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A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

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Abstract

We present the existence, uniqueness, and regularity of a strong solution to a super-linear stochastic partial differential equation (SPDE) with the random fractional Laplacians:

$$\begin{aligned} u_t = a\Delta ^{\alpha /2}u + b u_x + cu + \xi |u|^{1+\lambda } \dot{W},\quad t>0\,;\quad u(0,\cdot ) = u_0(\cdot ), \end{aligned}$$

where \(\dot{W}\) is a space-time white noise, \(\alpha \in (1,2)\), and \(\lambda \in [0,\alpha /2-1/2)\). The leading coefficient a satisfies the ellipticity condition and depends on \((\omega ,t)\). The lower-order coefficients b, c, and \(\xi \) depend on \((\omega ,t,x)\). The coefficients a ,b, c, and \(\xi \) are bounded. The initial data \(u_0\) depends on \((\omega ,x)\). The unique existence of local solutions to the SPDE follows from the unique solvability of a general Lipschitz case. We prove a Hölder embedding theorem for solution space \(\mathcal {H}_p^\gamma (\tau )\) and maximum principle for SPDEs with the random fractional Laplacians to extend local solutions to a global one. The range of \(\lambda \in [0,\alpha /2-1/2)\) depending on the highest order of the fractional Laplacian is given as a sufficient condition for the existence. When \(\alpha \uparrow 2\), the condition is in accordance with the one for unique solvability of Laplacian case. Moreover, the Hölder embedding theorem provides maximal Hölder regularity of the solution \(u(\omega ,t,x)\), which has \(\alpha \) times as much regularity in space as in time; for \(T\in (0,\infty )\) and small \(\varepsilon >0\), almost surely

$$\begin{aligned} u \in C^{\frac{1}{2}-\frac{1}{2\alpha }-\frac{\lambda }{\alpha } - \varepsilon ,\frac{\alpha }{2}-\frac{1}{2} - \lambda - \varepsilon }_{t,x}([0,T]\times \mathbb {R}). \end{aligned}$$

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Acknowledgements

The author is sincerely grateful to Professor Kyeong-Hun Kim for many valuable comments and Junhee Ryu, Hee-Sun Choi for finding few errors and typos in the earlier version of this draft. The author also would like to thank Editor Professor Arnaud Debussche and an anonymous reviewer for their precious comments and suggestions.

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Correspondence to Beom-Seok Han.

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The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324)

Appendices

A. Proof of Theorem 2.2 (ii)

This section provides a proof of Theorem 2.2 (ii). It should be remarked all the results hold for the spatial domain \(\mathbb {R}^d\), instead of \(\mathbb {R}\). The definitions of stochastic Banach spaces for \(\mathbb {R}^d\) are the same as those for \(\mathbb {R}\). The statements and proofs are motivated by [10, Section 7] and [9].

For \(\alpha \in (0,2)\), consider an SPDE with the fractional Laplacian;

$$\begin{aligned} du(t,x) = \left( \Delta ^{\alpha /2}u(t,x) + f(t,x) \right) \, dt + g dw_t,\quad (t,x)\in (0,\infty )\times \mathbb {R}^d;\quad u(0,\cdot ) = h(\cdot )\nonumber \\ \end{aligned}$$
(A.1)

where \(h\in C_c^\infty (\mathbb {R}^d)\), \(f,g\in C_c^\infty ((0,\infty )\times \mathbb {R}^d)\), and \(w_t\) is the Wiener process. It is well known that there is a fundamental solution p(tx) (e.g. [9, Theorem 2.1]) and the solution of (A.1) is given by

$$\begin{aligned}&\int _{\mathbb {R}^d}h(y)p(t,x-y)dy + \int _0^t \int _{\mathbb {R}^d}f(s,y)p(t-s,x-y)dyds \nonumber \\&+ \int _0^t \int _{\mathbb {R}^d}g(s,y)p(t-s,x-y)dydw_s. \end{aligned}$$

It turns out that the fundamental solution p(tx) satisfies the following properties.

Lemma A.1

  1. (i)

    There exists a fundamental solution p(tx) such that

    $$\begin{aligned} p(t,\cdot )\in L_1\left( \mathbb {R}^d \right) . \end{aligned}$$
  2. (ii)

    Let \(\gamma \in [0,\infty )\). There exists \(N = N(\alpha ,\gamma )\) such that for \((t,x)\in (0,\infty )\times \mathbb {R}^d\),

    $$\begin{aligned} |(\Delta ^{\gamma } p)(t,x)| \le N1_{t^{-1}|x|^{\alpha } \le 1}t^{-\frac{d+2\gamma }{\alpha }}+N1_{t^{-1}|x|^{\alpha } \ge 1}|x|^{-d-2\gamma }. \end{aligned}$$
    (A.2)
  3. (iii)

    Let \(\gamma \in [0,\infty )\). For \((t,x)\in (0,\infty )\times \mathbb {R}^d\),

    $$\begin{aligned} (\Delta ^{\gamma }p)(t,x) = t^{-\frac{d+2\gamma }{\alpha }}(\Delta ^\gamma p)(1,t^{-\frac{1}{\alpha }}x). \end{aligned}$$
    (A.3)

Proof

For (i) and (ii), see [9, Theorem 2.1, Theorem 2.3, Theorem 2.4]. For (iii), see (5.2) of [9]. \(\square \)

For convenience, set

$$\begin{aligned} T_t h(x) := \int _{\mathbb {R}}h(y)p(t,x-y)dy. \end{aligned}$$
(A.4)

To prove Theorem 2.2 (ii), we need Lemmas A.2 and A.3. Lemma A.2 describes the instant smoothing effect of \(T_t\).

Lemma A.2

For any \(h\in C_c^\infty (\mathbb {R}^d)\), \(\theta \in [0,1]\), and \(t>0\), we have

$$\begin{aligned} \Vert e^{-t}T_t h \Vert _{L_p}\le Nt^{-\theta }\Vert h \Vert _{H_p^{-\alpha \theta }}\quad \text{ and } \quad \Vert (T_t-1)h \Vert _{L_p}\le Nt^\theta \Vert h \Vert _{H^{\alpha \theta }_{p}}, \end{aligned}$$
(A.5)

where \(N= N(\alpha ,d,p,\theta )\).

Proof

We prove the first inequality of (A.5). If \(\theta = 0\), by Minköwski’s inequality, we have

$$\begin{aligned} \begin{aligned} \Vert T_t h \Vert _{L_p} \le \Vert p(t,\cdot )*h \Vert _{L_p} \le \Vert p(t,\cdot )\Vert _{L_1}\Vert h \Vert _{L_p}\le N\Vert h \Vert _{L_p}. \end{aligned} \end{aligned}$$
(A.6)

For \(\theta \in (0,1]\), by the classical multiplier theorem (e.g. Theorems 1.3.6 and 1.3.8 of [4]), we have

$$\begin{aligned} \begin{aligned} \Vert e^{-t}T_th \Vert _{H_p^{\alpha \theta }}&= \Vert (1-\Delta )^{\frac{\alpha \theta }{2}}(e^{-t}T_th)\Vert _{L_p} \\&\le N\Vert e^{-t}T_th \Vert _{L_p} + N\Vert \Delta ^{\frac{\alpha \theta }{2}}(e^{-t}T_th) \Vert _{L_p}. \\ \end{aligned} \end{aligned}$$

