Abstract
In this paper, we prove that stochastic porous media equations over \(\sigma \)-finite measure spaces \((E,\mathcal {B},\mu )\), driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function \(\Psi \) of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions \(\Psi \), for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an \(L^p(\mu )\)-Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where E is a manifold or a fractal, and to non-local operators L, as e.g. \(L=-f(-\Delta )\), where f is a Bernstein function.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Aronson, D.G.: Lecture Notes in Math, p. 1224. Springer, Berlin (1986)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics, Springer, New York (2010)
Barbu, V., Da Prato, G., Röckner, M.: Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57(1), 187–211 (2008)
Barbu, V., Da Prato, G., Röckner, M.: Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37(2), 428–452 (2009)
Barbu, V., Da Prato, G., Röckner, M.: Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise. J. Math. Anal. Appl. 389, 147–164 (2012)
Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Springer International Publishing Switzerland (2016)
Barbu, V., Da Prato, G., Röckner, M.: Stochastic porous media equations and self-organized criticality. Commun. Math. Phys. 285, 901–923 (2009)
Barbu, V., Röckner, M., Russo, F.: Stochastic porous media equation in \(\mathbb{R}^d\). J. Math. Pures Appl. 9103, (4), 1024–1052 (2015)
Bauer, H.: Measure and Integration Theory. Studies in Mathematics, Walter de Gruyter and Co (2001)
Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics, 14. Walter de Gruyter and Co, Berlin (1991)
Da Prato, G., Röckner, M.: Weak solutions to stochastic porous media equations. J. Evol. Equ 4(2), 249–271 (2004)
Da Prato, G., Röckner, M., Rozovskii, B.L., Wang, F.Y.: Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31(1–3), 277–291 (2006)
Fukushima, M.: Two topics related to Dirichlet forms: quasi-everywhere convergences and additive functionals, Dirichlet forms (Varenna, 1992), p. 21–53, Lecture Notes in Math., 1563, Springer, Berlin (1993)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin/New York (2011)
Gess, B.: Finite time extinction for stochastic sign fast diffusion and self-organized criticality. Comm. Math. Phys. 335(1), 309–344 (2015)
Gess, B., Röckner, M.: Singular-degenerate multivalued stochastic fast diffusion equations. SIAM J. Math. Anal. 47(5), 4058–4090 (2015)
Gess, B., Tölle, J.M.: Multi-valued, singular stochastic evolution inclusions. J. Math. Pures Appl. 9 101, (6), 789–827 (2014)
Grigor’yan, A., Yang, M.: Determination of the walk dimension of the Sierpiński gasket without using diffusion. J. Fractal Geom. 5(4), 419–460 (2018)
Herrero, M.A., Pierre, M.: The Cauchy problem for \(u_t=\Delta u^m\) when \(0<m<1\). Trans. Amer. Math. Soc. 291(1), 145–158 (1985)
Hsu, E.P.: Stochastic Analysis on Manifolds, Graduate Studies in Mathematics, 38. American Mathematical Society, Providence, RI, p. xiv+281 (2002)
Hu, Z.C., Ma, Z.M.: Beurling-Deny formula of semi-Dirichlet forms, English, French summary. C. R. Math. Acad. Sci. Paris 338(7), 521–526 (2004)
Itô, K.: Lectures on Stochastic Processes, Notes by K. Muralidhara Rao. Second edition. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 24. Distributed for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, p. iii+233 (1984)
Kim, J.U.: On the stochastic porous media equation. J. Diff. Equ. 220, 163–194 (2006)
Krylov, N.V.: Itô’s formula for the \(L_p\)-norm of stochastic \(W^1_p\)-valued processes. Probab. Theory Related Fields 147(3–4), 583–605 (2010)
Liu, W., Röckner, M.: Stochastic Partial Differential Equations: an Introduction, Springer International Publishing Switzerland (2015)
Liu, W., Stephan, M.: Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple. J. Math. Anal. Appl. 410(1), 158–178 (2014)
Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin Heidelberg (1992)
Neuss, M.: Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organized criticality. Stoch. Dyn. 21(5), 34 (2021). Paper No. 2150029
Okuda, H., Dawson, J.M.: Theory and numerical simulation on plasma diffusion across a magnetic field. Phys. Fluids 16, 408–426 (1973)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York-London (1967)
Picard, J.: The manifold-valued Dirichlet problem for symmetric diffusions. Potential Anal. 14(1), 53–72 (2001)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equation, vol. 1905 of Lecture Notes in Mathematics, Springer, Berlin (2007)
Ren, J.G., Röckner, M., Wang, F.Y.: Stochastic generalized porous media and fast diffusion equations. J. Differ. Equ. 238(1), 118–152 (2007)
Röckner, M., Wang, F.Y.: Non-monotone stochastic generalized porous media equations. J. Differ. Equ. 245(12), 3898–3935 (2008)
Röckner, M., Wu, W.N., Xie, Y.C.: Stochastic porous media equations on general measure spaces with increasing Lipschitz nonlinearities. Stoch. Process. Their Appl. 128(6), 2131–2151 (2018)
Schilling, R.L., Song, R.M., Vondrac̆ek, Z: Bernstein Functions. Theory and Applications, Second edition. De Gruyter Studies in Mathematics, 37. Walter de Gruyter Co., Berlin (2012)
Schnaubelt, R.: Lecture Notes: Functional Analysis (2014/15). https://www.math.kit.edu/iana3/schnaubelt/media/fa14-skript.pdf
Taylor, M.E.: Pseudodifferential Operators, Princeton Mathematical Series, No. 34. Princeton University Press, Princeton, N.J., p. xi+452 (1981)
Vazquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lect. Ser. Math. Appl., vol. 33, Oxford University Press, Oxford (2006)
Vazquez, J.L.: The Porous Medium Equation, Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford. p. xxii+624 (2007)
Funding
Michael Röckner is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through CRC 1283. Weina Wu is supported by the National Natural Science Foundation of China (NSFC) (No.11901285), China Scholarship Council (CSC) (No.202008320239) and DFG through CRC 1283. Yingchao Xie is supported by the NSFC (No.11931004).
Author information
Authors and Affiliations
Contributions
All authors contributed to the content of this article.
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 A.1 Auxiliary Results
In this part we aim to prove Eq. A.4, which has been used in the proof of Claims 4.1 and 4.2.
Lemma A.1
Let \(\nu ,{\varepsilon ,\lambda } \in (0,1]\). For all \(x\in F^*_{1,2}\), we have
For all \(x\in L^2(\mu )\),
Proof
Recall from Eq. 4.4 that
For \(x\in F^*_{1,2}\), to prove Eq. A.1, we rewrite
The proof of Eq. A.2 is analogous due to the fact that \(J_\varepsilon \) is \(\frac{1}{\sqrt{\nu \varepsilon \lambda }}\)-Lipschitz in \(L^2(\mu )\), so \(A^{\nu ,\varepsilon }_\lambda \in L^2(\mu )\) if \(x\in L^2(\mu )\).
Lemma A.2
Let \(x\in L^2(\mu )\). Then for \(\nu ,\varepsilon ,\lambda \in (0,1]\), \(t\in [0,T]\), we have
Proof
Applying Itô’s formula to \(|X^{\nu ,\varepsilon }_\lambda |^2_2\), we obtain
which by Eq. A.2 yields,
Taking expectation of both sides, by (H3)(i) we get
Then by Eq. 4.16 and Gronwall’s lemma we get Eq. A.3 as claimed.
Proposition A.1
Let \(x\in L^2(\mu )\). Then for \(\nu ,\varepsilon ,\lambda \in (0,1]\), \(t\in [0,T]\), we have
Proof
Let \(x\in L^2(\mu )\). Then
where in the first inequality we used (H4), the fact that \(r(\Psi _\lambda (r)+\lambda r)\ge 0\) for all \(r\in \mathbb {R}\) and \(\Psi _\lambda \) is \(\frac{1}{\lambda }\)-Lipschitz ([3, Page:41, Proposition 2.3 (ii)]), the last equality comes from the fact that \((\Psi _\lambda +\lambda I)(J_\varepsilon (x))\in D(L)\). Now from Eq. A.3, we get the assertion.
1.2 A.2 The \(L^p\)-Itô Formula in Expectation
The purpose in this section is to prove Theorem 7.1 below, which has been used in Lemmas 4.2 and 4.3.
