphi ),$ where $p_{t}=p(\theta ,\varphi ,t)$ is given by Theorem 1 and T is given by (23). The complex Gaussian random variables $a_{lm}^{(t)}$ are given by

$$\begin{aligned} a_{lm}^{(t)}= \sqrt{\frac{4\pi }{2l+1}}a_{lm}a_{l0}^{(p_{t})}, \end{aligned}$$

where $a_{l0}^{(p_{t})}=Y_{l0}^{*}({\mathbf {0}})d_{l}(\theta ,\varphi ,t)$ and

$$\begin{aligned} d_{l}(\theta ,\varphi ,t)= & {} \exp \left( -\frac{c^{2}t}{2D}\right) \biggl \{\biggl [ \cosh \left( tK_l \right) + \frac{c^{2}}{2DK_l}\;\sinh \left( t K_l\right) \biggr ] \\&\quad \times {\mathbf {1}} _{\left\{ l\le \frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}\right\} } + \biggl [ \cos \left( tK_l'\right) +\frac{c^{2}}{2DK_l'} \sin \left( t K_l'\right) \biggr ] {\mathbf {1}} _{\left\{ l>\frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}\right\} } \biggr \}. \end{aligned}$$

It gives the first statement of the theorem.

By the addition formula for spherical harmonics (see, i.e., [30, p.66])

$$\begin{aligned} \sum _{m=-l}^{l}Y_{lm}(\theta ,\varphi )Y_{lm}^{*}(\theta ^{\prime },\varphi ^{\prime }) = \frac{2l+1}{4\pi } P_l(\cos \varTheta ), \end{aligned}$$

(37)

where $P_l(\cdot )$ is the lth Legendre polynomial [see (8)], and $\cos \varTheta$ is the angular distance between the points $(\theta ,\varphi )$ and $(\theta ^{\prime },\varphi ^{\prime })$ on ${\mathbb {S}}^2.$

Using (10) we obtain that the random field $u(\theta ,\varphi ,t)$ is isotropic if and only if the covariance structure of the solution (25) can be written in the form

$$\begin{aligned}&\mathbf {Cov}(u(\theta ,\varphi ,t),u(\theta ^{\prime },\varphi ^{\prime },t^{\prime })) = \exp \left( -\frac{c^{2}}{2D}(t+t^{\prime })\right) \\&\quad \times \sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}(\theta ,\varphi )Y_{lm}^{*}(\theta ^{\prime },\varphi ^{\prime }) {\mathbf {E}}\xi _{lm}(t)\xi ^{*} _{lm}(t^{\prime }), \end{aligned}$$

which gives the result in (29) provided the series (29) converges for every fixed t and $t^{\prime },$ that is

$$\begin{aligned} \sum _{l=0}^{\infty }(2l+1)C_{l} P_l(\cos \varTheta ) [A_{l}(t)A_{l}(t^{\prime })+B_{l}(t)B_{l}(t^{\prime })]<\infty . \end{aligned}$$

(38)

Noting that $\left| P_l(\cos \varTheta )\right| \le 1,$ only a finite number of terms $A_{l}$ is non-zero, and there is a constant C such that $\sup _{t\ge 0} |B(t)|<C,$ we obtain that condition (38) follows from (13). This condition on the angular spectrum $C_{l}, l\ge 0,$ guarantees the convergence of the series (36) in the Hilbert space $L_{2}(\varOmega \times {{\mathbb {S}}}^{2},\sin \theta d\theta d\varphi ).$$\square$

Proof of Theorem 3

The approximation $u_L(\theta ,\varphi ,t)$ is a centered Gaussian random field, i.e. ${\mathbf {E}}u_L(\theta ,\varphi ,t)=0$ for all $L\in {\mathbb {N}}, \theta \in [0,\pi ), \varphi \in [0,2\pi ),$ and $t>0.$ Therefore,

$$\begin{aligned}&\left\| u(\theta ,\varphi ,t)-u_L(\theta ,\varphi ,t)\right\| _{L_2(\varOmega \times {{\mathbb {S}}}^{2})} = \exp \left( -\frac{c^{2}t}{2D}\right) \nonumber \\&\qquad \times \left( \sum _{l=L}^{\infty }\sum _{m=-l}^{l}Y_{lm}(\theta ,\varphi )Y_{lm}^{*}(\theta ,\varphi ) {\mathbf {E}}\xi _{lm}(t)\xi ^{*}_{lm}(t)\right) ^{1/2} \nonumber \\&\quad =\frac{1}{2\sqrt{\pi }}\exp \left( -\frac{c^{2}t}{2D}\right) \left( \sum _{l=L}^{\infty }(2l+1)C_{l}\cdot \left[ A_{l}^2(t)+B_{l}^2(t)\right] \right) ^{1/2}. \end{aligned}$$

