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:
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
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References
Azerad, P., Mellouk, M.: On a stochastic partial differential equation with nonlocal diffusion. Potential Anal. 27(2), 183–197 (2007)
Burdzy, K., Mueller, C., Perkins, E.: Nonuniqueness for nonnegative solutions of parabolic stochastic partial differential equations. Ill. J. Math. 54(4), 1481–1507 (2010)
Gomez, A., Lee, K., Mueller, C., Wei, A., Xiong, J.: Strong uniqueness for an SPDE via backward doubly stochastic differential equations. Stat. Probab. Lett. 83(10), 2186–2190 (2013)
Grafakos, L.: Modern Fourier Analysis, vol. 250. Springer, Berlin (2009)
Han, B., Kim, K.: Boundary behavior and interior Hölder regularity of solution to nonlinear stochastic partial differential equations driven by space-time white noise. Differ. Equ. 269(11), 9904–9935 (2020)
Kim, I., Kim, K.: An \(L_p\)-theory for stochastic partial differential equations driven by Lévy processes with pseudo-differential operators of arbitrary order. Stoch. Process. Their Appl. 126(9), 2761–2786 (2016)
Kim, I., Kim, K., Lim, S.: A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives. Ann. Probab. 47(4), 2087–2139 (2019)
Kim, K., Kim, P.: An \(L_p\)-theory of a class of stochastic equations with the random fractional Laplacian driven by Lévy processes. Stoch. Process. Their Appl. 122(12), 3921–3952 (2012)
Kim, K., Lim, S.: Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations. J. Korean Math. Soc. 53(4), 929–967 (2016)
Krylov, N.: An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives: (1999)
Krylov, N.: Introduction to the Theory of Diffusion Processes. AMS, Providence (1995)
Krylov, N.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, vol. 96. American Mathematical Society, Cambridge (2008)
Krylov, N.: Maximum principle for SPDEs and its applications. In: Rozovskii, B.L. (ed.) Stochastic Differential Equations: Theory and Applications: A Volume in Honor of Professor, pp. 311–338. World Scientific, Singapore (2007)
Krylov, N.: On a result of C. Mueller and E. Perkins. Probab. Theory Relat. Fields 108(4), 543–557 (1997)
Krylov, N.: On SPDE’s and superdiffusions. Ann. Probab. 25, 1789–1809 (1997)
Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)
Neervan, J., Veraar, M., Weis, L.: Maximal \(L_p\)-regularity for stochastic evolution equations. SIAM J. Math. Anal. 44(3), 1372–1414 (2012)
Mikulevicius, R., Pragarauskas, H.: On \(L_p\) theory for Zakai equation with discontinuous observation process (2010). arXiv preprint arXiv:1012.5816
Mikulevicius, R., Pragarauskas, H.: On Hölder solutions of the integro-differential Zakai equation. Stoch. Process. Their Appl. 119(10), 3319–3355 (2009)
Mueller, C.: Long time existence for the heat equation with a noise term. Probab. Theory Relat. Fields 90(4), 505–517 (1991)
Mueller, C.: The critical parameter for the heat equation with a noise term to blow up in finite time. Ann. Probab. 28, 1735–1746 (2000)
Mueller, C., Mytnik, L., Perkins, E.: Nonuniqueness for a parabolic SPDE with \(\frac{3}{4}-\varepsilon \)-Hölder diffusion coefficients. Ann. Probab. 42(5), 2032–2112 (2014)
Mytnik, L.: Weak uniqueness for the heat equation with noise. Ann. Probab. 26, 968–984 (1998)
Mytnik, L., Perkins, E.: Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. Probab. Theory Relat. Fields 149(1–2), 1–96 (2011)
Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)
Walsh, J.: An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour XIV-1984. pp. 265–439. Springer, Berlin (1986)
Xiong, J.: Super-Brownian motion as the unique strong solution to an SPDE. Ann. Probab. 41(2), 1030–1054 (2013)
Zhang, X.: \(L_p\)-solvability of nonlocal parabolic equations with spatial dependent and non-smooth kernels (2012). arXiv preprint arXiv:1206.2709
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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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;
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(t, x) (e.g. [9, Theorem 2.1]) and the solution of (A.1) is given by
It turns out that the fundamental solution p(t, x) satisfies the following properties.
