Abstract
We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly nonlinear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong–Zakai-type approximation. After a coordinate transformation, the problems are reformulated and analyzed in terms of stochastic evolution equations on domains of fractional powers of linear operators.
Similar content being viewed by others
References
H. Amann. Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J. Math. Anal. Appl., 65(2):432–467, 1978.
J. Appell and P. P. Zabrejko. Nonlinear superposition operators, volume 95 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990.
R. Cont and M. S. Müller. A stochastic PDE model for limit order book dynamics. arXiv preprint arXiv:1904.03058, 2019.
G. Da Prato, S. Kwapień, and J. Zabczyk. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics, 23(1):1–23, 1987.
G. Da Prato and J. Zabczyk. A note on stochastic convolution. Stochastic Analysis and Applications, 10(2):143–153, 1992.
G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1992.
K. Deimling. Ordinary differential equations in Banach spaces. Lecture Notes in Mathematics, Vol. 596. Springer-Verlag, Berlin-New York, 1977.
N. Dunford and J. T. Schwartz. Linear operators. Part II. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication.
K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.
D. Filipović, S. Tappe, and J. Teichmann. Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financial Math., 1(1):523–554, 2010.
P. Grisvard. Caractérisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal., 25:40–63, 1967.
B. Hambly, J. Kalsi, and J. Newbury. Limit order books, diffusion approximations and reflected spdes: from microscopic to macroscopic models. arXiv preprint arXiv:1808.07107, 2018.
D. Henry. Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1981.
N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.
W. Jachimiak. A note on invariance for semilinear differential equations. Bull. Polish Acad. Sci. Math., 45(2):181–185, 1997.
O. Kallenberg. Foundations of Modern Probability. Applied probability. Springer, 2002.
M. Keller-Ressel and M. S. Müller. A Stefan-type stochastic moving boundary problem. Stochastics and Partial Differential Equations: Analysis and Computations, 4(4):746–790, 2016.
M. Kunze and J. van Neerven. Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations. J. Differential Equations, 253(3):1036–1068, 2012.
J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications - 1. Springer, Springer, 1972.
A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Basel, 1995.
A. Lunardi. Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, second edition, 2009.
A. Milian. Comparison theorems for stochastic evolution equations. Stoch. Stoch. Rep., 72(1-2):79–108, 2002.
M. S. Müller. Semilinear stochastic moving boundary problems. Doctoral thesis, TU Dresden, 2016.
M. S. Müller. A stochastic Stefan-type problem under first-order boundary conditions. The Annals of Applied Probability, 28(4):2335–2369, 2018.
M. S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete and Continuous Dynamical Systems - B, 24(8):4317–4339, 2019.
T. Nakayama. Support theorem for mild solutions of SDE’s in Hilbert spaces. J. Math. Sci. Univ. Tokyo, 11(3):245–311, 2004.
T. Nakayama. Viability theorem for SPDE’s including HJM framework. J. Math. Sci. Univ. Tokyo, 11(3):313–324, 2004.
N. Pavel. Invariant sets for a class of semi-linear equations of evolution. Nonlinear Anal., 1(2):187–196, 1976/77.
N. Pavel. Differential equations, flow invariance and applications, volume 113. Pitman Pub., 1984.
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Number 44 in Applied Mathematical Sciences. Springer, 1992.
J. Prüss. On semilinear parabolic evolution equations on closed sets. J. Math. Anal. Appl., 77(2):513–538, 1980.
M. Sauer and W. Stannat. Analysis and approximation of stochastic nerve axon equations. Mathematics of Computation, 2016.
J. Stefan. Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Wien. Ber. XCVIII, Abt. 2a (965–983), 1888.
D. W. Stroock and S. R. S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 333–359. Univ. California Press, Berkeley, Calif., 1972.
G. Tessitore and J. Zabczyk. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ., 6(4):621–655, 2006.
K. Twardowska. Wong-Zakai approximations for stochastic differential equations. Acta Appl. Math., 43(3):317–359, 1996.
T. Valent. Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, volume 31 of Springer Tracts in Natural Philosophy. Springer New York, 1988.
T. Valent. A property of multiplication in Sobolev spaces. Some applications. Rend. Sem. Mat. Univ. Padova, 74:63–73, 1985.
J. M. A. M. van Neerven, M. C. Veraar, and L. Weis. Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal., 255(4):940–993, 2008.