Since \(e^{-t}\le Nt^{-\theta }\wedge 1\), we have

$$\begin{aligned} \begin{aligned} \Vert e^{-t}T_th \Vert _{H_p^{\alpha \theta }}&\le N(\alpha ,d,p,\theta )\left( t^{-\theta }\Vert T_th \Vert _{L_p} + \Vert \Delta ^{\frac{\alpha \theta }{2}}T_th \Vert _{L_p}\right) . \end{aligned} \end{aligned}$$
(A.7)

Inequality (A.6) implies that

$$\begin{aligned} t^{-\theta }\Vert T_th \Vert _{L_p} \le Nt^{-\theta }\Vert h \Vert _{L_p}. \end{aligned}$$
(A.8)

For the second term of the right hand side in (A.7), by using Minköwski’s inequality, we have

$$\begin{aligned} \Vert \Delta ^{\frac{\alpha \theta }{2}}T_th \Vert _{L_p} = \Vert \Delta ^{\frac{\alpha \theta }{2}}(p(t,\cdot )*h) \Vert _{L_p} \le \Vert (\Delta ^{\frac{\alpha \theta }{2}}p)(t,\cdot ) \Vert _{L_1}\Vert h \Vert _{L_p}. \end{aligned}$$
(A.9)

Then, by (A.3) and (A.2),

$$\begin{aligned} \begin{aligned} \Vert (\Delta ^{\frac{\alpha \theta }{2}}p)(t,\cdot ) \Vert _{L_1}&= \int _{\mathbb {R}^d} |(\Delta ^{\frac{\alpha \theta }{2}}p)(t,x)|dx \\&\le t^{-\theta }\int _{\mathbb {R}^d}|(\Delta ^{\frac{\alpha \theta }{2}}p)(1,x)|dx \\&\le N(\alpha ,d,p,\theta )t^{-\theta }. \end{aligned} \end{aligned}$$
(A.10)

By combining (A.9) and (A.10), we have

$$\begin{aligned} \Vert \Delta ^{\frac{\alpha \theta }{2}}T_th \Vert _{L_p} \le N(\alpha ,d,p,\theta )t^{-\theta }\Vert _{L_1}\Vert h \Vert _{L_p}. \end{aligned}$$
(A.11)

Thus, by applying (A.11) and (A.8) to (A.7), we have

$$\begin{aligned} \Vert e^{-t}T_th \Vert _{H_p^{\alpha \theta }} = \Vert (1-\Delta )^{\frac{\alpha \theta }{2}}e^{-t}T_th \Vert _{L_p}\le N(\alpha ,d,p,\theta )t^{-\theta }\Vert h \Vert _{L_p}. \end{aligned}$$
(A.12)

Therefore, by Lemma 2.1 (iv), we have the first inequality in (A.5).

Now, we prove the second inequality of (A.5). If \(\theta = 0\), by (A.6), we have

$$\begin{aligned} \Vert T_th - h \Vert _{L_p} \le \Vert T_th \Vert _{L_p} + \Vert h \Vert _{L_p} \le 2\Vert h \Vert _{L_p}. \end{aligned}$$

For \(\theta \in (0,1]\), notice that \(T_th\) satisfies (A.1) with \(f = g = 0\). For \(t\in (0,1)\), by (A.12), we have

$$\begin{aligned} \begin{aligned} \Vert (T_t-1)h \Vert _{L_p}&= \left\| \int _0^t \Delta ^\frac{\alpha }{2} (T_sh)ds \right\| _{L_p} \\&\le \int _0^t\Vert \Delta ^{\frac{\alpha }{2}}(1-\Delta )^{-\frac{\alpha }{2}}(1-\Delta )^{\frac{\alpha }{2}(1-\theta )}T_s(1-\Delta )^{\frac{\alpha \theta }{2}}h \Vert _{L_p}ds \\&\le N\int _0^t \Vert (1-\Delta )^{\frac{\alpha }{2}(1-\theta )}T_s(1-\Delta )^{\frac{\alpha \theta }{2}}h \Vert _{L_p}ds \\&\le N\int _0^t s^{\theta -1}e^sds\Vert (1-\Delta )^{\frac{\alpha \theta }{2}}h \Vert _{L_p} \\&\le N t^\theta \Vert h \Vert _{H_p^{\alpha \theta }}. \end{aligned} \end{aligned}$$

For \(t\ge 1\), observe that

$$\begin{aligned} \Vert (T_t-1)h \Vert _{L_p} \le N\Vert h \Vert _{L_p} \le Nt^\theta \Vert h \Vert _{H_p^{\alpha \theta }}. \end{aligned}$$

The lemma is proved. \(\square \)

To proceed further, one of the embedding theorems for Slobodetskii’s spaces (Lemma A.3) is introduced; see [25].

Lemma A.3

Let \(p\ge 1\) and \(\mu >1/p\). For any continuous \(L_p\)-valued function h(t) and \(s\le t\), we have

$$\begin{aligned} \begin{aligned}&\Vert h(t) - h(s) \Vert ^p_{L_p}\le N(\mu ,p)(t-s)^{\mu p - 1}\int _s^t\int _s^t 1_{r_2>r_1}\frac{\Vert h(r_2)-h(r_1)\Vert ^p_{L_p}}{|r_2-r_1|^{1+\mu p}}dr_1dr_2 \\&\quad =N(\mu ,p)(t-s)^{\mu p-1}\int _0^{t-s}\frac{1}{\rho ^{1+\mu p}}\int _s^{t-\rho }\Vert h(r+\rho )-h(r)\Vert _{L_p}^pdrd\rho ,\quad \left( \frac{0}{0}:=0\right) . \end{aligned} \end{aligned}$$
(A.13)

From (A.13), we have

$$\begin{aligned} \mathbb {E}\sup _{0\le s < t \le T}\frac{\Vert h(t) - h(s) \Vert _{L_p}^p}{(t-s)^{\mu p -1}} \le N(\mu ,p)\int _0^T\int _0^T 1_{r_2>r_1}\frac{\mathbb {E}\Vert h(r_2) - h(r_1) \Vert _{L_p}^p}{|r_2-r_1|^{1+\mu p}}dr_2dr_1.\nonumber \\ \end{aligned}$$
(A.14)

Now, we prove Theorem 2.2 (ii).

Proof of Theorem 2.2 (ii)

It suffices to show that Theorem 2.2 (ii) holds for \(\tau = T\). To see this, suppose the assertion holds for \(\tau = T\). Let \(\tau \le T\) be a bounded stopping time and \(u\in \mathcal {H}_{p}^{\gamma +\alpha }(\tau )\). Define \(f := (\mathbb {D}u - \Delta ^{\alpha /2} u)1_{t\le \tau }\) and \(g := (\mathbb {S}u)1_{t\le \tau }\). Then, u satisfies

$$\begin{aligned} du = (\Delta ^{\alpha /2} u +f)dt + g^k dw_t^k\,;\quad u(0,\cdot ) = u_0(\cdot ) \end{aligned}$$
(A.15)

on \(t\in [0,\tau ]\). On the other hand, by [6, Theorem 2.7], there exists \(v\in \mathcal {H}_p^{\gamma +\alpha }(T)\) such that v satisfies Eq. (A.15) on [0, T]. Note that \(w:=u-v\) satisfies

$$\begin{aligned} w_t = \Delta ^{\alpha /2}w,\quad t\in [0,\tau ];\quad w(0,\cdot ) = 0. \end{aligned}$$

Then, by the deterministic version of [8, Theorem 2.10], we have

$$\begin{aligned} u(t,\cdot )=v(t,\cdot ) \quad \text{ on }\quad [0,\tau ]. \end{aligned}$$