Lemma A.1
-
(i)
There exists a fundamental solution p(t, x) such that
$$\begin{aligned} p(t,\cdot )\in L_1\left( \mathbb {R}^d \right) . \end{aligned}$$ -
(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) -
(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
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
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
For \(\theta \in (0,1]\), by the classical multiplier theorem (e.g. Theorems 1.3.6 and 1.3.8 of [4]), we have
Since \(e^{-t}\le Nt^{-\theta }\wedge 1\), we have
Inequality (A.6) implies that
For the second term of the right hand side in (A.7), by using Minköwski’s inequality, we have
By combining (A.9) and (A.10), we have
Thus, by applying (A.11) and (A.8) to (A.7), we have
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
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
For \(t\ge 1\), observe that
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
From (A.13), we have
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
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
Then, by the deterministic version of [8, Theorem 2.10], we have
Since [8, Theorem 2.10] implies that
we have
Therefore, we only need to show
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
and
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
Observing that
yields
where
By inequality (A.13), we have
where
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
where \(N = N(T,p)\). Thus, we have
The last inequality follows from \(\mu <\nu \). In the case of \(I_1\), by the above result and Fubini’s theorem, we have
Let \(u_2:=u - u_1\). Notice that
where
By inequality (A.13),
where
By Burkholder-Davis-Gundy inequalities, \(\frac{(2\nu -1)p}{p-2}>-1\), \(\sigma <T\), and Lemma A.2, we obtain
where \(N = N(T,p)\). Thus,
Similarly, for \(I_2\), we have
Therefore, we have
The theorem is proved.
B. Maximum principle
In this section, we introduce the maximum principle for an SPDE with the random fractional Laplacian. Consider
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
-
(i)
The coefficients \(a = a(t)\) is predictable and nonnegative.
-
(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.
-
(iii)
The coefficient \(b = b(t,x)\) is continuously differentiable with respect to x.
-
(iv)
The function \(f = f(t,\cdot )\) is a \(L_2\)-valued \(\mathcal {F}_t\)-adapted, jointely measurable process.
-
(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.
-
(vi)
The function \(u = u(t,\cdot )\) is a \(L_2\)-valued \(\mathcal {F}_t\)-adapted, jointely measurable process satisfying Eq. (B.1).
-
(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 \(\omega \in \Omega \), \(k = 1,2,\dots \),
t-almost everywhere on \((0,\tau )\). If \(u_{0}\le 0\), then almost surely
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 \(g\in H_2^{\alpha /2}\),
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.
-
(i)
The function r(z) is a real-valued function on \(\mathbb {R}\).
-
(ii)
The function r(z) is continuously differentiable and \(r(0) = 0\).
-
(iii)
The function \(r'(z)\) is absolutely continuous and \(r'(0) = 0\).
-
(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
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
By Itô’s formula, we have
where
and \(\gamma \in (0,\infty )\) will be specified later. To deal with \(I_{2,s,n,\varepsilon }\), observe that
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
Therefore, by (B.5), we have
Now, notice that
and \(M_{t,n,\varepsilon }\) converges to
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 \(\varepsilon >0\),
Notice that
Therefore, we have
Observe that
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
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
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
Then, for any \(u,v\in H^{\alpha /2-1/2-\kappa }_{p}\),
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
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
and
Thus, by (B.6), (4.6), and Theorem 4.3 with \(\tau =t\), we have, for any \(t \le T\),
Therefore, for any \(t\le T\),
Inequalities (B.8) and (2.12), and Gronwall’s inequality imply
where N is independent of m and u. By (4.7) and (5.2), we have
By calculations in (4.8) and the dominated convergence theorem, we have
for almost all \((\omega ,t)\). Then, by using the dominated convergence theorem again, we have
Now we show \(u_m \ge 0\). For \(\varepsilon >0\), set
By Theorem 4.3, there exists \(u_{m,\varepsilon }\in \mathcal {H}_2^{2}(T)\) such that \(u_{m,\varepsilon }\) satisfies
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
Thus, Theorem 3.7 and triangle inequality imply that for \(t\le T\)
As in (4.9), we have
By (4.9) and Theorem 2.2 (iii), we get
where N is indepdent of \(\varepsilon \). By applying (B.12) and (B.13) to (B.11), we have
By Gronwall’s inequality, we have
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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DOI: https://doi.org/10.1007/s40072-021-00189-8
Keywords
- Stochastic partial differential equation
- Nonlinear
- Fractional Laplacian
- Space-time white noise
- Multiplicative noise
- Hölder regularity