E. Wong and M. Zakai. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci., 3:213–229, 1965.
J. Zabczyk. Stochastic invariance and consistency of financial models. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11(2):67–80, 2000.
Z. Zheng. STOCHASTIC STEFAN PROBLEMS: EXISTENCE, UNIQUENESS, AND MODELING OF MARKET LIMIT ORDERS. PhD thesis, Graduate College of the University of Illinois at Urbana-Champaign, 2012.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Both authors acknowledge support by the German Research Foundation (DFG) under the Grant ZUK 64. MM also acknowledges support by the Swiss National Science Foundation through Grant SNF \(205121\_163425\) at ETH Zürich. Most of the work of MM was carried out within the scope of his dissertation [24]
Appendices
A Nemytskii operators on Sobolev spaces
We continue with the analysis on the Sobolev spaces \(H^k({\mathbb {R}}_+)\), \(k\in {\mathbb {N}}\). In this section we prove some regularity results on the nonlinear Nemytskii operator
where, \(\mu :{\mathbb {R}}_+ \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) and \(x\in {\mathbb {R}}_+\). Note that these operators are well understood but most of the literature focuses on bounded domains, see, e.g., [2, 38, 39]. However, in the case of unbounded domains several additional conditions on \(\mu \) are necessary to make them work. First, we state a result on the spaces \(H^k\) which guarantees that, under certain assumptions on \(\mu \), N will map \(H^k\) into \(H^k\). For a proof, we refer to [39, Theorem 1], of which it is a special case.
Lemma A.1
For each integer \(k\ge 1\), the space \(H^k({\mathbb {R}}_+)\) is a Banach algebra. In particular, there exists a constant c such that for all u, \(v\in H^k({\mathbb {R}}_+)\) it holds that \(uv\in H^k({\mathbb {R}}_+)\) and
1.1 A.1 Continuity
We now adapt the proof of the continuity result [39, Theorem 2] to our setting, but with some corrections in the proof. For notational reasons, we also introduce the Nemytskii operators
for \(\quad u\in H^k({\mathbb {R}}_+; {\mathbb {R}}^d),\;x\in {\mathbb {R}}_+\). In order for N to map \(H^k\) into \(H^k\) again, we need certain growth restrictions, which is not the case on bounded domains. For a multi-index \(\alpha \), we denote by \(D^\alpha \) the respective partial derivative operator.
Assumption A.2
Assume \(\mu \in C^m({\mathbb {R}}_{\ge 0}\times {\mathbb {R}}^d, {\mathbb {R}})\) and
-
(a)
For each integer l, \(0\le l\le m\), there exists an \( a_l\in L^2({\mathbb {R}}_+)\) and some \( b_l:{\mathbb {R}}^d\rightarrow {\mathbb {R}}_+\) locally bounded, such that
$$\begin{aligned} \left|D^{(l,0,\ldots ,0)} \mu (x,y)\right| \le b_l(y)\left( a_l(x) + \left|y\right|\right) ,\,\forall \,x\in {\mathbb {R}}_+,\,y\in {\mathbb {R}}^d \end{aligned}$$ -
(b)
For each multi-index \(\alpha \) with \(\alpha _1 < \left|\alpha \right| \le m\), the family of functions \((D^\alpha \mu (x,.))_{x\in {\mathbb {R}}_{\ge 0}}\) is equicontinuous and \(\sup _{x\in {\mathbb {R}}_+} \left|D^\alpha \mu (x,.)\right|\) is locally bounded.
Assumption A.3
Assume that \(\mu \in C^{m}({\mathbb {R}}_{\ge 0} \times {\mathbb {R}}^d, {\mathbb {R}})\) and \(D^{\alpha }\mu (x,.)\) is locally Lipschitz for all multi-indices \(\alpha \), \(\left|\alpha \right|\le m\) with Lipschitz constants uniform in \(x\in {\mathbb {R}}_{\ge 0}\), i.e., we assume that for all \(r\ge 0\) there exists \(L_r\ge 0\) such that
holds for all x, y, \(z\in {\mathbb {R}}^d\) with \(\left|y\right|\), \(\left|z\right|\le r\) and \(\alpha \), \(\left|\alpha \right|\le m\)
Remark A.4
If \(\mu \) satisfies Assumption A.2 for some integer \(m\ge 1\), then \(\mu \) satisfies Assumption A.3 for \(m-1\).