Since [8, Theorem 2.10] implies that

$$\begin{aligned}&\Vert v \Vert _{\mathcal {H}_p^{\gamma +\alpha }(T)} \le N\left( \Vert f \Vert _{\mathbb {H}_p^{\gamma }(\tau )} + \Vert g \Vert _{\mathbb {H}_p^{\gamma +\alpha /2}(\tau ,\ell _2)}\right. \\&\quad \left. +\Vert u(0) \Vert _{U_p^{\gamma +\alpha ,\alpha }} \right) \le N\Vert u \Vert _{\mathcal {H}_p^{\gamma +\alpha }(\tau )}, \end{aligned}$$

we have

$$\begin{aligned} \mathbb {E}| u |^p_{C^{\mu -1/p}([0,\tau ];H_{p}^{\gamma +\alpha -\nu \alpha } )} \le \mathbb {E}| v |^p_{C^{\mu -1/p}([0,T];H_{p}^{\gamma +\alpha -\nu \alpha } )} \le N\Vert v \Vert ^p_{\mathcal {H}_{p}^{\gamma +\alpha }(T)}\le N\Vert u \Vert ^p_{\mathcal {H}_{p}^{\gamma +\alpha }(\tau )}. \end{aligned}$$

Therefore, we only need to show

$$\begin{aligned} \mathbb {E}| v |^p_{C^{\mu -1/p}([0,T];H_{p}^{\gamma +\alpha -\nu \alpha } )} \le N \Vert v \Vert ^p_{\mathcal {H}_p^{\gamma +\alpha }(T)}. \end{aligned}$$

Because of mollification and multiplying by cut off functions, without loss of generality, we may assume that \(u_0\) is infinitely differentiable and compactly supported in x. Furthermore, by [10, Theorem 3.10], we may assume that f and g are functions of the forms

$$\begin{aligned} f(t,x) = \sum _{i = 1}^{n} f_{i-1}(x) 1_{\left( \tau _{i-1},\tau _{i}\right] }(t) \end{aligned}$$

and

$$\begin{aligned} g^k(t,x) = {\left\{ \begin{array}{ll} \sum _{i = 1}^{n} g^k_{i-1}(x) 1_{\left( \tau ^k_{i-1},\tau ^k_{i}\right] }(t) &{}\quad \text{ for }\quad k = 1,2,\dots ,m,\\ \quad \quad \quad \quad 0 &{}\quad \text{ for }\quad k = m+1,\dots , \end{array}\right. } \end{aligned}$$

for some \(m\in \mathbb {N}\), where \(\tau _i,\tau _i^k\) are bounded stopping times and \(f_{i}, g^k_{i}\in C_c^\infty (\mathbb {R}^d)\).

Due to Lemma 2.1 (iv), we may assume \(\gamma = \alpha \nu - \alpha \). Set

$$\begin{aligned} u_1(t) := T_t u_0 + \int _0^t T_{t-s}f(s)ds. \end{aligned}$$

Observing that

$$\begin{aligned} u_1(r+\rho ) - u_1(r) = (T_\rho -1)u_1(r) + \int _0^{\rho } T_{\rho -\sigma }f(r+\sigma )d\sigma \end{aligned}$$

yields

$$\begin{aligned} \mathbb {E}\Vert u_1(r+\rho )-u_1(r) \Vert _{L_p}^p\le N(A_1(r,\rho )+B_1(r,\rho )), \end{aligned}$$

where

$$\begin{aligned} A_1(r,\rho ) := \mathbb {E}\Vert (T_\rho -1)u_1(r) \Vert _{L_p}^p \quad \text{ and }\quad B_1(r,\rho ) := \mathbb {E}\left\| \int _0^\rho T_{\rho -\sigma }f(r+\sigma )d\sigma \right\| _{L_p}^p. \end{aligned}$$

By inequality (A.13), we have

$$\begin{aligned} \mathbb {E}\sup _{0\le s < t \le T}\frac{\Vert u_1(t)-u_1(s) \Vert _{L_p}^p}{(t-s)^{\mu p -1}} \le N (I_1+J_1), \end{aligned}$$

where

$$\begin{aligned}&I_1 := \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}}A_1(r,\rho ) drd\rho \quad \text{ and }\quad \\&J_1 := \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}}B_1(r,\rho ) drd\rho . \end{aligned}$$

Let \(q := \frac{p}{p-1}\). By change of variable, Hölder’s inequality, \((\nu -1)q>-1\), \(\rho \le T\) and Lemma A.2, we have

$$\begin{aligned} \begin{aligned} B_1(r,\rho )&= \mathbb {E}\int _{\mathbb {R}^d}\left| \int _0^\rho \sigma ^{\nu -1}\sigma ^{1-\nu }T_\sigma f(r+\rho -\sigma ) d\sigma \right| ^pdx \\&\le \mathbb {E}\left( \int _0^\rho \sigma ^{(\nu -1)q}d\sigma \right) ^{p/q}\int _0^\rho \sigma ^{(1-\nu )p}\int _{\mathbb {R}^d} |T_\sigma f(r+\rho -\sigma )|^p dxd\sigma \\&\le N(T)\rho ^{\nu p -1}\mathbb {E}\int _0^\rho \Vert f(r+\rho -\sigma ) \Vert _{H^{\alpha (\nu -1)}_p}^p d\sigma \\&= N\rho ^{\nu p-1}\mathbb {E}\int _0^\rho \Vert f(r+\sigma ) \Vert _{H^{\alpha (\nu -1)}_p}^pd\sigma , \end{aligned} \end{aligned}$$

where \(N = N(T,p)\). Thus, we have

$$\begin{aligned} \begin{aligned} J_1&\le N\int _0^T\int _0^{T-\rho } \frac{1}{\rho ^{2+(\mu -\nu )p}} \mathbb {E}\int _0^\rho \Vert f(r+\sigma ) \Vert _{H^{\alpha (\nu -1)}_p}^p d\sigma dr d\rho \\&\le N\int _0^T \frac{1}{\rho ^{2+(\mu -\nu )p}}\int _0^\rho \mathbb {E}\int _0^T\Vert f(r) \Vert _{H^{\alpha (\nu -1)}_p}^p dr d\sigma d\rho \\&\le N\Vert f \Vert _{\mathbb {H}_p^{\alpha (\nu -1)}(T)}. \end{aligned} \end{aligned}$$

The last inequality follows from \(\mu <\nu \). In the case of \(I_1\), by the above result and Fubini’s theorem, we have

$$\begin{aligned} \begin{aligned} I_1&\le \int _0^T \int _0^{T-\rho } \frac{1}{\rho ^{1+\mu p}} A_1(r,\rho )dr d\rho \\&\le \int _0^T \int _0^{T-\rho } \frac{1}{\rho ^{1+\mu p}} \mathbb {E}\Vert (T_\rho -1)u_1(r) \Vert _{L_p(\mathbb {R}^d)}^p dr d\rho \\&\le \int _0^T \frac{1}{\rho ^{1+(\mu -\nu ) p}} \mathbb {E}\int _0^{T-\rho } \Vert u_1(r) \Vert _{H_p^{\alpha \nu }}^p dr d\rho \\&\le N \Vert u(0) \Vert _{U_p^{\alpha \nu ,\alpha }}+N\Vert f \Vert _{\mathbb {H}_p^{\alpha (\nu -1)}(T)}. \end{aligned} \end{aligned}$$