(39)

By (27) and (28) we get

$$\begin{aligned} |A_{l}(t)|\le C\exp \left( \frac{c^{2}t}{2D}\right) \quad \text{ and } \quad \sup _{t\ge 0}|B_{l}(t)|\le C. \end{aligned}$$

(40)

Hence, for all $L\in {\mathbb {N}}$ it holds

$$\begin{aligned} \Vert u(\theta ,\varphi ,t)-u_L(\theta ,\varphi ,t)\Vert _{L_2(\varOmega \times {{\mathbb {S}}}^{2})}\le C \left( \sum _{l=L}^{\infty }(2l+1)C_{l}\right) ^{1/2}. \end{aligned}$$

For $l>\frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}$ it follows from (27) that $A_{l}(t)\equiv 0.$ Therefore, by (39) and (40) we obtain

$$\begin{aligned} \Vert u(\theta ,\varphi ,t)-u_L(\theta ,\varphi ,t)\Vert _{L_2(\varOmega \times {{\mathbb {S}}}^{2})}\le C\exp \left( -\frac{c^{2}t}{2D}\right) \left( \sum _{l=L}^{\infty }(2l+1)C_{l}\right) ^{1/2}. \end{aligned}$$

$\square$

Proof of Corollary 2

The statement (i) immediately follows from (30) and the estimate

$$\begin{aligned} \sum _{l=L}^{\infty }(2l+1)C_{l}\le C\sum _{l=L}^{\infty }l^{-(\alpha -1)}=C {L}^{-(\alpha -2)}. \end{aligned}$$

Then, applying Chebyshev’s inequality, we get the upper bound in (ii).

Finally, (iii) follows from statement (ii) and the Borel–Cantelli lemma as

$$\begin{aligned} \sum _{l=L}^{\infty } \frac{1}{L^{\alpha -2}L^{-2\beta }}<\infty . \end{aligned}$$

$\square$

Proof of Theorem 4

Let h belong to a bounded neighbourhood of the origin. It follows from (14), (26), (27), (28) and (37) that

$$\begin{aligned}&\left\| u(\theta ,\varphi ,t+h)-u(\theta ,\varphi ,t)\right\| _{L_2(\varOmega \times {{\mathbb {S}}}^{2})} = \left\| \exp \left( -\frac{c^{2}(t+h)}{2D}\right) \sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}(\theta ,\varphi )\right. \nonumber \\&\qquad \left. \times \,\xi _{lm}(t+h)- \exp \left( -\frac{c^{2}t}{2D}\right) \sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}(\theta ,\varphi )\xi _{lm}(t)\right\| _{L_2(\varOmega \times {{\mathbb {S}}}^{2})}\nonumber \\&\quad =\frac{1}{2\sqrt{\pi }}\exp \left( -\frac{c^{2}t}{2D}\right) \left( \sum _{l=0}^{\infty }(2l+1)C_{l}\right. \left[ \left( \exp \left( -\frac{c^{2}h}{2D}\right) A_{l}(t+h)-A_{l}(t)\right) ^2\right. \nonumber \\&\qquad \left. \left. +\left( \exp \left( -\frac{c^{2}h}{2D}\right) B_{l}(t+h)-B_{l}(t)\right) ^2\right] \right) ^{1/2}. \end{aligned}$$

(41)

We start by showing how to estimate the first summand in (41). By (27), for the case $l=0$ we obtain

$$\begin{aligned}&\left( \exp \left( -\frac{c^{2}h}{2D}\right) A_{0}(t+h)-A_{0}(t)\right) ^2\\&\quad =\left( \exp \left( -\frac{c^{2}h}{2D}\right) \exp \left( \frac{c^{2}(t+h)}{2D}\right) \right. \left. -\exp \left( \frac{c^{2}t}{2D}\right) \right) ^2=0. \end{aligned}$$