Remark A.5
Recall the Sobolev imbeddings
As usual \(BUC^m({\mathbb {R}}_+)\) is equipped with the \(C^m_b\)-norm. In the following, we will work with the \(BUC^m\) representative of the elements in \(H^{m+1}\) without further comment.
Note that Assumption A.2 is stronger than [18, Assumption 6.2] so that we get the following two results from [18, Appendix 1].
Theorem A.6
If Assumption A.2 holds for some integer \(m\ge 1\), then the operator N is continuous from \((H^m({\mathbb {R}}_+))^d\) into \(H^m({\mathbb {R}}_+)\).
Theorem A.7
Let \(\mu \) satisfy Assumptions A.2 and A.3 for some positive integer m. Then, N is Lipschitz continuous from bounded subsets of \((H^m({\mathbb {R}}_+))^d\) into \(H^m({\mathbb {R}}_+)\).
1.2 A.2 Differentiability
We now discuss differentiability of N in Fréchet sense. Here we run into the following problem compared with the literature. To get continuity of the Fréchet derivatives, Valent [38] uses that \(H^m\) is a Banach algebra and the Nemytskii operator corresponding to \((\tfrac{\partial }{\partial y_j} \mu )\) maps into \(H^m\). On unbounded domains, this would exclude the linear case \(\mu (x,y):= y\) which is of particular interest for applications in this work. We resolve this problem in Lemma A.9. Note that multiplication is not only bilinear continuous on \(H^k\), but also from \(C^k_b\times H^k\) into \(H^k\). More precisely, see Lemma B.1, for all \(k\ge 0\) there exists \(c>0\) such that for all \(g\in C^k_b({\mathbb {R}}_+)\), \(u\in H^k({\mathbb {R}}_+)\) it holds that
We start with a result on continuity of the nonlinear operators, adapting [39, Theorem 2]. We now write shortly \(H^m({\mathbb {R}}_+)^d\) for \(H^m({\mathbb {R}}_+;{\mathbb {R}}^d)\).
Theorem A.8
If Assumption A.2.(b) holds for some integer \(m\ge 1\), then the operator \(N_{y_j}\) is continuous from \(H^m({\mathbb {R}}_+)^d\) into \(C^{m-1}_b({\mathbb {R}}_+)\) and maps bounded sets into bounded sets.
Proof
We prove the continuity in a similar way as done for Theorem A.6. First, let \(m=1\), and \((u_n) \subset H^1({\mathbb {R}}_+)^d\) converging to some \(u\in H^1({\mathbb {R}}_+)^d\). Sobolev imbeddings imply that \(u_n\), \(u \in BUC({\mathbb {R}}_+)^d\), \(n\in {\mathbb {N}}\) and \(u_n \rightarrow u\) uniformly, as \(n\rightarrow \infty \). Thus, \(x\mapsto \frac{\partial }{\partial y_i} \mu (x,u(x))\) is continuous and bounded.
Define \(R:= \sup _{n\in {\mathbb {N}}} \left||u_n\right||_{\infty }<\infty \) and observe that the family of functions
is uniformly equicontinuous on the \({\mathbb {R}}^d\)-ball of radius R.
