2.1 Notation

Throughout, $(M,g)$ denotes a smooth connected closed (compact without boundary) Riemannian manifold of dimension $n\ge 2$ . The Laplace–Beltrami operator $-\Delta _g$ is nonnegative, with $0$ as a simple eigenvalue, and its fractional power $(-\Delta _g)^\beta (0<\beta <1)$ is defined spectrally. For a function $u(t,x)$ , $\partial _t^\alpha u$ denotes the Caputo derivative of order $\alpha \in (0,1)$ . The $L^2$ inner product on M is written $\langle \cdot ,\cdot \rangle $ , and $\|\cdot \|$ denotes the corresponding norm. The symbol C denotes a generic constant that may change from line to line.

2.2 Spectral theory and function spaces

Let $(M,g)$ be a closed Riemannian manifold and $V\in L^\infty (M)$ be real-valued. The operator

$$\begin{align*}H:=(-\Delta_g)^\beta+V \end{align*}$$

is self-adjoint on $L^2(M)$ with discrete spectrum $\{\mu _k\}_{k=1}^\infty $ and eigenbasis $\{\psi _k\}_{k=1}^\infty $ .

Definition 2.1 (H-Sobolev scale)

For $s\ge 0$ , define

$$\begin{align*}D_H^s:=\Big\{u\in L^2(M):\sum_{k=1}^\infty(1+{|}\mu_k{|})^{s/\beta} |\langle u,\psi_k\rangle|^2<\infty\Big\}, \end{align*}$$

with norm $\|u\|_{D_H^s}^2=\sum _{k=1}^\infty (1+{|}\mu _k{|})^{s/\beta } |\langle u,\psi _k\rangle |^2$ .

2.3 Unique continuation property

In this section, we discuss the UCP needed in the proof of Theorem 1.2. The operator appearing in our problem is

$$\begin{align*}H=(-\Delta_g)^\beta+V, \end{align*}$$

where $(M,g)$ is a smooth connected closed Riemannian manifold, $\beta \in (0,1)$ , and $V\in L^\infty (M)$ is real-valued. The principal symbol of $(-\Delta _g)^\beta $ is the real, elliptic symbol $|\xi |_g^{2\beta }$ ; the potential term is of order zero and hence does not affect ellipticity. Ellipticity alone, however, does not yield unique continuation for a nonlocal operator, and we argue instead from the rigidity of $(-\Delta _g)^\beta $ itself.

We use the following local unique continuation result; part (ii) is the version applied to eigenfunctions.

Proof (i) Since u vanishes a.e. in W and $V\in L^\infty (M)$ , we have $Vu=0$ a.e. in W, and hence

$$\begin{align*}(-\Delta_g)^\beta u=Hu-Vu=0\quad\text{a.e.\ in }W. \end{align*}$$

Thus, $u\in H^{2\beta }(M)$ satisfies $u=(-\Delta _g)^\beta u=0$ in W, and the unique continuation property of the fractional Laplace–Beltrami operator on closed manifolds yields $u\equiv 0$ on M; see [Reference Li14, Proposition 2.1], proved there for $u\in H^{2r}(M)$ , $r\in (0,\infty )\left\backslash\right. \mathbb {N}$ , with Euclidean antecedents [Reference Ghosh, Salo and Uhlmann6, Reference Rüland18]. Note that $D(H)=D((-\Delta _g)^\beta )=H^{2\beta }(M)$ , since V is a bounded perturbation.

(ii) Apply part (i) with V replaced by $V-\mu \in L^\infty (M;\mathbb {R})$ , noting that $(-\Delta _g)^\beta +(V-\mu )=H-\mu $ and that $(H-\mu )u=0$ on all of M. The last statement follows by applying the first to a linear combination $\sum _p a_p\psi _{k,p}$ .

2.4 Caputo derivative and Mittag-Leffler functions

We recall the definition of the Caputo derivative of order $\alpha \in (0,1)$ :

$$\begin{align*}\partial_t^\alpha u(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^{\! t }\frac{u'(\tau)}{(t-\tau)^\alpha}\,d\tau, \end{align*}$$

and its Laplace transform formula $\mathcal L(\partial _t^\alpha u)(s)=s^\alpha \mathcal Lu(s)$ for $u(0)=0$ . The two-parameter Mittag-Leffler function is

$$\begin{align*}E_{a,b}(z)=\sum_{k=0}^\infty\frac{z^k}{\Gamma(ak+b)},\qquad a,b>0. \end{align*}$$

We will use the following asymptotic estimate on the imaginary axis (see [Reference Li14, Reference Podlubny17]).

Proposition 2.5 Let $0<a<1$ and $\dfrac {\pi a}{2}<\vartheta < \min \Big \{\dfrac {\pi }{2},\pi a\Big \}$ . Then, there exists $C>0$ such that for $|z|\to \infty $ with $\vartheta \le |\arg z|\le \pi $ ,

$$\begin{align*}|E_{a,1}(z)|\le\frac{C}{1+|z|},\qquad E_{a,1}(z)=-\frac{z^{-1}}{\Gamma(1-a)}+O(|z|^{-2}). \end{align*}$$

In the same sector, $E_{a,a}(z)=O(|z|^{-2})$ since the coefficient of $z^{-1}$ in the corresponding expansion is $1/\Gamma (0)=0$ . Because $\vartheta <\pi /2$ , both imaginary rays lie in the sector; together with continuity on bounded sets, this shows that $E_{a,1}$ and $E_{a,a}$ are bounded on the imaginary axis.

3.1 Function spaces generated by H

We use throughout the Hilbert scale $D_H^s$ introduced in Definition 2.1. In particular, $D_H^0=L^2(M)$ and $D_H^{2\beta }=D(H)$ .

Proposition 3.1 (Well-posedness)

Let $f\in C_c^2\big ((0,\infty );L^2(M)\big )$ . Then, there exists a unique solution

$$\begin{align*}u\in C^1\big([0,\infty);D_H^{2\beta}\big)\cap L^\infty\big(0,\infty;D_H^{2\beta}\big),\qquad \partial_tu\in L^\infty\big(0,\infty;D_H^{2\beta}\big) \end{align*}$$

of problem (1.1). The solution is given by the spectral representation

(3.1) $$ \begin{align} u(t)=\sum_{k=1}^\infty u_k(t)\psi_k, \end{align} $$

where $\{\psi _k\}_{k=1}^\infty $ is the orthonormal eigenbasis of H with eigenvalues $\mu _k$ , and the Fourier coefficients $u_k(t)$ satisfy the fractional ordinary differential equation

(3.2) $$ \begin{align} \begin{cases} (i\partial_t^\alpha+\mu_k)u_k=f_k, &t>0,\\[2pt] u_k(0)=0, \end{cases} \end{align} $$

with $f_k(t)=\langle f(t),\psi _k\rangle $ . Moreover,

(3.3) $$ \begin{align} u_k(t)=\int_0^{\! t}K_k(t-\tau)f_k(\tau)\,d\tau,\qquad K_k(t)=-i\,t^{\alpha-1}E_{\alpha,\alpha}(i\mu_kt^\alpha). \end{align} $$