Let \(\ell _2\) be the space of all square-summable sequences in \(\mathbb {R}\) and \(p\in [2,\infty )\). In addition, to the real-valued \(L^p\)-space, \(L^p(\mu ):=L^p(E,\mu )\) we consider the \(\ell _2\)-valued \(L^p\)-space \(L^p(\mu ;\ell _2):=L^p(E,\mu ;\ell _2)\). We set
Let \(\mathscr {P}\) denote the predictable \(\sigma \)-algebra on \([0,T]\times \Omega \) corresponding to \((\Omega ,\mathscr {F},(\mathscr {F}_t)_{t\ge 0},\mathbb {P})\). For \(p\in [2,\infty )\) we set
and
equipped with their standard \(L^p\)-norms. Since \((E,\mathcal {B})\) is a standard measurable space, by definition there exists a complete metric d on E, such that (E, d) is separable, i.e., a Polish space, whose Borel \(\sigma \)-algebra coincides with \(\mathcal {B}\). Below we fix this metric d and denote the corresponding set of all bounded continuous functions by \(C_b(E)\).
Let \(\mathcal {E}\) be all \(g=(g_k)_{k\in \mathbb {N}}\in L^\infty ([0,T]\times \Omega ;L^\infty (\mu ;\ell _2)\cap L^1(\mu ;\ell _2))\) such that there exists \(j\in \mathbb {N}\) and bounded stopping times \(\tau _0\le \tau _1\le \cdots \le \tau _j\le T\) such that
where \(g_k^i\in C_b(E)\cap L^1(\mu )\), \(1\le i\le j\).
Claim A.1
\(\mathcal {E}\) is dense in \(\mathbb {L}^p(T;\ell _2)\) for all \(p\in [2,\infty )\).
Proof
Let \(f=(f_k)_{k\in \mathbb {N}}\in L^q(T;\ell _2)\), with \(q:=\frac{p}{p-1}\), be such that
Now let \(\sigma \le \tau \) be two stopping times and \(k\in \mathbb {N}\). Define \(g\in \mathbb {L}^p(T;\ell _2)\) by \(g=(g_k\delta _{ik})_{i\in \mathbb {N}}\), where
and \(g_k^k\in C_b(E)\cap L^1(\mu )\). Then \(g\in \mathcal {E}\), hence
which implies that
since all sets of the type \((\sigma ,\tau ]\) generate the \(\sigma \)-algebra \(\mathscr {P}\) and since \(f_k\) is \(\mathscr {P}\)-measurable. Therefore, since \(C_b(E)\cap L^1(\mu )\) is dense in \(L^p(\mu )\),
Now the assertion follows by the Hahn-Banach theorem ([38, page: 61, Corollary 4.23]).
Remark A.1
Let \(\mathcal {S}\) be the set of all functions \(f\in L^\infty ([0,T]\otimes \Omega ;L^\infty (\mu )\cap L^1(\mu ))\) such that there exist \(l\in \mathbb {N}\) and bounded stopping times \(\tau '_0\le \tau '_1\le \cdots \le \tau '_l\le T\) such that \(f=\sum _{i=1}^lf^i1_{(\tau '_{i-1},\tau '_i]}\), where \(f^i\in C_b(E)\cap L^1(\mu )\), \(1\le i\le l\). Similarly to Claim 7.1, one can prove that \(\mathcal {S}\) is dense in \(\mathbb {L}^p(T)\) for all \(p\in [2,\infty )\).
Define \(\textbf{M}:\mathcal {E}\longmapsto \bigcap _{p\ge 1}L^p(\Omega ;C([0,T];L^p(\mu )))\) as follows:
Let us note that the right hand-side of Eq. A.5 is \(\mathbb {P}\)-a.s. for every \(t\in [0,T]\) a continuous \(\mu \)-version of \(\textbf{M}(g)(t)\in L^p(E,\mu )\), which for every \(x\in E\) is a continuous real-valued martingale and is equal to
Claim A.2
Let \(p\in [2,\infty )\). Then \(\textbf{M}\) extends to a linear continuous map \(\overline{\textbf{M}}\) from \(\mathbb {L}^p(T;\ell _2)\) to \(L^p(\Omega ;C([0,T];L^p(\mu )))\), such that \(\overline{\textbf{M}}(g)\) is a continuous martingale in \(L^p(\mu )\) for all \(g\in \mathbb {L}^p(T;\ell _2)\).