Let \(\epsilon >0\) be arbitrary and \(\delta = \delta _{\epsilon ,R}>0\) such that for all y, \({\tilde{y}}\in {\mathbb {R}}^d\) with \(\left|y\right|\), \(\left|{\tilde{y}}\right| \le R\) and \(\left|{\tilde{y}}-y\right|<\delta \) it holds that
Now, let \(N_\delta \in {\mathbb {N}}\) such that for all \(n\ge N\) it holds that
Hence, \(\sup _{x\in {\mathbb {R}}_+} \left|| \frac{\partial }{\partial y_i} \mu (x,u(x)) - \frac{\partial }{\partial y_i} \mu (x,u_n(x))\right||_{\infty } <\epsilon \) for all \(n\ge N_\delta \), and thus
Let \(M\subset H^1({\mathbb {R}}_+)\) be bounded and \(R>0\) so that M is contained in the radius R ball of \(C_b({\mathbb {R}}_+)\). Then, for all \(u\in M\),
We finish the proof by induction, so assume the claim holds true for \(m\in {\mathbb {N}}\). By induction hypothesis, \(N_{y_j}\) is continuous from \(H^{m+1}\) into \(C^{m-1}_b\), so it remains to show that the same holds true for \(\frac{\,{\text {d}}}{\,{\text {d}}x} N_{y_j}\). Chain rule yields
for \(u\in H^{m+1}({\mathbb {R}}_+)^d\hookrightarrow BUC^m({\mathbb {R}}_+)^d\). The function \({\bar{\mu }}\) defined as
for \(x\in {\mathbb {R}}_{\ge 0}\), \((y,z)\in {\mathbb {R}}^{d+d}\), satisfies Assumption A.2.(b) for m. Hence, by induction hypothesis, the Nemytskii operators corresponding to the \(y_j\) (and \(z_i\))-derivatives of \({\bar{\mu }}\) are continuous and map bounded sets into bounded sets, from \(H^{m}({\mathbb {R}}_+)^{d+d}\) into \(C^{m-1}_b({\mathbb {R}}_{\ge 0})^{d+d}\). Since the map \(u\mapsto \nabla u_i\) is linear continuous from \(H^{m+1}({\mathbb {R}}_+)^d\) into \(H^{m}({\mathbb {R}}_+)\), we get the properties for \(\frac{\,{\text {d}}}{\,{\text {d}}x} N_{y_j}\). \(\square \)
In the following, we write for \(j=1,\ldots ,d\), \(u\in H^m({\mathbb {R}}_+)^d\), \(v\in H^m({\mathbb {R}}_+)\),
Lemma A.9
Let \(m\ge 1\) and \(\mu \) fulfilling Assumption A.2.(b) for \(m+1\), then, the mapping
is continuous from \(H^m({\mathbb {R}}_+)^d\) into \({\mathscr {L}}(H^m({\mathbb {R}}_+))\), for all \(j= 1,\ldots ,d\). Moreover, \(\varPhi _i\) maps bounded sets into bounded sets.
Proof
Note that \({\tilde{\mu }}(x,y,z) := \mu (x,y)z\) fulfills Assumption A.2.(a) for m so that Theorem A.6 yields continuity of
Of course, \({\widetilde{N}}_{y_j}\) is linear in its second argument so that \({\widetilde{N}}_{y_j}(u,.)\in {\mathscr {L}}(H^m({\mathbb {R}}_+))\), for each \(u\in H^m({\mathbb {R}}_+)^d\). It remains to prove continuity in the uniform operator topology for which we proceed by induction, again.
-
Step I.
With \(m=1\) let \((u^{(n)})\subset H^1({\mathbb {R}}_+)^d\), \(u\in H^1({\mathbb {R}}_+)^d\) and \(v\in H^1({\mathbb {R}}_+)\). First note that by Theorem A.8, \(N_{y_j}\) is continuous from \(H^1\) into \(C_b\), so that
$$\begin{aligned} \left||{\widetilde{N}}_{y_j}(u^{(n)},v) - {\widetilde{N}}_{y_j}(u,v)\right||_{L^2} \le c \left||N_{y_j} (u^{(n)})- N_{y_j}(u^{(n)})\right||_{C_b} \left||v\right||_{L^2}, \end{aligned}$$converges to 0, as \(n\rightarrow \infty \), uniformly in v. Similar we get uniform \(L^2\)-convergence for \(N_{y_j}(u^{(n)})\tfrac{\partial }{\partial x}v\) and for the operator
$$\begin{aligned} (u,v) \mapsto (\tfrac{\partial ^2}{\partial x\partial y_j} \mu )(x,u(x))v(x). \end{aligned}$$Moreover, for all i, \(j\in \{1,\ldots ,d\}\), by Sobolev imbeddings
$$\begin{aligned}&\int _{{\mathbb {R}}_+} \left|\tfrac{\partial ^2}{\partial y_j\partial y_i} \mu (x,u^{(n)}(x)) \nabla u_i^{(n)}(x) - \tfrac{\partial ^2}{\partial y_j\partial y_i}\mu (x,u(x))\nabla u_i(x)\right|^2 \left|v(x)\right|^2 \,{\text {d}}x \le \nonumber \\&\le K \left||v\right||_{H^1}^2 \int _{{\mathbb {R}}_+} \left|\tfrac{\partial }{\partial y_j\partial y_i} \mu (x,u^{(n)}(x)) \nabla u_i^{(n)}(x) - \tfrac{\partial }{\partial y_j\partial y_i}\mu (x,u(x))\nabla u_i(x)\right|^2 \,{\text {d}}x .\nonumber \\ \end{aligned}$$(A.4)Note that \(\tfrac{\partial }{\partial y_i}\mu \) fulfills Assumption A.2.(b) and recall that multiplication is continuous from \(C_b\times L^2\) into \(L^2\). Hence, the integral converges to 0, as \(n\rightarrow \infty \) by application of Theorem A.8 to the corresponding Nemytskii operator, and so we conclude continuity of \(\varPhi \). Using almost the same estimates and applying the corresponding of part Theorem A.8, we get that \(\varPhi _i\) maps bounded sets into bounded sets again.