Let \(u_2:=u - u_1\). Notice that

$$\begin{aligned} \begin{aligned} u_2(r+\rho )-u_2(r) =&(T_\rho -1)u_2(r)+\int _0^{\rho }T_{\rho -\sigma }g^k(r+\sigma )dw_\sigma ^k\\&\Vert u_2(r+\rho ) - u_2(r) \Vert _{L_p}^p \le N(A_2(r,\rho )+B_2(r,\rho )), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} A_2(r,\rho ) := \mathbb {E}\Vert (T_\rho -1)u_2(r) \Vert _{L_p}^p,\quad B_2(r,\rho ) := \mathbb {E}\left\| \int _0^{\rho }T_{\rho -\sigma }g^k(r+\sigma )dw_\sigma ^k \right\| _{L_p}^p. \end{aligned}$$

By inequality (A.13),

$$\begin{aligned} \mathbb {E}\sup _{0\le s<t\le T}\frac{\Vert u_2(t) - u_2(s) \Vert _{L_p}^p}{(t-s)^{\mu p - 1}} \le N(I_2 + J_2), \end{aligned}$$

where

$$\begin{aligned}&I_2 = \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}}A_2(r,\rho ) dr d\rho ,\\&J_2 = \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}}B_2(r,\rho )dr d\rho . \end{aligned}$$

By Burkholder-Davis-Gundy inequalities, \(\frac{(2\nu -1)p}{p-2}>-1\), \(\sigma <T\), and Lemma A.2, we obtain

$$\begin{aligned} B_2= & {} \mathbb {E}\left\| \int _0^{\rho }T_{\rho -\sigma }g^k(r+\sigma ) dw_\sigma ^k\right\| _{L_p(\mathbb {R}^d)}^p \\\le & {} \int _{\mathbb {R}^d}\mathbb {E}\left| \int _0^{\rho }T_{\rho -\sigma }g^k(r+\sigma ) dw_\sigma ^k\right| ^pdx \\\le & {} \int _{\mathbb {R}^d}\mathbb {E}\left( \int _0^{\rho }\sum _{k}|T_{\rho -\sigma }g^k(r+\sigma )|^2 d\sigma \right) ^{p/2}dx \\\le & {} N \mathbb {E}\int _{\mathbb {R}^d}\left( \int _0^\rho | T_\sigma g^k(r+\rho -\sigma ) |_{\ell _2}^2d\sigma \right) ^{p/2}dx \\\le & {} N\mathbb {E}\int _{\mathbb {R}^d}\left( \int _0^\rho \sigma ^{2\nu -1}\sigma ^{1-2\nu } | T_\sigma g(r+\rho -\sigma )|^2_{\ell _2} d\sigma \right) ^{p/2}dx \\\le & {} N\rho ^{\nu p -1} \mathbb {E}\int _{\mathbb {R}^d} \sigma ^{(1-2\nu )p/2}\Vert T_\sigma g(r+\rho -\sigma ) \Vert _{L_p(\mathbb {R}^d,\ell _2)}^p d\sigma \\\le & {} N\rho ^{\nu p -1}\mathbb {E}\int _0^\rho \Vert g(r+\rho -\sigma ) \Vert ^p_{H^{\alpha \nu -\alpha /2}_p(\ell _2)} d\sigma , \end{aligned}$$

where \(N = N(T,p)\). Thus,

$$\begin{aligned} \begin{aligned} J_2&\le N \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{2+(\mu -\nu )p }}\mathbb {E}\int _0^\rho \Vert g(r+\rho -\sigma ) \Vert ^p_{H_p^{\alpha \nu -\alpha /2}(\ell _2)}d\sigma dr d\rho \\&\le \int _0^T \frac{1}{\rho ^{2+(\mu -\nu )p}} \int _0^\rho \mathbb {E}\int _\sigma ^{T-\rho +\sigma } \Vert g(r) \Vert ^p_{H_p^{\alpha \nu -\alpha /2}(\ell _2)}drd\sigma d\rho \\&\le N \Vert g \Vert ^p_{\mathbb {H}_p^{\alpha \nu -\alpha /2}(T,\ell _2)}. \end{aligned} \end{aligned}$$

Similarly, for \(I_2\), we have

$$\begin{aligned} \begin{aligned} I_2&= \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}}A_2(r,\rho ) drd\rho \\&\le \int _0^T\int _0^{T-\rho }\frac{1}{\rho ^{1+\mu p}} \mathbb {E}\Vert (T_\rho - 1)u_2(r) \Vert _{L_p(\mathbb {R}^d)}^p drd\rho \\&\le \int _0^T \frac{1}{\rho ^{1+(\mu -\nu ) p}} \int _0^{T-\rho } \mathbb {E}\Vert u_2(r) \Vert _{H_p^{\alpha \nu }}^p drd\rho \\&\le N\Vert u_2 \Vert ^{p}_{\mathcal {H}_p^{\alpha \nu }(T)} \\&\le N\Vert g \Vert _{\mathbb {H}_p^{\alpha \nu - \alpha /2}(T,\ell _2)}. \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} E\Vert u \Vert ^p_{C^{\mu -1/p}([0,T];L_p(\mathbb {R}^d))}&\le N \left( \Vert f \Vert ^p_{\mathbb {H}_p^{\alpha \nu - \alpha }(T)} + \Vert g \Vert ^p_{\mathbb {H}_p^{\alpha \nu - \alpha /2}(T,\ell _2)} + \Vert u(0) \Vert ^p_{U_p^{\alpha \nu ,\alpha }} \right) \\&\le N\Vert u \Vert ^p_{\mathcal {H}_p^{\alpha \nu }(T)}. \end{aligned} \end{aligned}$$

The theorem is proved.

B. Maximum principle

In this section, we introduce the maximum principle for an SPDE with the random fractional Laplacian. Consider

$$\begin{aligned} \begin{aligned}&du = \left( a\Delta ^{\alpha /2} u + b u_x + cu + f\right) \,dt \\&\quad + \left( \nu ^k u + g^k\right) \, dw_t^k, \quad t>0\,; \quad u(0,\cdot ) = u_0(\cdot ), \end{aligned} \end{aligned}$$
(B.1)

where \(\alpha \in (1,2)\). It should be remarked that a similar argument can be made for the case of \(\mathbb {R}^d\), instead of \(\mathbb {R}\). The settings and ideas of our proof for the maximum principle are motivated by [13].

Assumption B.1

  1. (i)

    The coefficients \(a = a(t)\) is predictable and nonnegative.

  2. (ii)

    The coefficients \(b = b(t,x)\), \(c = c(t,x)\), and \(\nu _s^k = \nu _s^k(s,x)\) are \(\mathcal {P}\times \mathcal {B}(\mathbb {R})\)-measurable.

  3. (iii)

    The coefficient \(b = b(t,x)\) is continuously differentiable with respect to x.

  4. (iv)

    The function \(f = f(t,\cdot )\) is a \(L_2\)-valued \(\mathcal {F}_t\)-adapted, jointely measurable process.

  5. (v)

    The function \(g = g(t,\cdot ) = (g^1(t,\cdot ),g^2(t,\cdot ),\dots )\) is a \(L_2(\ell _2)\)-valued \(\mathcal {F}_t\)-adapted, jointely measurable process.

  6. (vi)

    The function \(u = u(t,\cdot )\) is a \(L_2\)-valued \(\mathcal {F}_t\)-adapted, jointely measurable process satisfying Eq. (B.1).