For $l>0$ we will use the upper bound

$$\begin{aligned}&\left( \exp \left( -\frac{c^{2}h}{2D}\right) A_{l}(t+h)-A_{l}(t)\right) ^2=\left( \exp \left( -\frac{c^{2}h}{2D}\right) \left( A_{l}(t+h)-A_{l}(t)\right) \right. \\&\quad \left. -\left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) A_{l}(t)\right) ^2\le 2\left( A_{l}(t+h)-A_{l}(t)\right) ^2\\&\quad +2\left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) ^2A^2_{l}(t). \end{aligned}$$

By properties of $\cosh (\cdot )$ and $\sinh (\cdot )$ we get

$$\begin{aligned} \cosh (x)-\cosh (y)= & {} \frac{\exp {(x)}}{2} \left( 1-\exp \left( -(x+y)\right) \right) \left( 1-\exp \left( -(x-y)\right) \right) ,\\ \sinh (x)-\sinh (y)= & {} \frac{\exp {(x)}}{2}\left( 1+\exp \left( -(x+y)\right) \right) \left( 1-\exp \left( -(x-y)\right) \right) . \end{aligned}$$

Then, applying (27) and noting that only a finite number of $A_l$ is non-vanished (namely, only if $l\in \left[ 0, \frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}\right]$) we obtain the following estimates

$$\begin{aligned}&\left( A_{l}(t+h)-A_{l}(t)\right) ^2\le \frac{\exp \left( 2(t+h)K_l\right) }{2}\Big [\left( 1- \exp \left( -(t+h/2)K_l\right) \right) ^2\\&\qquad \times \left( 1- \exp \left( -h K_l/2\right) \right) ^2+\frac{c^4}{4D^2K_l^2}\left( 1+ \exp \left( -(t+h/2)K_l\right) \right) ^2\\&\qquad \times \left( 1- \exp \left( -h K_l/2\right) \right) ^2\Big ]\le C\exp \left( 2hK_l\right) \exp \left( 2tK_l\right) \\&\qquad \times \left( 1- \exp \left( -h K_l/2\right) \right) ^2\le C\exp \left( 2tK_l\right) h^2,\\&\quad \left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) ^2A^2_{l}(t)\le \frac{c^{4}}{4D^2}h^2A^2_{l}(t)=\frac{c^{4}}{8D^2}h^2\exp \left( 2tK_l\right) \\&\qquad \times \Big [\left( 1+ \exp \left( -2tK_l\right) \right) ^2+\frac{c^4}{4D^2K_l^2}\left( 1- \exp \left( -2tK_l\right) \right) ^2\Big ]\le C\exp \left( 2tK_l\right) h^2. \end{aligned}$$

Now we estimate the second summand in (41) as

$$\begin{aligned}&\left( \exp \left( -\frac{c^{2}h}{2D}\right) B_{l}(t+h)-B_{l}(t)\right) ^2\le \left( \exp \left( -\frac{c^{2}h}{2D}\right) \left( B_{l}(t+h)-B_{l}(t)\right) \right. \\&\quad \left. -\left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) B_{l}(t)\right) ^2\le 2\left( B_{l}(t+h)-B_{l}(t)\right) ^2\\&\quad +2\left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) ^2B^2_{l}(t). \end{aligned}$$

Using (40) and applying the inequalities $|\cos (x)-\cos (y)|\le 2\left| \sin \left( \frac{x-y}{2}\right) \right| \le |x-y|$ and $|\sin (x)-\sin (y)|\le |x-y|$ we obtain

$$\begin{aligned}&\left( B_{l}(t+h)-B_{l}(t)\right) ^2\le 2 \left( (K_l')^2+\frac{c^4}{4D^2}\right) h^2,\\&\left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) ^2B^2_{l}(t)\le \left( 1-\exp \left( -\frac{c^{2}h}{2D}\right) \right) ^2\left( 1+\frac{c^2}{2D}\right) ^2\le C h^2. \end{aligned}$$

Note that for all $l\ge 0$ it holds

$$\begin{aligned} K_l\le \frac{c^{2}}{2D}\quad \text{ and }\quad K_l'\le C(2l+1). \end{aligned}$$