-
Step II:
By induction hypothesis, the Lemma holds true for \(m\in {\mathbb {N}}\) fixed, so assume that \(\mu \) fulfills Assumption A.2.(b) for \(m+2\). Then, \(\varPhi _j\) is continuous from \(H^{m+1}({\mathbb {R}}_+)^d\) into \({\mathscr {L}}(H^m({\mathbb {R}}_+))\). Let \((u^{(n)})\subset H^{m+1}({\mathbb {R}}_+)^d\) converging to \(u\in H^{m+1}({\mathbb {R}}_+)^d\). Note that
$$\begin{aligned} \left||\varPhi (u^{(n)}) - \varPhi (u)\right||_{{\mathscr {L}}(H^{m+1})}^2\le & {} \sup _{w\in H^{m}} \frac{\left||\varPhi (u^{(n)})w - \varPhi (u)w\right||_{H^m}^2}{\left||w\right||_{H^m}^2}\nonumber \\&+ \sup _{v\in H^{m+1}}\frac{\left||\frac{\,{\text {d}}^{m+1}}{\,{\text {d}}x^{m+1}} (\varPhi (u^{(n)})v - \varPhi (u)v)\right||_{L^2}^2}{\left||v\right||_{H^{m+1}}^2}.\nonumber \\ \end{aligned}$$(A.5)The first term vanishes as \(n\rightarrow \infty \) by induction hypothesis. For all \(v\in H^{m+1}\), we can write the latter one can be estimated by
$$\begin{aligned} \frac{\,{\text {d}}^{m+1}}{\,{\text {d}}x^{m+1}} (\varPhi (u^{(n)})v - \varPhi (u)v)&= \frac{\,{\text {d}}^{m}}{\,{\text {d}}x^{m}} [(\tfrac{\partial }{\partial x}N_{y_j}(u^{(n)}) - \tfrac{\partial }{\partial x}N_{y_j}(u)) v] \\&\qquad + \frac{\,{\text {d}}^{m}}{\,{\text {d}}x^{m}} [(N_{y_j}(u^{(n)}) - N_{y_j}(u))w], \end{aligned}$$for \(w:= \tfrac{\partial }{\partial x}v \in H^m\). By induction hypothesis, the latter term converges to 0 in \(L^2\), uniformly over all \(w\in H^m({\mathbb {R}}_+)\). For the first summand, we observe that \(D_x \mu (x,y)\) fulfills Assumption A.2.(b) for \(m+1\) so that the induction hypothesis applied on
$$\begin{aligned} \varPsi (u)v := (\tfrac{\partial ^2}{\partial x \partial y_j} \mu )(.,u(.)) v \end{aligned}$$yields \(L^2\) convergence. Plugging in into (A.5) finally yields the convergence uniform in \({\mathscr {L}}(H^{m+1})\). With the same decomposition, we deduce from induction hypothesis that \(\varPhi _i\) maps bounded sets from \(H^{m+1}\) into bounded sets of \({\mathscr {L}}(H^{m+1})\).
\(\square \)
Based on the continuity in the uniform topology, we are now able to prove Fréchet differentiability.