Proof We proceed as in [Reference Li14], adapted to the scale $D_H^s$ . For each $k\in \mathbb {N}$ , the function $f_k$ belongs to $C_c^2((0,\infty ))$ . Applying the Laplace transform to (3.2) and using the property $\mathcal L(\partial _t^\alpha u_k)(s)=s^\alpha \mathcal Lu_k(s)$ (valid for $u_k(0)=0$ ) yields

$$\begin{align*}(is^\alpha+\mu_k)\mathcal Lu_k(s)=\mathcal Lf_k(s),\qquad \mathrm{Re}\,s>0. \end{align*}$$

Hence,

$$\begin{align*}\mathcal Lu_k(s)=\frac{1}{is^\alpha+\mu_k}\mathcal Lf_k(s). \end{align*}$$

The inverse Laplace transform of $(is^\alpha +\mu _k)^{-1}$ is known:

$$\begin{align*}\mathcal L^{-1}\Big[\frac{1}{is^\alpha+\mu_k}\Big](t) =-it^{\alpha-1}E_{\alpha,\alpha}(i\mu_kt^\alpha)=:K_k(t). \end{align*}$$

Convolution in time gives the representation (3.3). Since $K_k\in L^1_{\mathrm {loc}}([0,\infty ))$ and $f_k(0)=0$ , we may write $u_k(t)=\int _0^{\! t}K_k(\sigma )f_k(t-\sigma )\,d\sigma $ and differentiate under the integral sign, the boundary term $K_k(t)f_k(0)$ being absent; this gives

(3.4) $$ \begin{align} u_k'(t)={-i\int_0^{\! t}(t-\tau)^{\alpha-1} E_{\alpha,\alpha}\big(i\mu_k(t-\tau)^\alpha\big)f_k'(\tau)\,d\tau}. \end{align} $$

Integrating by parts in (3.3) gives the further representation

(3.5) $$ \begin{align} \mu_ku_k(t)=f_k(t)-\int_0^{\! t}E_{\alpha,1}\big(i\mu_k(t-\tau)^\alpha\big) f_k'(\tau)\,d\tau. \end{align} $$

Now we establish convergence in $D_H^{2\beta }$ . Let $T_0>0$ be such that $\mathrm {supp} f\subset (0,T_0)\times M$ . From (3.5), we have

$$\begin{align*}{|}\mu_ku_k(t){|}\le|f_k(t)|+\int_0^{\! t} \big|E_{\alpha,1}\big(i\mu_k(t-\tau)^\alpha\big)\big|\,|f_k'(\tau)|\,d\tau. \end{align*}$$

Since $E_{\alpha ,1}$ is bounded on the imaginary axis by Proposition 2.5, there exists $C_\alpha>0$ such that

$$\begin{align*}{|}\mu_ku_k(t){|}\le(1+C_\alpha)\int_0^{T_0} |f_k'(\tau)|\,d\tau. \end{align*}$$

Applying the Cauchy–Schwarz inequality yields

$$\begin{align*}\mu_k^2|u_k(t)|^2\le(1+C_\alpha)^2T_0\int_0^{T_0}|f_k'(\tau)|^2\,d\tau. \end{align*}$$

Summing over k and using Parseval’s identity gives

$$ \begin{align*} \sum_{k=1}^\infty\mu_k^2|u_k(t)|^2 &\le(1+C_\alpha)^2T_0\sum_{k=1}^\infty\int_0^{T_0}|f_k'(\tau)|^2\,d\tau\\ &=(1+C_\alpha)^2T_0\int_0^{T_0}\|f'(\tau)\|_{L^2(M)}^2\,d\tau<\infty. \end{align*} $$

Recalling that $\|u(t)\|_{D_H^{2\beta }}^2=\sum _{k=1}^\infty (1+{|}\mu _k{|})^2|u_k(t)|^2$ , that $(1+|\mu _k|)^2\le 2(1+\mu _k^2)$ , and that $\|u(t)\|_{L^2(M)}$ is bounded directly from (3.3): by $|K_k(r)|\le C_\alpha r^{\alpha -1}$ (Proposition 2.5), Minkowski’s integral inequality, and $\int _0^{\min \{t,T_0\}}(t-\tau )^{\alpha -1}d\tau \le T_0^\alpha /\alpha $ for every $t\ge 0$ ,

$$\begin{align*}\|u(t)\|_{L^2(M)}\le C_\alpha\int_0^{\min\{t,T_0\}}(t-\tau)^{\alpha-1} \|f(\tau)\|_{L^2(M)}\,d\tau \le\frac{C_\alpha T_0^\alpha}{\alpha}\,\|f\|_{C^0([0,T_0];L^2(M))}; \end{align*}$$

altogether, we obtain

(3.6) $$ \begin{align} \sup_{t\ge0}\|u(t)\|_{D_H^{2\beta}}\le C(T_0,\alpha) \|f\|_{C^1([0,T_0];L^2(M))}. \end{align} $$

An identical estimate holds for $u_k'$ : differentiating (3.5) in the same way yields

$$\begin{align*}\mu_ku_k'(t)=f_k'(t)-\int_0^{\! t}E_{\alpha,1}\big(i\mu_k(t-\tau)^\alpha\big) f_k"(\tau)\,d\tau , \end{align*}$$

and the argument just given applies verbatim with $f'$ in place of f, at the cost of replacing $\|f\|_{C^1([0,T_0];L^2(M))}$ by $\|f\|_{C^2([0,T_0];L^2(M))}$ in (3.6). Let $P_N$ denote the orthogonal projection onto $\mathrm {span}\{\psi _1,\dots ,\psi _N\}$ . Since $P_N$ commutes with the solution operator, $(I-P_N)u$ is the solution with source $(I-P_N)f$ ; so, (3.6) and its analog for $u'$ give, as $N\to \infty $ ,

$$ \begin{align*} \sup_{t\ge0}\|(I-P_N)u(t)\|_{D_H^{2\beta}}&\longrightarrow0,\\ \sup_{t\ge0}\|(I-P_N)u'(t)\|_{D_H^{2\beta}}&\longrightarrow0 , \end{align*} $$

the right-hand sides being controlled by the tails of $f,f'$ in $C([0,T_0];L^2(M))$ and of $f',f"$ in $L^2(0,T_0;L^2(M))$ . The latter tend to zero by dominated convergence; so do the former because $f,f'$ are continuous on the compact interval $[0,T_0]$ , and hence have compact range in $L^2(M)$ , and $\|(I-P_N)v\|_{L^2(M)}\to 0$ for each fixed v with $\|I-P_N\|\le 1$ ; so, the convergence is uniform on compact subsets of $L^2(M)$ . Since the finite spectral sums are continuously differentiable, this proves convergence in $C^1([0,\infty );D_H^{2\beta })$ . Moreover, for each $T>0$ ,