  7. (vii)

    There is a finite constant \(K > 0\) such that

    $$\begin{aligned}&K^{-1}< a < K \quad \text{ for } \text{ all } \quad \omega , t, \\&\quad b| + |b_x| + |c| + |\nu |^2_{\ell _2} \le K \quad \text{ for } \text{ all } \quad \omega ,t,x, \end{aligned}$$

    and

    $$\begin{aligned} \int _0^t \Vert u(s,\cdot ) \Vert _{L_2}^2 ds + \int _0^t \Vert f(s,\cdot ) \Vert _{L_2}^2 ds + \int _0^t \Vert g(s,\cdot ) \Vert _{L_2(\ell _2)}^2 ds < K \quad \text{ for } \text{ all } \quad \omega ,t. \end{aligned}$$

Theorem B.2

Let \(\tau \le T\) be a bounded stopping time. Suppose Assumption B.1 holds. Assume that for any \(\omega \in \Omega \), \(k = 1,2,\dots \),

$$\begin{aligned} 1_{u(t,\cdot )>0}f(t,\cdot ) \le 0 \quad \text{ and } \quad 1_{u(t,\cdot )>0}g^k(t,\cdot ) = 0 \end{aligned}$$

t-almost everywhere on \((0,\tau )\). If \(u_{0}\le 0\), then almost surely

$$\begin{aligned} u(t,\cdot )\le 0\quad \text{ for } \text{ all }\quad t\in [0,\tau ]. \end{aligned}$$

Remark B.3

It is well-known that there are several equivalent definitions for the fractional Laplacian. For example, for \(\alpha \in (0,2)\), \(f\in L_2\) and \(g\in H_2^{\alpha /2}\),

$$\begin{aligned} \int _{\mathbb {R}} \left( \Delta ^{\alpha /2}f(x)\right) \,g(x) dx := -c_{1,\alpha }\int _{\mathbb {R}}\int _{\mathbb {R}}\frac{(f(x)-f(y))(g(x)-g(y))}{|x-y|^{1+\alpha }}dydx,\nonumber \\ \end{aligned}$$
(B.2)

where \(c_{1,\alpha } = \frac{2^\alpha \Gamma (\frac{1+\alpha }{2})}{2\pi ^{1/2}|\Gamma (-\frac{\alpha }{2})|}\); see [16].

Remark B.4

Let \(\mathcal {R}\) be a set of functions satisfying the following conditions.

  1. (i)

    The function r(z) is a real-valued function on \(\mathbb {R}\).

  2. (ii)

    The function r(z) is continuously differentiable and \(r(0) = 0\).

  3. (iii)

    The function \(r'(z)\) is absolutely continuous and \(r'(0) = 0\).

  4. (iv)

    The function \(r''(z)\) is bounded and left continuous.

Then, for \(r\in \mathcal {R}\), there exists an infinitely differentiable function \(r_n(z)\) such that

$$\begin{aligned} |r_n(z)|\le N|z|^2,\quad |r'_n(z)|\le N|z|,\quad |r''_n(z)|\le N,\quad \text{ and }\quad r_n,r_n',r_n''\rightarrow r,r,'r'',\nonumber \\ \end{aligned}$$
(B.3)

where N is independent of z and n. For more details, see [13, Remark 2.1].

Proof of Theorem B.2

Define \(r(z) := |z|^2 1_{z>0}\) for \(z\in \mathbb {R}\). Then by Remark B.4, there exists a sequence of functions \(r_n(z)\) satisfying (B.3). Take a nonnegative function \(\zeta \in C_c^\infty (\mathbb {R})\) such that \(\int _{\mathbb {R}}\zeta dx = 1\), and define \(\zeta _\varepsilon (x) := \varepsilon ^{-1}\zeta (x/\varepsilon )\). For \(v\in L_{1,loc}(\mathbb {R})\), set

$$\begin{aligned} v^{(\varepsilon )}(x) := (v*\zeta _\varepsilon )(x) = \int _{\mathbb {R}} v(y)\zeta _\varepsilon (x-y) dy = \int _{\mathbb {R}} v(x-\varepsilon y)\zeta (y)dy. \end{aligned}$$

By Itô’s formula, we have

$$\begin{aligned} \begin{aligned}&e^{-\gamma K(\tau \wedge t)}\int _{\mathbb {R}}r_n\left( u^{(\varepsilon )}(\tau \wedge t,x) \right) dx \\&\quad = \int _0^{\tau \wedge t} I_{1,s,n,\varepsilon } - I_{2,s,n,\varepsilon } + I_{3,s,n,\varepsilon } + I_{4,s,n,\varepsilon } - I_{5,s,n,\varepsilon }\, ds + M_{\tau \wedge t,\varepsilon }(n)\\&\quad \le \int _0^{\tau \wedge t} I_{1,s,n,\varepsilon } + \left| I_{2,s,n,\varepsilon }\right| + I_{3,s,n,\varepsilon } + I_{4,s,n,\varepsilon } \\&\qquad - I_{5,s,n,\varepsilon }\, ds + M_{\tau \wedge t,\varepsilon }(n), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} I_{1,s,n,\varepsilon }&= e^{-\gamma Ks}\int _{\mathbb {R}} r_n'\left( u^{(\varepsilon )}(s,x)\right) a(s)(\Delta ^{\alpha /2}u(s,\cdot ))^{(\varepsilon )}(x) dx, \\ I_{2,s,n,\varepsilon }&= e^{-\gamma Ks}\int _{\mathbb {R}}r'_n\left( u^{(\varepsilon )}(s,x)\right) \int _\mathbb {R}u(s,y)\left( b(s,y)\zeta _\varepsilon (x-y) \right) _y dydx \\ I_{3,s,n,\varepsilon }&= e^{-\gamma Ks}\int _{\mathbb {R}} r_n'\left( u^{(\varepsilon )}(s,x) \right) \left( (c(s,\cdot ) u(s,\cdot ))^{(\varepsilon )}(x) + f^{(\varepsilon )}(s,x)\right) dx \\ I_{4,s,n,\varepsilon }&= \frac{1}{2}e^{-\gamma Ks}\int _{\mathbb {R}}r_n''\left( u^{(\varepsilon )}(s,x)\right) \left| (\nu ^k(s,\cdot ) u(s,\cdot ))^{(\varepsilon )}(x) + g^{(\varepsilon )}(s,x) \right| _{\ell _2}^2 dx, \\ I_{5,s,n,\varepsilon }&= \gamma Ke^{-\gamma Ks}\int _{\mathbb {R}}r_n\left( u^{(\varepsilon )}(s,x)\right) dx,\\ M_{t,n,\varepsilon }&= \int _0^t e^{-\gamma Ks}\int _{\mathbb {R}} r_n'\left( u^{(\varepsilon )}(s,x) \right) \left( (\nu ^{k}(s,\cdot ) u(s,\cdot ))^{(\varepsilon )}(x) + g^{k(\varepsilon )}(s,x) \right) dxdw_s^k, \end{aligned} \end{aligned}$$

and \(\gamma \in (0,\infty )\) will be specified later. To deal with \(I_{2,s,n,\varepsilon }\), observe that

$$\begin{aligned} \begin{aligned}&\left| \left( b(s,\cdot )u(s,\cdot )\right) _x^{(\varepsilon )}(x) - b(s,x)u_x^{(\varepsilon )}(s,x) \right| \\&\quad = \frac{1}{\varepsilon }\left| \int _\mathbb {R}( b(s,x-\varepsilon y) - b(s,x))u(s,x-\varepsilon y)\zeta '(y)dy \right| \\&\quad \le N_1 \int _\mathbb {R}|u(s,x-\varepsilon y)||y\zeta '(y)|dy, \end{aligned} \end{aligned}$$
(B.4)

where \(N_1 = N_1(K)\). The last inequality follows from the fact that b is continuously differentiable with respect to x. Thus, by (B.4) and integration by parts, we have

$$\begin{aligned} \begin{aligned}&\left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( b(s,\cdot )u(s,\cdot ) \right) _x^{(\varepsilon )}(x)dx \right| \\&\quad \le \left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( \left( b(s,\cdot )u(s,\cdot ) \right) _x^{(\varepsilon )}(x) - b(s,x)u^{(\varepsilon )}_x(s,x)\right) dx \right| \\&\qquad + \left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( u^{(\varepsilon )}(s,x)\right) _xb(s,x)dx \right| \\&\quad \le N_1\int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \int _\mathbb {R}|u(s,x-\varepsilon y)||y\zeta '(y)|dydx \\ {}&+ \left| \int _\mathbb {R}r_n\left( u^{(\varepsilon )}(s,x)\right) b_{x}(s,x)dx \right| . \end{aligned} \end{aligned}$$
(B.5)