Theorem A.10
If Assumption A.2 holds for some integer \(m+1\), \(m\ge 1\), then the operator N defined above is in \(C^1(H^m({\mathbb {R}}_+)^d, H^m({\mathbb {R}}_+))\) with derivative
Proof
From Theorem A.6, we already know that N maps \(H^m({\mathbb {R}}_+;{\mathbb {R}}^d)\) continuously into \(H^m({\mathbb {R}}_+)\). Moreover, Lemma A.9 tells us that DN, defined as above, is continuous from \(H^m({\mathbb {R}}_+)^d\) into \({\mathscr {L}}(H^m({\mathbb {R}}_+))\). Thus, it remains to verify that DN is at least the Gâteaux derivative of N, i.e., that for any u, \(v\in H^{m +1}({\mathbb {R}}_+;{\mathbb {R}}^d)\),
By fundamental theorem of calculus, we get for fixed u, \(v\in H^{m}\), and all \(x\in {\mathbb {R}}_+\),
The map \( t\mapsto {\widetilde{N}}_{y_i}(u+t\epsilon v, v_i)\) is continuous from [0, 1] into \(H^{m}({\mathbb {R}}_+)\) by Theorem A.6. Therefore, the integral in Eq. (A.7) can be considered as an Bochner integral and (A.6) follows from Lemma A.9 and the estimate
\(\square \)
If \(\mu \in C^2\), then we define for \(u\in H^m({\mathbb {R}}_+)^d\), v, \(w\in H^m({\mathbb {R}}_+)\), \(x\in {\mathbb {R}}_+\),
Theorem A.11
Assume that \(\mu \) satisfies Assumption A.2 for \(m+2\), \(m\ge 1\), then the Nemytskii operator \(N: H^m({\mathbb {R}}_+)^d \rightarrow H^m({\mathbb {R}}_+)\) is of class \(C^2\) with second derivative
for u, v, \(w\in H^m({\mathbb {R}}_+)^d\).
Proof
By the previous theorem, N is of class \(C^1\), so we have to show the same for the map
Since \(H^m\) is a Banach algebra, we get for u, \({\bar{u}} \in H^m({\mathbb {R}}_+)^d\),
Now, we apply Lemma A.9 to \(\frac{\partial }{\partial y_i} \mu (x,u(x))\), \(i=1,\ldots ,d\), which indeed fulfill Assumption A.2.(b) for \(m+1\). This yields continuity of \(D^2N\).
To finish the proof, it again suffices to show differentiability in Gâteaux sense. Fix u, \(w\in H^m({\mathbb {R}}_+)^d\) and let \(\epsilon >0\). As in the proof of Theorem A.10, cf. (A.7), we get by fundamental theorem of calculus, for all \(v\in H^m({\mathbb {R}}_+)^d\)
From the first part of this proof, we know that \(\epsilon \mapsto D^2N(u+ \epsilon w)\) is uniformly continuous from [0, 1] into the space of continuous bilinear operators. Hence, the right-hand side is in \(o(\epsilon )\), uniformly in \(v\in H^m\). \(\square \)
The following conclusion is a combination of Theorem A.8 and the representations of DN and \(D^2N\).
Corollary A.12
Under the assumptions of, respectively, Theorems A.10 and A.11, the maps
and
map bounded sets into bounded sets.
B The noise operator
We will now study the operator-valued map \({\mathcal {C}}\), defined previously by
for \(u\in {\mathcal {D}}({\mathcal {C}})\subset L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_+) \oplus {\mathbb {R}}\), \(w\in L^2({\mathbb {R}})=: U\) and \(x\in {\mathbb {R}}\). We can reduce the problem to the operator
for \(\sigma : {\mathbb {R}}^2\rightarrow {\mathbb {R}}\), and \(\zeta \) and an integral kernel \(\zeta :{\mathbb {R}}^2\rightarrow {\mathbb {R}}\), which we aim to take values in spaces of Hilbert–Schmidt operators like \({\mathscr {L}}_2(U; L^2({\mathbb {R}}_+))\). As above, we write
and define the Nemytskii operator
Naturally, it will make sense to separate the study of \(\varPsi \) into the operators \(N_\sigma \) and \(T_\zeta \). Recall that we have discussed the Nemytskii operators \(N_\sigma \) in Appendix A.
1.1 B.1 The Hilbert–Schmidt property
Note that on \(L^2(D)\), for a domain \(D\subset {\mathbb {R}}^d\), \(d\in {\mathbb {N}}\), every Hilbert–Schmidt operator is of the form \(T_\kappa \), for an integral kernel \(\kappa \) satisfying
see, e.g., [8, Section XI.6]. When D has infinite Lebesgue measure, this condition is obviously violated for convolution kernels \(\kappa (x,y) = \kappa (x-y)\), in which have been interested in Example 1.4 for instance. Hence, \(T_\zeta \) itself will typically not be Hilbert–Schmidt on the spaces of interest. We skip the proofs in the following three lemmas since they will be the same as the proofs of, respectively, Lemmas 7.1, 7.2 and 7.4 in [18].