$$\begin{align*}\sup_{0\le t\le T}\|\partial_t^\alpha(I-P_N)u(t)\|_{L^2(M)} \le \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \sup_{0\le t\le T}\|(I-P_N)u'(t)\|_{L^2(M)}\longrightarrow0, \end{align*}$$

and so, the Caputo derivative may be taken term by term. For each k,

$$\begin{align*}i\partial_t^\alpha u_k(t)+\mu_ku_k(t)=f_k(t) \end{align*}$$

since $u_k\in C^1([0,\infty ))$ is bounded with $u_k(0)=0$ , so that $\mathcal L(\partial _t^\alpha u_k)=s^\alpha \mathcal Lu_k$ , and hence $\mathcal L\big (i\partial _t^\alpha u_k+\mu _ku_k\big ) =(is^\alpha +\mu _k)\mathcal Lu_k=\mathcal Lf_k$ for $\mathrm {Re}\,s>0$ ; both sides being continuous in t, uniqueness of Laplace transforms gives the identity pointwise. Summing over k and using the boundedness of $H=(-\Delta _g)^\beta +V:D_H^{2\beta }\to L^2(M)$ yields

$$\begin{align*}i\partial_t^\alpha u(t)+\big((-\Delta_g)^\beta+V\big)u(t) =\sum_{k=1}^\infty f_k(t)\psi_k=f(t). \end{align*}$$

The initial condition $u(0)=0$ follows from $u_k(0)=0$ for all k and the uniform convergence. Finally, if u and v are two solutions with the same source f, then $w=u-v$ satisfies

$$\begin{align*}i\partial_t^\alpha w+\big((-\Delta_g)^\beta+V\big)w=0,\qquad w(0)=0. \end{align*}$$

Taking the Fourier coefficients $w_k(t)=\langle w(t),\psi _k\rangle $ gives

$$\begin{align*}i\partial_t^\alpha w_k+\mu_kw_k=0,\qquad w_k(0)=0. \end{align*}$$

The unique solution of this fractional ODE is $w_k\equiv 0$ . Hence, $w\equiv 0$ , proving uniqueness.

4.1 Determination of the time-fractional power $\boldsymbol\alpha $

Proof of Theorem 1.2. We now recover $\alpha $ . Rather than the large-time asymptotics of a compactly supported pulse, we use the high-frequency behavior of the Laplace transform of the solution; this makes no use of the sign or the invertibility of H, and no mean-zero restriction on the sources is imposed here or in Section 4.2.

Choose $\xi \in C_c^\infty (W_1)$ with $\xi \neq 0$ , write $q:=e(\xi )\in C^\infty (M)$ for its extension by zero from $W_1$ to M, and fix a nonnegative $a\in C_c^2((0,\infty ))$ , $a\not \equiv 0$ , so that $\mathcal La(s)>0$ for every $s>0$ . Apply the source $f(t)=a(t)q$ . Taking the Laplace transform of the spectral representation (3.1) exactly as in the proof of Proposition 3.1, and using $\mathcal Lf_k(s)=\mathcal La(s)\langle q,\psi _k\rangle $ , we obtain, for $\mathrm {Re}\,s>0$ ,

$$\begin{align*}\mathcal Lu_f(s)=\mathcal La(s)\sum_{k\ge1} \frac{\langle q,\psi_k\rangle}{\mu_k+is^\alpha}\,\psi_k =\mathcal La(s)\,\big(H+is^\alpha\big)^{-1}q. \end{align*}$$

The a priori bound (3.6) guarantees that the transform converges for $\mathrm {Re}\,s>0$ ; so, the source-to-solution map determines the $L^2(W_2)$ -valued function

$$\begin{align*}F(s):=\frac{1}{\mathcal La(s)}\big(\mathcal Lu_f(s)\big)\big|_{W_2} =\Big(\big(H+is^\alpha\big)^{-1}q\Big)\Big|_{W_2},\qquad s>0. \end{align*}$$

We claim that, with $\rho :=s^\alpha $ ,

(4.1) $$ \begin{align} \Big\|\big(H+i\rho\big)^{-1}q-\frac{q}{i\rho}+\frac{Hq}{(i\rho)^2} \Big\|_{L^2(M)}=o\big(\rho^{-2}\big),\qquad\rho\to\infty. \end{align} $$

Indeed, H is self-adjoint with real spectrum, $H+i\rho $ is boundedly invertible for $\rho>0$ , and for every $v\in L^2(M),$ the spectral theorem and dominated convergence give

$$ \begin{align*} \Big\|\big(H+i\rho\big)^{-1}v-\frac{v}{i\rho}\Big\|_{L^2(M)} &=\frac{1}{\rho} \big\|H\big(H+i\rho\big)^{-1}v\big\|_{L^2(M)}\\ &=\frac{1}{\rho}\Big(\sum_{k\ge1} \frac{\mu_k^2}{\mu_k^2+\rho^2} \big|\langle v,\psi_k\rangle\big|^2\Big)^{1/2} =o\big(\rho^{-1}\big). \end{align*} $$

Since $q\in C^\infty (M)\subset D(H)$ , the identity $(H+i\rho )^{-1}=\frac {1}{i\rho }I-\frac {1}{i\rho }(H+i\rho )^{-1}H$ on $D(H)$ , followed by the estimate above applied to $v=Hq$ , yields

$$\begin{align*}\big(H+i\rho\big)^{-1}q=\frac{q}{i\rho} -\frac{1}{i\rho}\big(H+i\rho\big)^{-1}Hq =\frac{q}{i\rho}-\frac{Hq}{(i\rho)^2}+o\big(\rho^{-2}\big), \end{align*}$$

which is (4.1).

Because $W_1\cap W_2=\emptyset ,$ we have $q|_{W_2}=0$ , and the first-order term of (4.1) disappears on restriction to $W_2$ ; and since V acts by multiplication, $(Hq)|_{W_2}=\big ((-\Delta _g)^\beta q\big )|_{W_2}$ . As $(i)^{-2}=-1$ , restricting (4.1) to $W_2$ therefore gives

(4.2) $$ \begin{align} F(s)=s^{-2\alpha}\big(\omega+o(1)\big)\quad\text{in }L^2(W_2), \qquad\omega:=\big((-\Delta_g)^\beta q\big)\big|_{W_2}. \end{align} $$

We claim that $\omega \neq 0$ . Indeed, if $\omega =0,$ then both q and $Hq=(-\Delta _g)^\beta q+Vq$ vanish a.e. on $W_2$ , and so Proposition 2.3(i) forces $q\equiv 0$ on M, contradicting $\xi \neq 0$ . Hence, (4.2) gives

$$\begin{align*}\lim_{s\to\infty}\frac{\log\|F(s)\|_{L^2(W_2)}}{\log s}=-2\alpha. \end{align*}$$