Therefore, by (B.5), we have

$$\begin{aligned}&\left| \int _{\mathbb {R}}r'_n\left( u^{(\varepsilon )}(s,x)\right) \int _\mathbb {R}u(s,y)\left( b(s,y)\zeta _\varepsilon (x-y) \right) _y dydx \right| \\&\quad = \left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( b_y(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}(x)dx\right. \\&\left. - \int _{\mathbb {R}}r'_n\left( u^{(\varepsilon )}(s,x)\right) \left( b(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}_x(x)dx \right| \\&\quad = \left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( b_y(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}(x)dx \right| \\&+ \left| \int _{\mathbb {R}}r'_n\left( u^{(\varepsilon )}(s,x)\right) \left( b(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}_x(x)dx \right| \\&\quad \le \left| \int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \left( b_y(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}(x)dx \right| \\&\qquad + N_1\int _\mathbb {R}r_n'\left( u^{(\varepsilon )}(s,x)\right) \int _\mathbb {R}|u(s,x-\varepsilon y)||y\zeta '(y)|dydx \\&+ \int _\mathbb {R}r_n\left( u^{(\varepsilon )}(s,x)\right) |b_{x}(s,x)|dx. \end{aligned}$$

Now, notice that

$$\begin{aligned} \begin{aligned}&I_{1,s,\varepsilon }:=\lim _{n\rightarrow \infty } I_{1,s,n,\varepsilon } = 2 e^{-\gamma Ks}\int _{\mathbb {R}} \left( u^{(\varepsilon )}(s,x)\right) ^+ a(\Delta ^{\alpha /2}u(s,\cdot ))^{(\varepsilon )}(x) dx, \\&I_{2,s,\varepsilon }:= \lim _{n\rightarrow \infty } I_{2,s,n,\varepsilon } \le \int _\mathbb {R}\left( u^{(\varepsilon )}(s,x)\right) ^+\left| \left( b_y(s,\cdot )u(s,\cdot ) \right) ^{(\varepsilon )}(x)\right| dx \\&\qquad + N_1\int _\mathbb {R}\left( u^{(\varepsilon )}(s,x)\right) ^+\int _\mathbb {R}|u(s,x-\varepsilon y)||y\zeta '(y)|dydx \\&\qquad + \int _\mathbb {R}\left( \left( u^{(\varepsilon )}(s,x)\right) ^+\right) ^2 |b_{x}(s,x)|dx\\&I_{3,s,\varepsilon }:= \lim _{n\rightarrow \infty }I_{3,s,n,\varepsilon } = 2 e^{-\gamma Ks}\int _{\mathbb {R}} \left( u^{(\varepsilon )}(s,x) \right) ^+\left( (c(s,\cdot ) u(s,\cdot ))^{(\varepsilon )}(x) + f^{(\varepsilon )}(s,x)\right) dx,\\&I_{4,s,\varepsilon }:=\lim _{n\rightarrow \infty } I_{4,s,n,\varepsilon } = e^{-\gamma Ks} \int _{\mathbb {R}} 1_{u^{(\varepsilon )}(s,x)\ge 0} \left| (\nu (s,\cdot )u(s,\cdot ))^{(\varepsilon )}(x) + g^{(\varepsilon )}(s,x)\right| _{\ell _2}^2 dx, \\&I_{5,s,\varepsilon }:=\lim _{n\rightarrow \infty } I_{5,s,n,\varepsilon } = \gamma Ke^{-\gamma Ks}\int _{\mathbb {R}}\left| \left( u^{(\varepsilon )}(s,x)\right) ^+ \right| ^2 dx,\\ \end{aligned} \end{aligned}$$

and \(M_{t,n,\varepsilon }\) converges to

$$\begin{aligned} M_{t,\varepsilon }:= 2\int _0^t \int _{\mathbb {R}}e^{-Ks}\left( u^{(\varepsilon )}(s,x)\right) ^+\left( (\nu ^{k}(s,\cdot )u(s,\cdot ))^{(\varepsilon )}(x) + g^{k(\varepsilon )}(s,x)\right) dxdw_s^k \end{aligned}$$

uniformly in t on a finite time interval in probability. Note that for each \(s\in (0,\tau \wedge t)\), we have \(\left( u^{(\varepsilon )}(s,\cdot )\right) ^+\in H_2^1\) (e.g. Exercise 1.3.18 of [12]). Since \(\left( \Delta ^{\alpha /2} u(s,\cdot ) \right) ^{(\varepsilon )}(x) = \Delta ^{\alpha /2} u^{(\varepsilon )}(s,x)\), by (B.2), for any \(\varepsilon >0\),

$$\begin{aligned} \begin{aligned}&a(s)\int _{\mathbb {R}}\left( u^{(\varepsilon )}(s,x)\right) ^+ \Delta ^{\alpha /2}u^{(\varepsilon )}(s,x)dx \\&\quad = -a(s)\int _{\mathbb {R}}\int _{\mathbb {R}} \frac{\left( \left( u^{(\varepsilon )}(s,x)\right) ^+ - \left( u^{(\varepsilon )}(s,y)\right) ^+\right) \left( u^{(\varepsilon )}(s,x) - u^{(\varepsilon )}(s,y)\right) }{|x-y|^{1+\alpha }} dxdy. \end{aligned} \end{aligned}$$

Notice that

$$\begin{aligned} \begin{aligned}&\left( \left( u^{(\varepsilon )}(s,x)\right) ^+ - \left( u^{(\varepsilon )}(s,y)\right) ^+\right) (u^{(\varepsilon )}(s,x) - u^{(\varepsilon )}(s,y)) \\&\quad = {\left\{ \begin{array}{ll} \left( u^{(\varepsilon )}(s,x) - u^{(\varepsilon )}(s,y)\right) ^2\quad &{}\text{ if } \quad u^{(\varepsilon )}(s,x)\ge 0,\,u^{(\varepsilon )}(s,y)\ge 0, \\ u^{(\varepsilon )}(s,x)\left( u^{(\varepsilon )}(s,x) - u^{(\varepsilon )}(s,y)\right) \quad &{}\text{ if } \quad u^{(\varepsilon )}(s,x)\ge 0,\,u^{(\varepsilon )}(s,y)<0, \\ u^{(\varepsilon )}(s,y)\left( u^{(\varepsilon )}(s,y) - u^{(\varepsilon )}(s,x)\right) \quad &{}\text{ if } \quad u^{(\varepsilon )}(s,x)<0,\,u^{(\varepsilon )}(s,y)\ge 0, \\ 0\quad &{}\text{ if } \quad u^{(\varepsilon )}(s,x)<0,\,u^{(\varepsilon )}(s,y)<0. \end{array}\right. } \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned} -a(s)\int _{\mathbb {R}}\left( u^{(\varepsilon )}(s,x)\right) ^+ \Delta ^{\alpha /2}u^{(\varepsilon )}(s,x)dx \le 0. \end{aligned}$$