Applying the above estimates to (41) we obtain

$$\begin{aligned}&\Vert u(\theta ,\varphi ,t+h)-u(\theta ,\varphi ,t)\Vert _{L_2(\varOmega \times {{\mathbb {S}}}^{2})} \le C\exp \left( -\frac{c^{2}t}{2D}\right) \left( \sum _{l=0}^{\infty }(2l+1)C_{l}\right. \\&\quad \left. \times \Big [\exp \left( 2tK_l\right) +(K_l')^2+C\Big ]\right) ^{1/2}h\le C\left( \sum _{l=0}^{\infty }(2l+1)^3C_{l}\right) ^{1/2}h, \end{aligned}$$

which completes the proof. $\square$

Proof of Corollary 4

Note that $u(\theta ,\varphi ,t)$ is a centered Gaussian random field and for any centered Gaussian random variable X it holds

$$\begin{aligned} {\mathbf {E}}|X|^{p}= \frac{2^{p/2}\varGamma \left( {\frac{p+1}{2}}\right) }{\sqrt{\pi }}\left( {\mathbf {E}}|X|^{2}\right) ^{p/2}. \end{aligned}$$

Applying this result to the statement of Theorem 4 we obtain

$$\begin{aligned} \Vert u(\theta ,\varphi ,t+h)-u(\theta ,\varphi ,t)\Vert _{L_p(\varOmega \times {{\mathbb {S}}}^{2})} = C \Vert u(\theta ,\varphi ,t+h)-u(\theta ,\varphi ,t)\Vert _{L_2(\varOmega \times {{\mathbb {S}}}^{2})}\le Ch. \end{aligned}$$

$\square$

Proof of Corollary 5

By (29) it holds

$$\begin{aligned} \mathbf {Var}\left( u(\theta ,\varphi ,t)-u(\theta ',\varphi ',t)\right)= & {} \mathbf {Var}\left( u(\theta ,\varphi ,t)\right) \\&\quad +\mathbf {Var}\left( u(\theta ',\varphi ',t)\right) -2\,\mathbf {Cov}(u(\theta ,\varphi ,t),u(\theta ',\varphi ',t))\\= & {} C\exp \left( -\frac{c^{2}t}{D}\right) \sum _{l=0}^{\infty }C_{l}\left( 2l+1\right) \\&\times \,\left( A_{l}^2(t)+B^2_{l}(t)\right) (1-P_l(\cos \varTheta )). \end{aligned}$$

Applying the next property of Legendre polynomials (see, for example, [27, p.16]) $|1 - P_l (x)| \le 2|1 - x|^\gamma (l(l + 1))^\gamma , \gamma \in [0,1],$ and the upper bounds (40), we obtain that uniformly in $t\ge 0$

$$\begin{aligned} \mathbf {Var}\left( u(\theta ,\varphi ,t)-u(\theta ',\varphi ',t)\right) \le C \sum _{l=0}^{\infty }C_{l}\left( 2l+1\right) ^{1+2\gamma } (1-\cos \varTheta )^\gamma . \end{aligned}$$

$\square$

Appendix C: Sensitivity to Parameters

To further understand the impact of time and the model parameters on the difference of the mean ${L_2(\varOmega \times {\mathbb S}^{2})}$-errors and their upper bound (30) we produced 3D-plots showing the difference as a function of the truncation degree L and each parameter provided that other parameters are fixed. These plots are displayed in Figs. 12, 13, 14, and 15.

Fig. 12

Difference of the mean ${L_2(\varOmega \times {\mathbb S}^{2})}$-errors and their upper bound (30) for $c=1, D=1$ and $k=0.1$

Fig. 13

Difference of the mean ${L_2(\varOmega \times {\mathbb S}^{2})}$-errors and their upper bound (30) for $D=1$ and $k=0.1$ at $t'=10$

Fig. 14

Difference of the mean ${L_2(\varOmega \times {\mathbb S}^{2})}$-errors and their upper bound (30) for $c=1$ and $k=0.1$ at $t'=10$

Fig. 15

Difference of the mean ${L_2(\varOmega \times {\mathbb S}^{2})}$-errors and their upper bound (30) for $c=1$ and $D=1$ at $t'=10$

In all cases the difference between the error and its upper bound asymptotically vanish when L increases. Figure 12 demonstrates that the difference is a decreasing function of time $t',$ which is expected as the series representation (25) of the solutions $u(\theta ,\varphi ,t')$ has the multiplication factor $\exp (-t')=\exp (-{c^{2}t}/(2D))$ exponentially decaying in time. The differences are extreme at the origin and decrease when time or the parameter c increases, see Fig. 13. For the parameter D the situation depicted in Fig. 14 is opposite and the difference is increasing in D which is expected as the multiplication factor is exponentially decaying in $D^{-1}.$ Finally, Fig. 15 suggests that the parameter k seems have no substantial impact on the difference.