Now run the same construction on $(\widetilde {M},\widetilde {g})$ with the parameters $\widetilde {\alpha },\widetilde {\beta },\widetilde {V}$ and the source $\Phi _*f=a(t)\,e(\Phi _*\xi )$ . Since $\Phi _*\xi \in C_c^\infty (\widetilde W_1)$ is nonzero and $\widetilde W_1\cap \widetilde W_2=\emptyset $ , the argument just given applies verbatim on $(\widetilde {M},\widetilde {g})$ and yields $\log \|\widetilde F(s)\|_{L^2(\widetilde W_2)}/\log s\to -2\widetilde {\alpha }$ . Finally, the same a is used on both manifolds, and the same factor $\mathcal La(s)$ is divided out; since $\Phi ^*\widetilde {g}=g$ on $W_2$ , $\Phi ^*$ is an isometry of $L^2(\widetilde W_2)$ onto $L^2(W_2)$ . Equality of the source-to-solution maps therefore gives $\|F(s)\|_{L^2(W_2)}=\|\widetilde F(s)\|_{L^2(\widetilde W_2)}$ for every $s>0$ , and comparing the two limits forces $\alpha =\widetilde {\alpha }$ . This completes the determination of $\alpha $ .

4.2 Recovery of the spectral data

Having identified $\alpha $ , we now recover the spectral data of the operator $H=(-\Delta _g)^\beta +V$ . Fix a nonnegative function $a\in C_c^2((0,\infty ))$ whose Laplace transform satisfies $\mathcal La(s)>0$ for all $s>0$ . For $\xi \in L^2(W_1)$ , consider the source

$$\begin{align*}f(t)=a(t)\,e(\xi). \end{align*}$$

Taking the Laplace transform in time of the solution $u_f$ and using the convolution formula (see the proof of Proposition 3.1) gives

$$\begin{align*}\mathcal Lu_f(s)=\mathcal La(s)\sum_{k=1}^\infty\frac{1}{\mu_k+is^\alpha} \langle e(\xi),\psi_k\rangle\psi_k,\qquad \mathrm{Re}\,s>0. \end{align*}$$

Restricting to the observation set $W_2$ and regrouping the terms according to the distinct eigenvalues of H, which from here on we order as

$$\begin{align*}\mu_1<\mu_2<\cdots, \qquad \mu_k\longrightarrow+\infty, \end{align*}$$

and continue to denote by $\mu _1,\mu _2,\dots $ , we obtain the operator-valued function

$$\begin{align*}\mathcal H(s)\xi:=\sum_{k=1}^\infty\frac{1}{\mu_k+is^\alpha} P^H_{W_1,W_2,k}\xi,\qquad \mathrm{Re}\,s>0, \end{align*}$$

where $P^H_{W_1,W_2,k}$ denotes the projection onto the k-th eigenspace of H, restricted from $W_1$ to $W_2$ :

$$\begin{align*}P^H_{W_1,W_2,k}\xi=\sum_{p=1}^{m_k}\langle e(\xi),\psi_{k,p}\rangle \psi_{k,p}|_{W_2}. \end{align*}$$

Here, $\{\psi _{k,p}\}_{p=1}^{m_k}$ is an orthonormal basis of the eigenspace corresponding to $\mu _k$ , and $m_k$ its multiplicity. In this section and in Section 4.4, the $\mu _k$ are enumerated without repetition, and $\Pi _k$ denotes the orthogonal projection of $L^2(M)$ onto the $\mu _k$ -eigenspace, so that $\Pi _kF=\sum _{p=1}^{m_k}\langle F,\psi _{k,p}\rangle \psi _{k,p}$ ; its Schwartz kernel $\Pi _k(x,y)=\sum _{p=1}^{m_k}\psi _{k,p}(x)\overline {\psi _{k,p}(y)}$ does not depend on the choice of orthonormal basis of the eigenspace. The source-to-solution map $L^{\alpha ,\beta ,V}_{W_1,W_2}$ determines $\mathcal H(s)$ for all $s>0$ . Because the $\Pi _k$ are mutually orthogonal and $P^H_{W_1,W_2,k}\xi =(\Pi _ke(\xi ))|_{W_2}$ , for every compact set $K\Subset \mathbb {C}\left\backslash\right. \{-\mu _k:k\ge 1\}$ ,

$$\begin{align*}\sup_{z\in K} \left\| \sum_{k>N}\frac{1}{\mu_k+z}P^H_{W_1,W_2,k} \right\| \le \sup_{\substack{z\in K\\ k>N}}\frac{1}{|\mu_k+z|} \longrightarrow0. \end{align*}$$

Thus, the series converges in operator norm locally uniformly away from its poles. Analytic continuation in the variable $z=is^\alpha $ yields a meromorphic operator-valued function

(4.3) $$ \begin{align} \mathcal H(z)=\sum_{k=1}^\infty\frac{1}{\mu_k+z}P^H_{W_1,W_2,k}, \qquad z\in\mathbb{C}\left\backslash\right.\{-\mu_k\}_{k=1}^\infty. \end{align} $$

The poles of $\mathcal H(z)$ are contained in the set $\{-\mu _k\}_{k\ge 1}$ , are simple, and the residue at $-\mu _k$ equals $P^H_{W_1,W_2,k}$ . The ray $\{is^\alpha :s>0\}$ lies in the connected open set $\mathbb {C}\left\backslash\right. \{-\mu _k\}_{k\ge 1}$ and has accumulation points there; so, by the identity theorem, the values of $\mathcal H$ on that ray – which the source-to-solution map supplies – determine $\mathcal H(z)$ for all z, and hence every pole with nonvanishing residue.

That every eigenvalue is indeed visible in the data, with the correct multiplicity – so that the poles are exactly the points $-\mu _k$ – is guaranteed by the following lemma.

Proof Let $\{\psi _{k,p}\}_{p=1}^{m_k}$ be an orthonormal basis of the $\mu _k$ -eigenspace and let $e(\cdot )$ denote extension by zero. Since $\langle e(\xi ),\psi _{k,p}\rangle _{L^2(M)} =\langle \xi ,\psi _{k,p}|_{W_1}\rangle _{L^2(W_1)}$ ,

$$\begin{align*}P^H_{W_1,W_2,k}\xi=\sum_{p=1}^{m_k} \big\langle\xi,\psi_{k,p}|_{W_1}\big\rangle_{L^2(W_1)}\,\psi_{k,p}|_{W_2}, \qquad\xi\in L^2(W_1). \end{align*}$$

By Proposition 2.3(ii), the restrictions $\psi _{k,p}|_{W_2}$ , $1\le p\le m_k$ , are linearly independent, and their span has dimension $m_k$ ; by the same proposition, the $\psi _{k,p}|_{W_1}$ are linearly independent, and the map $\xi \mapsto (\langle \xi ,\psi _{k,p}|_{W_1}\rangle )_{p=1}^{m_k}$ is onto $\mathbb {C}^{m_k}$ and the range of $P^H_{W_1,W_2,k}$ is that entire span. Hence, $\mathrm {rank}P^H_{W_1,W_2,k}=m_k\ge 1$ , and in particular, $P^H_{W_1,W_2,k}\neq 0$ . The case $W_1=W_2=O$ is identical.