Lemma B.1
For any integer \(n\ge 0\), multiplication is bilinear continuous from \(H^n({\mathbb {R}}_+) \times C_b^n({\mathbb {R}}_{\ge 0})\) into \(H^n({\mathbb {R}}_+)\).
The lemma is the first step in the direction to separate our discussion of \(\varPsi \) into the operators \(N_\sigma \) and \(T_\zeta \). Provided that \(\zeta \) is sufficiently nice, \(T_\zeta \) will indeed map into the space of bounded and uniformly continuous functions.
Assumption B.2
Let \(n\ge 1\), \(\zeta (.,y) \in C^{n+1}({\mathbb {R}})\) for all \(y\in {\mathbb {R}}\) and \(\tfrac{\partial ^{i}}{\partial x^i}\zeta (x,.)\in L^2({\mathbb {R}})\) for all \(x\in {\mathbb {R}}\), \(i\in \{0,\dots , n+1\}\). Moreover,
In the following, we use the notation \(\zeta ^{(i)}:=\tfrac{\partial ^{i}}{\partial x^i} \zeta \).
Remark B.3
For convolution kernels \(\zeta (x,y):= \zeta (x-y)\), this assumption is satisfied when \(\zeta \in H^{n+1}({\mathbb {R}})\cap C^{n+1}({\mathbb {R}})\).
Lemma B.4
Let Assumption B.2 be fulfilled for \(n\in {\mathbb {N}}\). Then, \(T_\zeta \) maps U into \(BUC^n({\mathbb {R}})\). Moreover, \(T_\zeta w\) and its first n derivatives are Lipschitz continuous for all \(w\in U\) and it holds that \(T_\zeta \in {\mathscr {L}}(U; BUC^2({\mathbb {R}}))\).
Lemma B.5
Let \(n\in {\mathbb {N}}\) and Assumption B.2 be satisfied. For \(u\in H^n({\mathbb {R}}_+)\) and \(x_*\in {\mathbb {R}}\) it holds that
For application in Sect. 4, we need to deal \({\mathcal {C}}\) on the domain of the Dirichlet Laplacian. In fact, Assumption \(({\hbox {Noise}}_0)\) and Lemma B.1 ensure \(N_\sigma (u)\in H^2({\mathbb {R}}_+)\cap H^1_0({\mathbb {R}}_+)\) for all \(u\in H^2({\mathbb {R}}_+) \cap H^1_0({\mathbb {R}}_+)\).
1.2 B.2 Lipschitz continuity and differentiability
In order to apply the results, let us introduce the translation group \((\theta _x)_{x\in {\mathbb {R}}}\) which is strongly continuous on \(BUC({\mathbb {R}})\).
Remark B.6
By the structure of the direct sum of Hilbert spaces, the following two results directly extend to \({\mathcal {C}}\) as a mapping from \(H^n({\mathbb {R}}_+)\oplus H^n({\mathbb {R}}_+) \oplus {\mathbb {R}}\) into \({\mathscr {L}}_2(U; H^n({\mathbb {R}}_+) \oplus H^n({\mathbb {R}}_+) \oplus {\mathbb {R}})\).
Remark B.7
Note that for \(x\in {\mathbb {R}}\)
where \(\zeta _x := \zeta (x+.,.)\) satisfies Assumption B.2, whenever \(\zeta \) does.
We impose the following conditions on \(\sigma \).
Assumption B.8
Let \(n\ge 1\) and assume that \(\sigma \in C^n({\mathbb {R}}^2;{\mathbb {R}})\) satisfies
-
(i)
For every multi-index \(I = (i,j) \in {\mathbb {N}}^2\) with \(\left|I\right| \le n\), there exist \(a_I\in L^2({\mathbb {R}}_+)\) and \(b_I \in L^\infty _{loc}({\mathbb {R}}, {\mathbb {R}}_+)\) such that
$$\begin{aligned} \left|\tfrac{\partial ^{\left|I\right|}}{\partial x^{i}\partial y^{j}} \sigma (x,y)\right| \le {\left\{ \begin{array}{ll} b_I(y)\left( a_I(|x|) +\left|y\right|\right) , &{} j=0, \\ b_I (y), &{} j\ne 0. \end{array}\right. } \end{aligned}$$ -
(ii)
\(\sigma \) and its partial derivatives (in x and y) are locally Lipschitz with Lipschitz constants independent of \(x\in {\mathbb {R}}\).