Proof
We have
where we have used the BDG inequality applied to the real-valued martingale in Eq. A.6 in the third step, the assumption that \(p\ge 2\) and Minkowski’s inequality in the sixth step and Hölder’s inequality in the last step. Hence the first part of the assertion follows.
To prove the second let \(g\in \mathbb {L}^p(T;\ell _2)\). It suffices to prove that for all \(f\in L^q(\mu )\) with \(q:=\frac{p}{p-1}\),
is a real-valued martingale (see e.g. [26, Remark 2.2.5]). But since for some \(g_n\in \mathcal {E}\), \(n\in \mathbb {N}\), we have \(\forall ~t\in [0,T]\) that
it follows that
So, we may assume that \(g\in \mathcal {E}\). But in this case by Eq. A.5 it follows immediately that \(\int _Ef~\textbf{M}(g)(t)d\mu \), \(t\in [0,T]\), is a real-valued martingale.
Below we define for \(g\in \mathbb {L}^p(T;\ell _2)\), \(p\in [2,\infty )\),
where \(\overline{\textbf{M}}\) is as in Claim 7.2.
Now we fix \(p\in [2,\infty )\) and consider the following process
defined by
where \(u(0)\in L^p(\Omega ,\mathcal {F}_0;L^p(\mu ))\), \(f\in \mathbb {L}^p(T)\) and \(g\in \mathbb {L}^p(T;\ell _2)\).
Theorem A.1
“Itô-formula in expectation" Let \(p\in [2,\infty )\), \(f\in \mathbb {L}^p(T)\), \(g\in \mathbb {L}^p(T;\ell _2)\). Let u be as in Eq. A.7. Then for all \(t\in [0,T]\),
Remark A.2
In the case \(E=\mathbb {R}^d\), \(\mu =\)Lebesgue measure, N. Krylov proved Itô’s formula for the \(L^p\)-norm of a large class of \(W^{1,p}\)-valued stochastic processes in his fundamental paper [25]. In particular, Lemma 5.1 in that paper gives a pathwise Itô formula for processes u as in Eq. A.7, which immediately implies Eq. A.8. The proof, however, uses a smoothing technique by convoluting the process u in x with Dirac-sequence of smooth functions, which is not available in our more general case, where \((E,\mathcal {B})\) is just a standard measurable space with a \(\sigma \)-finite measure \(\mu \), without further structural assumptions that we wanted to avoid to cover applications e.g. to underlying spaces E which are fractals. Fortunately, the above Itô formula in expectation is enough to prove all main results in this paper without any further assumptions. After the preparations above, its proof is quite simple.
We recall the following well-known result (see e.g. Theorem 21.7 in [10]):
Lemma A.3
Let \(p\in [1,\infty )\), \(v_n,v\in L^p(\mu )\) such that \(v_n\rightarrow v\) in \(\mu \)-measure as \(n\rightarrow \infty \) and
Then
Proof of Theorem 7.1
By Claim 7.1 and Remark 7.1, we can find \(f_n\in \mathcal {S}\), \(n\in \mathbb {N}\), and \(g_n\in \mathbb {L}^p(T;\ell _2)\), \(n\in \mathbb {N}\), such that as \(n\rightarrow \infty \)
and
For \(n\in \mathbb {N}\), define
By Eqs. A.7, A.9, A.10 and Claim 7.2, it follows that as \(n\rightarrow \infty \),
in \(L^p(\Omega ;C([0,T];L^p(\mu )))\).
Applying the Itô formula to the real-valued semi-martingale \(|u_n(t,x)|_p^p\) for each \(x\in E\), and integrating w.r.t. \(x\in E\) and \(\omega \in \Omega \), we obtain
Note that by Lemma 7.3 and Eq. A.11
as \(n\rightarrow \infty \). Hence by Eqs. A.9 and A.10 we may pass to the limit \(n\rightarrow \infty \) in Eq. A.12 to get Eq. A.8. \(\square \)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Röckner, M., Wu, W. & Xie, Y. Stochastic Generalized Porous Media Equations Over \(\sigma \)-finite Measure Spaces with Non-continuous Diffusivity Function. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10127-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-024-10127-7
Keywords
- Wiener process
- Porous media equation
- Dirichlet form
- Maximal monotone graph
- Yosida approximation
- \(L^p\)-Itô formula in expectation