Therefore, from $\mathcal H(z),$ we recover the full spectral dataset

$$\begin{align*}\mathcal S=\big\{(\mu_k,P^H_{W_1,W_2,k}):k\in\mathbb{N}\big\}. \end{align*}$$

4.3 Determination of $\boldsymbol \beta $

With the eigenvalues $\{\mu _k\}_{k=1}^\infty $ of $H=(-\Delta _g)^\beta +V$ , their multiplicities, and the cross projections recovered from the source-to-solution map, we now complete the proof of Theorem 1.2.

Lemma 4.2 Let $\lambda _1\le \lambda _2\le \cdots $ be the eigenvalues of $-\Delta _g$ and $\mu _1\le \mu _2\le \cdots $ those of H, both repeated according to multiplicity (in contrast with the enumeration by distinct eigenvalues used in Theorem 1.2(2)). Then,

$$\begin{align*}|\mu_k-\lambda_k^\beta|\le\|V\|_{L^\infty(M)}\qquad\text{for all }k\ge1. \end{align*}$$

Consequently, the counting function $N_H(\mu ):=\#\{k:\mu _k\le \mu \}$ satisfies

(4.4) $$ \begin{align} N_H(\mu)=\frac{\omega_n}{(2\pi)^n}\,\mathrm{vol}_g(M)\,\mu^{n/(2\beta)} \big(1+o(1)\big)\qquad(\mu\to\infty), \end{align} $$

where $\omega _n$ denotes the volume of the unit ball in $\mathbb {R}^n$ .

Proof By the spectral theorem, the eigenvalues of $(-\Delta _g)^\beta $ are $\lambda _k^\beta $ , in the same order. By the Courant–Fischer min–max characterization, for a self-adjoint, semibounded operator A with discrete spectrum and a bounded self-adjoint perturbation $B,$ one has $|\mu _k(A+B)-\mu _k(A)|\le \|B\|$ : indeed, $\mu _k(A+B)=\min _{\dim S=k}\max _{u\in S,\,\|u\|=1}\langle (A+B)u,u\rangle $ and $|\langle Bu,u\rangle |\le \|B\|$ for every unit vector u, so each Rayleigh quotient – and hence the min–max value – moves by at most $\|B\|$ . Apply this with $A=(-\Delta _g)^\beta $ and $B=V$ , which is self-adjoint and bounded because $V\in L^\infty (M)$ is real-valued. Writing $c=\|V\|_{L^\infty }$ , it follows that, for $\mu \ge c$ ,

$$\begin{align*}N_{-\Delta_g}\big((\mu-c)^{1/\beta}\big)\le N_H(\mu) \le N_{-\Delta_g}\big((\mu+c)^{1/\beta}\big), \end{align*}$$

and Weyl’s law for $-\Delta _g$ on the closed manifold $(M,g)$ , namely, $N_{-\Delta _g}(\lambda )=\frac {\omega _n}{(2\pi )^n}\mathrm {vol}_g(M) \lambda ^{n/2}(1+o(1))$ , gives the stated asymptotics since $(\mu \pm c)^{n/(2\beta )}=\mu ^{n/(2\beta )}(1+o(1))$ .

Completion of the proof of Theorem 1.2. Conclusion (1) for $\alpha $ was obtained in Section 4.1. Since $\Phi $ is a diffeomorphism between nonempty open subsets of M and $\widetilde {M}$ , we have $n:=\dim M=\dim \widetilde {M}$ . By Section 4.2 and Lemma 4.1, equality of the source-to-solution maps implies (after the identification by $\Phi $ ) that the meromorphic families (4.3) coincide; hence, $\mu _k=\widetilde {\mu }_k$ for all k with equal multiplicities, together with the stated identification of the cross projections. This is conclusion (2). Applying Lemma 4.2 on both manifolds,

$$\begin{align*}\frac{n}{2\beta}=\lim_{\mu\to\infty}\frac{\log N_H(\mu)}{\log\mu} =\lim_{\mu\to\infty}\frac{\log N_{\widetilde{H}}(\mu)}{\log\mu}=\frac{n}{2\widetilde{\beta}}, \end{align*}$$

and so $\beta =\widetilde {\beta }$ , completing conclusion (1).

For a general nonconstant potential, H need not commute with $-\Delta _g$ , and the recovered H-spectral projectors cannot be identified with Laplace–Beltrami spectral projectors. Therefore, the disjoint-data argument stops at the cross-spectral data of H and does not recover $(g,V)$ ; the next section treats coinciding measurements using [Reference Feizmohammadi, Krupchyk and Uhlmann5].

4.4 Coinciding source and observation regions: Proof of Theorem 1.3

We place ourselves in the setting of Theorem 1.3: $W_1=W_2=O$ , the potentials are smooth and vanish on O, and $(M,g,O')$ , $(\widetilde {M},\widetilde {g},O')$ satisfy Hypothesis (H) of [Reference Feizmohammadi, Krupchyk and Uhlmann5] for a fixed nonempty open connected $O'$ with $\overline {O'}\subset O$ . Definition 1.1 applies verbatim with $W_1=W_2=O$ .

Step 1: Determination of $\alpha $ . The argument of Section 4.1 applies with $W_1= W_2=O$ and is in fact simpler: here, $q=e(\xi )$ satisfies $q|_O=\xi \neq 0$ , and so the first-order term of (4.1) survives the restriction and neither the identification of $\omega $ nor the unique continuation step is needed. Thus,

$$\begin{align*}\|F(s)\|_{L^2(O)}=s^{-\alpha}\big(\|\xi\|_{L^2(O)}+o(1)\big), \qquad\text{whence}\qquad \lim_{s\to\infty}\frac{\log\|F(s)\|_{L^2(O)}}{\log s}=-\alpha. \end{align*}$$

Comparing with the same limit computed on $(\widetilde {M},\widetilde {g})$ gives $\alpha =\widetilde {\alpha }$ .

Step 2: Spectral data and $\beta $ . Running Section 4.2 with $W_1=W_2=O$ , the source-to-solution map determines the meromorphic family (4.3), and hence, by Lemma 4.1, the same-set spectral data

$$\begin{align*}\mathcal S_O:=\big\{(\mu_k,\ \Pi_k|_{O\times O}):k\in\mathbb{N}\big\}, \end{align*}$$

where $\Pi _k|_{O\times O}$ denotes the Schwartz kernel of $\Pi _k$ restricted to $O\times O$ (equivalently, the operator $P^H_{O,O,k}$ ); here, the eigenfunctions, and hence the kernels $\Pi _k$ , are smooth, by bootstrapping $(-\Delta _g)^\beta \psi =(\mu _k-V)\psi $ with $V\in C^\infty (M)$ . Equality of the maps gives $\mu _k=\widetilde {\mu }_k$ and $\Pi _k|_{O\times O}=\widetilde {\Pi }_k|_{O\times O}$ for all k; the multiplicities agree as well since $m_k=\mathrm {rank}P^H_{O,O,k}$ by Lemma 4.1. Since M and $\widetilde {M}$ contain the common open set O, we have $\dim M=\dim \widetilde {M}=:n$ , and Lemma 4.2 yields $\beta =\widetilde {\beta }$ exactly as in Section 4.3.