Theorem B.9
Let \(n\in {\mathbb {N}}\) and assume that Assumption B.2 is fulfilled for \(n+1\) and, respectively, B.8 for \(n+2\). Then, \(\varPsi \) is of class \(C^2\) from \(H^n({\mathbb {R}}_+)\oplus {\mathbb {R}}\) into \({\mathscr {L}}_2:= {\mathscr {L}}_2(U; H^n({\mathbb {R}}_+))\), with derivatives
Moreover, \(\varPsi \), \(D\varPsi \), and \(D^2\varPsi \) map bounded sets into bounded sets.
Remark B.10
For \(n=2\) and under the additional assumption that \(\sigma (0,0) = 0\), it holds that \(\varPsi (u,x)\in H^2\cap H^1_0({\mathbb {R}}_+)\), when \(u\in H^2\cap H^1_0({\mathbb {R}}_+)\). This even translates to \(D\varPsi \) and \(D^2\varPsi \).
Proof
Let u, \(v\in H^n({\mathbb {R}}_+)\), x, \(y\in {\mathbb {R}}\) and \(\epsilon >0\), then with Lemma B.5,
In fact, the first term is in \(o(\epsilon )\) because of differentiability of \(N_\sigma \) we get from Theorem A.10. For the second summand, we have defined
Thus,
Using fundamental theorem of calculus again, we obtain
which goes to 0, as \(\epsilon \rightarrow 0\). Using that (B.3) holds for \(i=0,\ldots ,n+2\), the same calculation can be done for \(\zeta ^{(i)}\), \(i=1,\dots , n\) which then shows that \(D\varPsi \) is at least the Gâteaux derivative of \(\varPsi \). To finish the proof, it is now enough to show that
is Gâteaux differentiable, and
is continuous. Let us start with the latter claim and show continuity of each summand separately. To this end, we first decompose as above
Consider u, \({\tilde{u}}\), v, \({\bar{v}}\in H^n\), \(x,{\tilde{x}},y,\bar{y}\in {\mathbb {R}}\). Because \(N_\sigma \in C^2\) by Theorem A.11, we get
Applying Lemma B.5, we see that both terms go to 0, as \(\left||u-{\tilde{u}}\right||_{H^n} + \left|x-{\tilde{x}}\right|\) does, and that the convergence is uniformly in v, \({\bar{v}}\in H^n\) with norm smaller than 1. Indeed, for the first term this is continuity of \(D^2N\), the second term can be estimated by
Convergence of the right-hand side follows with the same procedure as in (B.5). For \(R_2\) and \(R_3\), we use continuity of \(DN_\sigma \), for \(R_4\) continuity of \(N_\sigma \) itself. With almost the same estimates, we observe that \(D^2\varPsi \) maps bounded sets into bounded sets. In fact, this property is inherited by N, DN and \(D^2N\), see Corollary A.12.
It remains to show that \(D^2\varPsi \) is indeed the derivative of \(D\varPsi \). The derivative of the second summand can be computed in the same way as \(D\varPsi \) itself has been computed. For the first summand, we have to be slightly more careful, but note that by Lemma B.5
which is in \(o(\epsilon )\) thanks to (B.5). The remaining estimates follow in the same way: First apply Lemma B.5, but then use that \(N_\sigma \) is of class \(C^2\). Hence, \(D^2\varPsi \) is the Gâteaux derivative of \(D\varPsi \). By continuity of \(D^2\varPsi \), the differentiability also holds true in Fréchet sense. \(\square \)
Rights and permissions
About this article
Cite this article
Keller-Ressel, M., Müller, M.S. Forward invariance and Wong–Zakai approximation for stochastic moving boundary problems. J. Evol. Equ. 20, 869–929 (2020). https://doi.org/10.1007/s00028-019-00550-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-019-00550-4
Keywords
- Stochastic partial differential equation
- Stefan problem
- moving boundary problem
- Phase separation
- Forward invariance
- Wong–Zakai approximation