Observe that

$$\begin{aligned} \begin{aligned}&I_{2,s}:= \lim _{\varepsilon \downarrow 0}I_{2,s,\varepsilon } \le N_2\int _\mathbb {R}|u^+(s,x)|^2dx \\&I_{3,s}:= \lim _{\varepsilon \downarrow 0}I_{3,s,\varepsilon } = 2 e^{-\gamma Ks}\int _{\mathbb {R}} u^+(s,x)\left( c(s,x) u(s,x) + f(s,x)\right) dx,\\&I_{4,s}:= \lim _{\varepsilon \downarrow 0} I_{4,s,\varepsilon } = e^{-\gamma Ks} \int _{\mathbb {R}} 1_{u(s,x)\ge 0} |\nu (s,x) u(s,x) + g(s,x)|_{\ell _2}^2 dx, \\&I_{5,s}:= \lim _{\varepsilon \downarrow 0}I_{5,s,\varepsilon } = \gamma Ke^{-\gamma Ks}\int _{\mathbb {R}}\left| u^+(s,x) \right| ^2 dx,\\ \end{aligned} \end{aligned}$$

where \(N_2 = 2\sup _{t,x}|b_x(t,x)| + N_1\int _\mathbb {R}|y\zeta '(y)|dy\). Also, note that \(M_{t,\varepsilon }\) converges to

$$\begin{aligned} M_{t}:= \lim _{\varepsilon \downarrow 0}M_{t,\varepsilon } = 2\int _0^t \int _{\mathbb {R}^d}e^{-\gamma Ks}u^+(s,x)\left( \nu ^k(s,x) u(s,x) + g^{k}(s,x)\right) dxdw_s^k \end{aligned}$$

uniformly in t on a finite time interval in probability. Thus, if we choose \(\gamma > N_2 + 4K\) at the beginning of the proof, we have

$$\begin{aligned} \begin{aligned} \mathbb {E}e^{-\gamma K(\tau \wedge t)}\int _{\mathbb {R}^d}\left| u^+(\tau \wedge t,x) \right| ^2dx \le I_{2,s} + I_{3,s} + I_{4,s} - I_{5,s}\le 0. \end{aligned} \end{aligned}$$

Thus, we have \(u^+(\tau \wedge t,x) = 0\). The theorem is proved. \(\square \)

Proof of Theorem 5.2

Even though the proof is similar to [5, Theorem 4.2], we prove this theorem for the completeness of the paper. As in the proof of Theorem 4.3, we may assume that \(\tau = T\). Take a sequence of functions \(\{u^m_0 \in U_p^{\gamma ,\alpha }:u^m_0\ge 0,m=1,2,\dots , \gamma = 1,2,\dots \}\) such that \(u^m_0 \rightarrow u_0\) in \(U^{\alpha /2-1/2-\kappa }_{p}\), and \(u^m_0\) has compact supports with respect to x. For \(m=1,2,\cdots \), let

$$\begin{aligned}&\varvec{\eta }=(\eta _1, \eta _2, \cdots ), \quad \varvec{\eta }_m=(\eta _1,\eta _2,\cdots ,\eta _m,0,0,\cdots ),\\&g(u)=\xi u\varvec{\eta }=(\xi u \eta _1, \xi u\eta _2, \cdots ), \quad g_m(u)\\&\quad =\xi u\varvec{\eta }_m=(\xi u \eta _1, \xi u\eta _2, \cdots ,\xi u \eta _m,0,0,\cdots ). \end{aligned}$$

Then, for any \(u,v\in H^{\alpha /2-1/2-\kappa }_{p}\),

$$\begin{aligned} \Vert g_m(u)-g_m(v)\Vert _{H^{-1/2-\kappa }_{p}(\ell _2)} \le \Vert g(u)-g(v)\Vert _{H^{-1/2-\kappa }_{p}(\ell _2)}. \end{aligned}$$
(B.6)

Thus, all the assumptions in Theorem 4.3 hold with \(g_m(u)\) instead of g(u). Therefore, there exists a unique solution \(u_m \in \mathcal {H}_{p}^{\alpha /2-1/2-\kappa }(T)\) to equation

$$\begin{aligned} du_m = \left( a\Delta ^{\alpha /2} u_m + bu_{mx} + cu_m\right) \,dt+\sum _{k=1}^{\infty } g^{k}_m (u_m)dw^k_t, \,\, 0<t\le T\, ; \quad v(0,\cdot )=u^m_0.\nonumber \\ \end{aligned}$$
(B.7)

Thus, it suffices to show \(u_m \rightarrow u\) in \(\mathcal {H}_{p}^{\alpha /2-1/2-\kappa }(T)\) and \(u_m \ge 0\).

Note that \(v_m:=u-u_m\) satisfies

$$\begin{aligned} \begin{aligned} dv_m =&\left( a\Delta ^{\alpha /2} v_m + bv_{mx} + cv_m \right) \,dt\\&+\sum _{k=1}^{\infty } \left( g^k(u)-g^{k}_m (u_m)\right) dw^k_t, \quad 0<t\le T\, ; \quad v(0,\cdot )=u_0-u^m_0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} g(u)-g_{m}(u_m)=g_m(u)-g_m(u_m)+ \xi u ( \varvec{\eta }-\varvec{\eta }_m) . \end{aligned}$$

Thus, by (B.6), (4.6), and Theorem 4.3 with \(\tau =t\), we have, for any \(t \le T\),

$$\begin{aligned} \begin{aligned}&\Vert u-u_m \Vert ^p_{\mathcal {H}_{p}^{\alpha /2-1/2-\kappa }(t)} \\&\quad \le N(\Vert u_0-u^m_0\Vert ^p_{U^{\alpha /2-1/2-\kappa ,\alpha }_{p}} + N\Vert g_m(u)-g_m(u_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(t,\ell _2)} \\&\qquad + N\Vert \xi u ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(t,\ell _2)}\\&\quad \le N\Vert u_0-u^m_0\Vert ^p_{U^{\alpha /2-1/2-\kappa ,\alpha }_{p}}+ \frac{1}{2}\Vert u-u_m \Vert ^p_{\mathbb {H}^{\alpha /2-1/2-\kappa }_{p}(t)}+N\Vert u-u_m \Vert ^p_{\mathbb {H}^{-\alpha /2-1/2-\kappa }_{p}(t)} \\&\qquad + N \Vert \xi u ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(t,\ell _2)}. \end{aligned} \end{aligned}$$

Therefore, for any \(t\le T\),

$$\begin{aligned} \begin{aligned}&\Vert u-u_m \Vert ^p_{\mathcal {H}_{p}^{\alpha /2-1/2-\kappa }(t)} \\&\quad \le N\Vert u_0-u^m_0\Vert ^p_{U^{\alpha /2-1/2-\kappa ,\alpha }_{p}}+ N\Vert u-u_m \Vert ^p_{\mathbb {H}^{-\alpha /2-1/2-\kappa }_{p}(t)} \\&\qquad +N \Vert \xi u ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(T,\ell _2)}. \end{aligned} \end{aligned}$$
(B.8)