Step 3: Reduction to the Calderón problem for fractional Schrödinger operators. Following [Reference Feizmohammadi, Krupchyk and Uhlmann5], define the Cauchy dataset on $O'$ :

$$\begin{align*}\mathcal C^{O'}_{M,g,V} :=\Big\{\big(u|_{O'},((-\Delta_g)^{\beta}u)|_{O'}\big): u\in C^\infty(M),\ Hu=0\ \text{in }M\left\backslash\right.\overline{O'}\Big\}. \end{align*}$$

The reduction is accomplished by the following lemma.

Proof Let

$$\begin{align*}\mathcal F:=\{F\in C^\infty(M):\mathrm{supp} F\subset\overline{O'}\}, \end{align*}$$

and let $\Pi _{\ker H}$ be the orthogonal projection onto $\ker H$ , with $\Pi _{\ker} H=0$ when $\ker H=\{0\}$ . Define

$$\begin{align*}\mathcal F_0^H:=\{F\in\mathcal F:\Pi_{\ker H}F=0\}, \qquad \mathcal N_{O'}^H:=\{w|_{O'}:w\in\ker H\}. \end{align*}$$

Because $\overline {O'}\subset O$ , restriction to O identifies $\mathcal F$ with the smooth functions on O supported in $\overline {O'}$ ; thus, $\mathcal F$ is determined by $O,O'$ , and $g|_O$ alone.

Since H is self-adjoint with discrete spectrum, its Moore–Penrose inverse is

$$\begin{align*}H^\dagger F=\sum_{\mu_k\ne0}\mu_k^{-1}\Pi_kF. \end{align*}$$

For $F\in \mathcal F_0^H$ ,

$$\begin{align*}\sum_{\mu_k\ne0}\mu_k^2 \big\|\mu_k^{-1}\Pi_kF\big\|_{L^2(M)}^2 =\sum_{\mu_k\ne0}\|\Pi_kF\|_{L^2(M)}^2<\infty. \end{align*}$$

Hence, $H^\dagger F\in D(H)$ and $HH^\dagger F=(I-\Pi _{\ker H})F=F$ . Writing $u:=H^\dagger F$ , the identity $(-\Delta _g)^\beta u=F-Vu$ and the boundedness of multiplication by the smooth function V on every $H^s(M)$ , $s\ge 0$ , bootstrap $u\in H^{2\beta }(M)\Rightarrow u\in H^{4\beta }(M)\Rightarrow \cdots $ , whence $u\in \bigcap _{s\ge 0}H^s(M)=C^\infty (M)$ .

We claim that

(4.5) $$ \begin{align} \mathcal C^{O'}_{M,g,V} =\left\{ \left( \left.\sum_{\mu_k\ne0}\mu_k^{-1}\Pi_kF\right|{}_{O'}+h,\ F|_{O'} \right): F\in\mathcal F_0^H,\ h\in\mathcal N_{O'}^H \right\}. \end{align} $$

Indeed, if $u\in C^\infty (M)$ satisfies $Hu=0$ in $M\left\backslash\right. \overline {O'}$ , then $F:=Hu$ belongs to $\mathcal F$ and

$$\begin{align*}\langle F,w\rangle=\langle Hu,w\rangle=\langle u,Hw\rangle=0 \qquad(w\in\ker H). \end{align*}$$

Thus, $F\in \mathcal F_0^H$ and $u-H^\dagger F\in \ker H$ . Conversely, every $u=H^\dagger F+w$ , with $F\in \mathcal F_0^H$ and $w\in \ker H$ , satisfies $Hu=F$ , and hence $Hu=0$ in $M\left\backslash\right. \overline {O'}$ . Since $V|_{O'}=0$ ,

$$\begin{align*}((-\Delta_g)^\beta u)|_{O'}=(Hu-Vu)|_{O'}=F|_{O'}, \end{align*}$$

which proves (4.5).

It remains to verify that every object in (4.5) is determined by $\mathcal S_O$ . For $F\in \mathcal F$ , the restriction $(\Pi _{\ker H}F)|_O$ is computed from the zero-eigenspace kernel on $O\times O$ . Moreover,

$$\begin{align*}(\Pi_{\ker H}F)|_O=0 \quad\Longleftrightarrow\quad \Pi_{\ker H}F=0 \end{align*}$$

because $\Pi _{\ker H}F\in \ker H$ and Proposition 2.3(ii) makes restriction to O injective on $\ker H$ . Hence, $\mathcal F_0^H$ is determined. Let $e_O:L^2(O)\to L^2(M)$ denote extension by zero. The same local kernel determines $\mathcal N_{O'}^H$ : indeed, $T:=\Pi _{\ker H}e_O:L^2(O)\to \ker H$ has adjoint $T^*w=w|_O$ . Proposition 2.3(ii) makes $T^*$ injective, and since $\ker H$ is finite dimensional, T is surjective. Therefore,

$$\begin{align*}\mathcal N_{O'}^H =\left\{\left.(\Pi_{\ker H}e_OG)\right|_{O'}: G\in L^2(O)\right\}. \end{align*}$$

Finally, for each $F\in \mathcal F_0^H$ ,

$$\begin{align*}(H^\dagger F)|_{O'} =\left.\sum_{\mu_k\ne0}\mu_k^{-1}\Pi_kF\right|_{O'}. \end{align*}$$

The series converges in $L^2(M)$ , and its sum is smooth by the preceding elliptic-regularity argument; hence, its $L^2(O')$ restriction determines the smooth function $(H^\dagger F)|_{O'}$ . Thus, it is determined from the kernels $\Pi _k|_{O\times O}$ because $\mathrm {supp} F\subset \overline {O'}\subset O$ . Thus, the entire Cauchy dataset is determined.

Completion of the proof of Theorem 1.3. By Steps 1 and 2, $\alpha =\widetilde {\alpha }$ , $\beta =\widetilde {\beta }$ , and $\mathcal S_O=\widetilde {\mathcal S}_O$ . By Lemma 4.3 applied on both manifolds (with the same O, $O'$ , and $g|_O=\widetilde {g}|_O$ ), $\mathcal C^{O'}_{M,g,V}=\mathcal C^{O'}_{\widetilde {M},\widetilde {g},\widetilde {V}}$ . All hypotheses of [Reference Feizmohammadi, Krupchyk and Uhlmann5, Theorem 1.2] are met: $M\left\backslash\right. \overline {O'}\supset M\left\backslash\right. \overline O\neq \emptyset $ (and similarly on $\widetilde {M}$ ), the potentials are smooth and vanish on $O'\subset O$ , $O'$ is open and connected and satisfies $g|_{O'}=\widetilde {g}|_{O'}$ because $\overline {O'}\subset O$ , Hypothesis (H) holds by assumption, and the fractional orders agree since $\beta =\widetilde {\beta }$ . Hence, [Reference Feizmohammadi, Krupchyk and Uhlmann5, Theorem 1.2] yields an isometry $\Psi :(M,g)\to (\widetilde {M},\widetilde {g})$ with $\Psi |_{O'}=\mathrm {Id}$ and $V=\Psi ^*\widetilde {V}$ .

5.1 Stochastic model and well-posedness

Throughout this section, we fix a filtered probability space $(\Omega ,{\mathcal F,}(\mathcal F_t)_{{t\ge 0}},\mathbb P)$ . We use $\mathbb {E}$ to denote the expectation and $\dot B$ to denote the formal derivative of a standard real-valued $(\mathcal F_t)$ -Brownian motion $B_t$ . For $\frac 12<\alpha <1$ (the restriction $\alpha>\frac 12$ is what makes the integral $\int _0^t(t-\tau )^{2(\alpha -1)} d\tau $ appearing in the Itô isometry below finite), we consider the fractional stochastic problem

(5.1) $$ \begin{align} \begin{cases} i\partial_t^\alpha u+\big((-\Delta_g)^\beta+V\big)u=f+\sigma\dot B, &\text{in }M\times(0,\infty),\\[2pt] u(0)=0. \end{cases} \end{align} $$

Here, $f,\sigma \in C_c^2\big ((0,\infty );L^2(M)\big )$ are deterministic, with $\mathrm {supp} f\cup \mathrm {supp}\sigma \subset (0,T_0)$ for some $T_0>0$ ; thus, the forcing $\sigma (t)\dot B_t$ is white in time and spatially correlated through the deterministic amplitude $\sigma (t,\cdot )$ and is not cylindrical space–time white noise. The spatial operator is

$$\begin{align*}H=(-\Delta_g)^\beta+V,\qquad V\in L^\infty(M)\ \text{real-valued}, \end{align*}$$

with spectrum $\{\mu _k\}$ and eigenfunctions $\{\psi _k\}$ as in Section 2.

Definition 5.1 (Mild solution)

For $f,\sigma \in C_c^2\big ((0,\infty );L^2(M)\big )$ , an $(\mathcal F_t)$ -adapted process u is called a mild solution of (5.1) if

(5.2) $$ \begin{align} \sup_{t>0}\mathbb{E}\|u(t)\|_{L^2(M)}^2<\infty, \end{align} $$

and, for every $t\ge 0$ ,

$$\begin{align*}u(t)=\sum_{k=1}^\infty u_k(t)\psi_k \quad {\text{in }L^2(\Omega;L^2(M))}, \end{align*}$$

where $u_k(t)=\langle u(t),\psi _k\rangle $ satisfies the fractional stochastic ODE

(5.3) $$ \begin{align} \begin{cases} (i\partial_t^\alpha+\mu_k)u_k=f_k+\sigma_k\dot B,\\[2pt] u_k(0)=0, \end{cases} \end{align} $$

with $f_k(t)=\langle f(t),\psi _k\rangle $ , $\sigma _k(t)=\langle \sigma (t), \psi _k\rangle $ , in the weak sense described below as an identity in $L^2(\Omega )$ .

We see that equation (5.3) is interpreted in the weak form, understood as an identity in $L^2(\Omega )$ for each $t\ge 0$ :

(5.4) $$ \begin{align} u_k(t)=\frac{1}{\Gamma(\alpha)}\int_0^{\! t}(t-\tau)^{\alpha-1} \big(i\mu_ku_k(\tau)-if_k(\tau)\big)\,d\tau +\frac{-i}{\Gamma(\alpha)}\int_0^{\! t}(t-\tau)^{\alpha-1}\sigma_k(\tau)\,dB_\tau. \end{align} $$

We now give the stochastic representation of the solution.

Proposition 5.2 (Representation formula)

The stochastic system (5.1) admits a unique mild solution

$$\begin{align*}u(t)=\sum_{k=1}^\infty u_k(t)\psi_k, \end{align*}$$

where

(5.5) $$ \begin{align} \begin{aligned} u_k(t)&=\int_0^{\! t}-i(t-\tau)^{\alpha-1}E_{\alpha,\alpha} \big(i\mu_k(t-\tau)^\alpha\big)f_k(\tau)\,d\tau\\ &\qquad+\int_0^{\! t}-i(t-\tau)^{\alpha-1}E_{\alpha,\alpha} \big(i\mu_k(t-\tau)^\alpha\big)\sigma_k(\tau)\,dB_\tau. \end{aligned} \end{align} $$

Moreover, the bound (5.2) holds.

Proof The argument is essentially the same as in the deterministic case by Proposition 3.1, combined with the Itô isometry. Write $g_\alpha (r)=r^{\alpha -1}/\Gamma (\alpha )$ and $K_k(r)=-ir^{\alpha -1}E_{\alpha ,\alpha }(i\mu _kr^\alpha )$ , so that (5.4) reads $u_k=i\mu _k(g_\alpha *u_k)-i(g_\alpha *f_k) -i\int _0^\cdot g_\alpha (\cdot -\tau )\sigma _k\,dB_\tau $ , and let $v_k$ denote the right-hand side of (5.5), that is, $v_k=K_k*f_k+\int _0^\cdot K_k(\cdot -\tau )\sigma _k\,dB_\tau $ . Comparing Laplace transforms gives the resolvent identity

$$\begin{align*}K_k=-ig_\alpha+i\mu_k(g_\alpha*K_k), \end{align*}$$

valid for every $\mu _k\in \mathbb {R}$ . Substituting it into $v_k$ and using stochastic Fubini to exchange $g_\alpha *$ with the Itô integral shows that $v_k$ satisfies (5.4). The exchange is justified as follows. Proposition 2.5 gives

$$\begin{align*}|K_k(r)|\le C_\alpha r^{\alpha-1}, \qquad \int_0^{\! T}|K_k(r)|^2\,dr \le\frac{C_\alpha^2T^{2\alpha-1}}{2\alpha-1}<\infty, \end{align*}$$

uniformly in k, while

$$\begin{align*}\int_\tau^t(t-s)^{\alpha-1}(s-\tau)^{\alpha-1}\,ds =B(\alpha,\alpha)(t-\tau)^{2\alpha-1}. \end{align*}$$

Consequently,

$$ \begin{align*} &\int_0^{\! t} \left( \int_\tau^t |g_\alpha(t-s)K_k(s-\tau)|\,ds \right)^2 |\sigma_k(\tau)|^2\,d\tau\\ &\qquad\le C_\alpha\int_0^{\min\{t,T_0\}} (t-\tau)^{4\alpha-2}|\sigma_k(\tau)|^2\,d\tau<\infty, \end{align*} $$

which gives the square integrability required by stochastic Fubini.