Inequalities (B.8) and (2.12), and Gronwall’s inequality imply

$$\begin{aligned} \Vert u-u_m\Vert ^p_{\mathcal {H}_{p}^{\alpha /2-1/2-\kappa }(T)}\le N\Vert u_0-u^m_0\Vert ^p_{U^{\alpha /2-1/2-\kappa ,\alpha }_{p}}+ N \Vert \xi u ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(T,\ell _2)},\nonumber \\ \end{aligned}$$
(B.9)

where N is independent of m and u. By (4.7) and (5.2), we have

$$\begin{aligned} \begin{aligned} \Vert \xi (t) u(t) ( \varvec{\eta }-&\varvec{\eta }_m)\Vert ^p_{H_{p}^{-1/2-\kappa }(\ell _2)} \le N \int _{\mathbb {R}^d} \left( \sum _{k>m} \left[ \int _{\mathbb {R}^d}{R_{1/2+\kappa }(x-y)} u(t,y) \eta _k(y) dy \right] ^2 \right) ^{p/2} dx. \end{aligned} \end{aligned}$$

By calculations in (4.8) and the dominated convergence theorem, we have

$$\begin{aligned} \Vert \xi (t) u(t) ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{H_{p}^{-1/2-\kappa }(\ell _2)} \rightarrow 0 \quad \text {as}\quad m\rightarrow \infty \end{aligned}$$

for almost all \((\omega ,t)\). Then, by using the dominated convergence theorem again, we have

$$\begin{aligned} \Vert \xi u ( \varvec{\eta }-\varvec{\eta }_m)\Vert ^p_{\mathbb {H}_{p}^{-1/2-\kappa }(T,\ell _2)} \rightarrow 0 \quad \text {as}\quad m\rightarrow \infty . \end{aligned}$$

Now we show \(u_m \ge 0\). For \(\varepsilon >0\), set

$$\begin{aligned} g_{m,\varepsilon }(u) = \left( \xi {\varvec{\eta }}_m \right) ^{(\varepsilon )} u = \left( (\xi \eta _1)^{(\varepsilon )}u,(\xi \eta _2)^{(\varepsilon )}u,\cdots ,(\xi \eta _m)^{(\varepsilon )}u,0,0,\dots \right) . \end{aligned}$$

By Theorem 4.3, there exists \(u_{m,\varepsilon }\in \mathcal {H}_2^{2}(T)\) such that \(u_{m,\varepsilon }\) satisfies

$$\begin{aligned} d u_{m,\varepsilon } = \left( a\Delta ^{\alpha /2} u_{m,\varepsilon } + b u_{mx} + cu_{m,\varepsilon }\right) \,dt+\sum _{k=1}^{m} g_{m,\varepsilon }^k(u_{m,\varepsilon })dw^k_t, \,\, 0<t\le T\nonumber \\ \end{aligned}$$
(B.10)

with initial data \(u_{m,\varepsilon }(0,\cdot ) = u_0^m\). By Theorem B.2, we have \(u_{m,\varepsilon }\ge 0\). On the other hand, by following proof of Theorem 3.3, we have \(u_{m,\varepsilon }\in \mathcal {H}_p^{\alpha /2-1/2-\kappa }(T)\). In addition, note that

$$\begin{aligned} d (u_m - u_{m,\varepsilon })&= \left( a\Delta ^{\alpha /2} (u_m - u_{m,\varepsilon }) + b (u_m - u_{m,\varepsilon })_x + c(u_m - u_{m,\varepsilon })\right) \,dt\\&\quad + \sum _{k=1}^{m} \left( g_{m}^k(u_{m}) - g_{m,\varepsilon }^k(u_{m,\varepsilon })\right) dw^k_t, \quad 0<t\le T. \end{aligned}$$

Thus, Theorem 3.7 and triangle inequality imply that for \(t\le T\)

$$\begin{aligned} \begin{aligned}&\Vert u_m - u_{m,\varepsilon } \Vert _{\mathcal {H}_p^{\alpha /2-1/2-\kappa }(t)}^p \\&\quad \le N\Vert g_{m}(u_{m}) - g_{m,\varepsilon }(u_{m,\varepsilon }) \Vert ^p_{\mathbb {H}_p^{-1/2-\kappa }(t,\ell _2)} \\&\quad \le N\left\| g_{m}(u_{m}) - g_{m,\varepsilon }(u_{m}) \right\| ^p_{\mathbb {H}_p^{-1/2-\kappa }(t,\ell _2)} + \left\| g_{m,\varepsilon }(u_{m}) - g_{m,\varepsilon }(u_{m,\varepsilon }) \right\| ^p_{\mathbb {H}_p^{-1/2-\kappa }(t,\ell _2)} \\ \end{aligned} \end{aligned}$$
(B.11)

As in (4.9), we have

$$\begin{aligned} \left\| g_{m}(u_{m}) - g_{m,\varepsilon }(u_{m}) \right\| ^p_{\mathbb {H}_p^{-1/2-\kappa }(t,\ell _2)} \le N\sum _{i=1}^m\left\| \left[ \left( \xi \eta _i \right) ^{(\varepsilon )} - \xi \eta _i\right] u_m \right\| ^p_{\mathbb {L}_p(t)}.\nonumber \\ \end{aligned}$$
(B.12)

By (4.9) and Theorem 2.2 (iii), we get

$$\begin{aligned} \begin{aligned}&\left\| g_{m,\varepsilon }(u_{m}) - g_{m,\varepsilon }(u_{m,\varepsilon }) \right\| ^p_{\mathbb {H}_p^{-1/2-\kappa }(t,\ell _2)} \\&\quad \le N \mathbb {E}\int _0^t \left\| u_m(s,\cdot ) - u_{m,\varepsilon }(s,\cdot ) \right\| ^p_{L_p}ds\\&\quad \le N \int _0^t \mathbb {E}\sup _{r\le s} \left\| u_m(r,\cdot ) - u_{m,\varepsilon }(r,\cdot ) \right\| ^p_{L_p}ds\\&\quad \le N\int _0^t\Vert u_m - u_{m,\varepsilon } \Vert ^p_{\mathcal {H}^{\alpha /2-1/2-\kappa }_p(s)}ds, \end{aligned} \end{aligned}$$
(B.13)

where N is indepdent of \(\varepsilon \). By applying (B.12) and (B.13) to (B.11), we have

$$\begin{aligned} \begin{aligned}&\Vert u_m - u_{m,\varepsilon } \Vert _{\mathcal {H}_p^{\alpha /2-1/2-\kappa }(t)}^p \\&\quad \le N\sum _{i=1}^m\left\| \left[ \left( \xi \eta _i \right) ^{(\varepsilon )} - \xi \eta _i\right] u_m \right\| ^p_{\mathbb {L}_p(T)} + N\int _0^t\Vert u_m - u_{m,\varepsilon } \Vert ^p_{\mathcal {H}^{\alpha /2-1/2-\kappa }_p(s)}ds. \end{aligned} \end{aligned}$$

By Gronwall’s inequality, we have

$$\begin{aligned} \begin{aligned}&\Vert u_m - u_{m,\varepsilon } \Vert _{\mathcal {H}_p^{\alpha /2-1/2-\kappa }(T)}^p \le N\sum _{i=1}^m\left\| \left[ \left( \xi \eta _i \right) ^{(\varepsilon )} - \xi \eta _i\right] u_m \right\| ^p_{\mathbb {L}_p(T)}, \end{aligned} \end{aligned}$$

where N is independent of \(\varepsilon \). Therefore, by dominated convergence theorem, we have \(u_{m,\varepsilon }\rightarrow u_m\) in \(\mathcal {H}_p^{\alpha /2-1/2-\kappa }(T)\) as \(\varepsilon \rightarrow 0\). The theorem is proved. \(\square \)

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Han, BS. A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians. Stoch PDE: Anal Comp 9, 940–983 (2021). https://doi.org/10.1007/s40072-021-00189-8

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