Conversely, let $w:=u_k-v_k$ be the difference of two solutions of (5.4) with the same data and set

$$\begin{align*}W_T:=\sup_{0\le t\le T}\big(\mathbb{E}|w(t)|^2\big)^{1/2}, \end{align*}$$

which is finite by Definition 5.1 for $u_k$ and by the estimate below for $v_k$ . Iterating $w=i\mu _k(g_\alpha *w)$ gives $w=(i\mu _k)^N(g_{N\alpha }*w)$ , and hence

$$\begin{align*}W_T\le \frac{|\mu_k|^N T^{N\alpha}}{\Gamma(N\alpha+1)}\,W_T. \end{align*}$$

The coefficient on the right tends to zero as $N\to \infty $ ; choosing N so that it is strictly smaller than one yields $W_T=0$ . Thus, $u_k=v_k$ .

It remains to pass from finite spectral sums to the series in (5.5) and to prove (5.2). Let $P_N$ be the projection onto the first N eigenfunctions and set $Q_N=I-P_N$ . Write $u=u^D+u^I$ for the deterministic and Itô parts. The uniform kernel bound above, the Cauchy–Schwarz inequality, and the Itô isometry give

$$ \begin{align*} \sup_{t\ge0}\|Q_Nu^D(t)\|_{L^2(M)}^2 &\le C_{\alpha,T_0} \sup_{0\le\tau\le T_0}\|Q_Nf(\tau)\|_{L^2(M)}^2,\\ \sup_{t\ge0}\mathbb{E}\|Q_Nu^I(t)\|_{L^2(M)}^2 &\le C_{\alpha,T_0} \sup_{0\le\tau\le T_0}\|Q_N\sigma(\tau)\|_{L^2(M)}^2. \end{align*} $$

The right-hand sides tend to zero because f and $\sigma $ have compact ranges in $L^2(M)$ . Hence, the finite spectral sums converge in $L^\infty ((0,\infty );L^2(\Omega ;L^2(M)))$ to an adapted process. This also justifies the infinite series, Parseval’s identity, the Bochner expectation, and the mean-zero identity used below.

For the uniform bound, set

$$\begin{align*}I(t):=\int_0^{\min\{t,T_0\}}(t-\tau)^{2\alpha-2}\,d\tau. \end{align*}$$

Since $2\alpha -2\in (-1,0)$ ,

$$\begin{align*}\sup_{t\ge0}I(t)=\frac{T_0^{2\alpha-1}}{2\alpha-1}. \end{align*}$$

The Cauchy–Schwarz inequality, the Itô isometry, and Parseval’s identity now give

$$ \begin{align*} \mathbb{E}\|u(t)\|_{L^2(M)}^2 &\le C_\alpha^2 I(t) \int_0^{T_0}\|f(\tau)\|_{L^2(M)}^2\,d\tau\\ &\quad+C_\alpha^2\int_0^{\min\{t,T_0\}} (t-\tau)^{2\alpha-2}\|\sigma(\tau)\|_{L^2(M)}^2\,d\tau. \end{align*} $$

Consequently,

$$\begin{align*}\sup_{t\ge0}\mathbb{E}\|u(t)\|_{L^2(M)}^2 \le C_{\alpha,T_0}\Big( \|f\|_{L^\infty(0,T_0;L^2(M))}^2+ \|\sigma\|_{L^\infty(0,T_0;L^2(M))}^2\Big)<\infty.\\[-38pt] \end{align*}$$

5.2 Inverse problem with stochastic source

Fix nonempty disjoint open subsets $W_1,W_2\subset M$ (or, in the setting of Theorem 1.3, $W_1=W_2=O$ ). For a prescribed deterministic $\sigma \in C_c^2\big ((0,\infty );L^2(W_1)\big )$ , define the stochastic source-to-solution map

$$\begin{align*}{\mathscr L^{\alpha,\beta,V,\sigma}_{W_1,W_2}}: f\mapsto u_f|_{W_2\times(0,\infty)},\qquad \mathrm{supp} f\subset W_1\times(0,\infty), \end{align*}$$

where $u_f$ is the mild solution of (5.1). Since $\mathbb {E}[u_f]$ solves the deterministic problem with source f, we have

$$\begin{align*}\mathbb{E}[u_f(t)]=(u_f^{\mathrm{det}})(t), \end{align*}$$

where $u_f^{\mathrm {det}}$ is the unique deterministic solution. Thus, all information in $\mathbb {E}[\mathscr L^{\alpha ,\beta ,V,\sigma }_{W_1,W_2}f]$ is identical to that of the deterministic source-to-solution map.

Theorem 5.3 (Stochastic uniqueness)

Let $(M,g)$ , $(\widetilde {M},\widetilde {g})$ be smooth connected closed Riemannian manifolds of common dimension $n\ge 2$ , and let $V\in L^\infty (M)$ and $\widetilde {V}\in L^\infty (\widetilde {M})$ be real-valued potentials. Assume $\frac 12<\alpha ,\widetilde {\alpha }<1$ and $0<\beta ,\widetilde {\beta }<1$ . Let $W_1,W_2$ be nonempty open sets contained in both M and $\widetilde {M}$ , and assume $g=\widetilde {g}$ on $W_1\cup W_2$ . Assume either (i) $W_1,W_2$ are disjoint or (ii) $W_1=W_2=O$ and the hypotheses of Theorem 1.3 hold. Fix $\sigma \in C_c^2\big ((0,\infty );L^2(W_1)\big )$ . Suppose

$$\begin{align*}\mathbb{E}\big[\mathscr L^{\alpha,\beta,V,\sigma}_{W_1,W_2}f\big] =\mathbb{E}\big[\widetilde{\mathscr L}^{\,\widetilde{\alpha},\widetilde{\beta},\widetilde{V},\sigma}_{W_1,W_2}f\big] \qquad\text{for all }f\in C_c^2\big((0,\infty);L^2(W_1)\big). \end{align*}$$

Then, in case (i), the conclusions (1) and (2) of Theorem 1.2 hold: $\alpha =\widetilde {\alpha }$ , $\beta =\widetilde {\beta }$ , and the cross spectral data of H and $\widetilde {H}$ associated with $(W_1,W_2)$ coincide. In case (ii), the conclusions of Theorem 1.3 hold: $\alpha =\widetilde {\alpha }$ , $\beta =\widetilde {\beta }$ , and there exists an isometry $\Psi :(M,g)\to (\widetilde {M},\widetilde {g})$ with $\Psi |_{O'}=\mathrm {Id}$ , such that $V=\Psi ^*\widetilde {V}$ .