Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space

Abstract

In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: \( du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum _{k = 1}^m g^k (u(t, x)) dw^k (t). \) We prove the convergence of a Wong–Zakai type approximation scheme of the above equation in the space \( C^{\theta } ([0, T], H^{\gamma }_p ({\mathbb {R}}^d)) \) in probability, for some \( \theta \in (0,1/2), \gamma \in (1, 2)\), and \(p > 2\). We also prove a Stroock–Varadhan’s type support theorem. To prove the results we combine V. Mackevičius’s ideas from his papers on Wong–Zakai theorem and the support theorem for diffusion processes with N. V. Krylov’s \(L_p\)-theory of SPDEs.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Acquistapace, P., Tereni, B.: An approach to linear equations in Hilbert spaces by approximation of white noise with colored noise. Stoch. Anal. Appl. 2, 131–186 (1984)

    Article  Google Scholar 

  2. 2.

    Breźniak, Z., Capinski, M., Flandoli, F.: A convergence result for stochastic partial differential equations. Stochastics 24, 423–445 (1988)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brzeźniak, Z., Flandoli, F.: Almost sure approximation of Wong–Zakai type for stochastic partial differential equations. Stoch. Process. Appl. 55(2), 329–358 (1995)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bally, V., Millet, A., Sanz-Solé, M.: Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23(1), 178–222 (1995)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Burkholder, D. L., Davis, B. J., Gundy, R. F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 223–240. University of California Press (1972)

  6. 6.

    Cardon-Weber, C., Millet, A.: A support theorem for a generalized Burgers SPDE. Potential Anal. 15(4), 361–408 (2001)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Deya, A., Gubinelli, M., Tindel, S.: Non-linear rough heat equations. Probab. Theory Relat. Fields 153(1–2), 97–147 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Friz, P., Oberhauser, H.: Rough path stability of (semi-)linear SPDEs. Probab. Theory Relat. Fields 158(1–2), 401–434 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gyöngy, I.: On the approximation of stochastic differential equations. Stochastics 23(3), 331–352 (1988)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gyöngy, I.: On the approximation of stochastic partial differential equations. II. Stochastics 26, 129–164 (1989)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Gyöngy, I.: The stability of stochastic partial differential equations. II. Stoch. Rep. 27(3), 189–233 (1989)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gyöngy, I.: On stochastic partial differential equations. Results on approximations. In: Gerencsér, L., Caines, P. E. (eds) Topics in Stochastic Systems: Modelling, Estimation and Adaptive Control, Lecture Notes in Control and Information Sciences, vol. 161, pp. 116–136. Springer, Berlin (1991)

  13. 13.

    Gyöngy, I., Shmatkov, A.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54(3), 315–341 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gyöngy, I., Stinga, P.R.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. In: Seminar on Stochastic Analysis, Random Fields and Applications VII, Progress in Probability, vol. 67, pp. 95–130. Birkhauser, Basel (2013)

  15. 15.

    Hairer, M., Pardoux, E.: A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67(4), 1551–1604 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space. J. Eur. Math. Soc. 20(4), 1005–1054 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Krylov, N.V.: Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, vol. 142. American Mathematical Society, Providence (1995)

    Google Scholar 

  18. 18.

    Krylov, N.V.: An Analytic Approach to SPDEs, Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64, pp. 185–242. American Mathematical Society, Providence (1999)

    Google Scholar 

  19. 19.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence (1996)

    Google Scholar 

  20. 20.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, vol. 96. American Mathematical Society, Providence (2008)

    Google Scholar 

  21. 21.

    Krylov, N.V.: A relatively short proof of Itô’s formula for SPDEs and its applications. Stoch. Partial Differ. Equ. Anal. Comput. 1(1), 152–174 (2013)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Mackevičius, V.: \(S^P\)-stability of solutions of symmetric stochastic differential equations. Liet. Matem. Rink. 25(4), 72–84 (1985). (in Russian); English translation in Lithuanian Math. J. 25(4), 343–352 (1985)

  23. 23.

    Mackevičius, V.: The support of the solution of a stochastic differential equation. Litovsk. Mat. Sb. 26(1), 91–98 (1986). (in Russian); English translation in Lith. Math. J. 26(1), 57–62 (1986)

  24. 24.

    Millet, A., Sanz-Solé, M.: A simple proof of the support theorem for diffusion processes. In: Séminaire de Probabilités, XXVIII, Lecture Notes in Mathematics, vol. 1583, pp. 36–48. Springer, Berlin (1994)

  25. 25.

    Nakayama, T.: Support theorem for mild solutions of SDE’s in Hilbert spaces. J. Math. Sci. Univ. Tokyo 11(3), 245–311 (2004)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Walter de Gruyter Co., Berlin (1996)

    Google Scholar 

  27. 27.

    Shmatkov, A.: Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs. Ph.D. thesis, University of Edinburgh (2005)

  28. 28.

    Tessitore, G., Zabczyk, J.: Wong–Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6(4), 621–655 (2006)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Twardowska, K.: Wong–Zakai approximations for stochastic differential equations. Acta Appl. Math. 43(3), 317–359 (1996)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Twardowska, K.: On support theorems for stochastic nonlinear partial differential equations. In: Csiszár, I., Michaletzky G. Y. (eds) Stochastic Differential and Difference Equations, Progress in Systems Control Theory, vol. 23, pp. 309–317. Birkhauser, Boston (1997)

  31. 31.

    Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Yastrzhembskiy, T.: Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line. Stoch. PDE Anal. Comp. (2019). https://doi.org/10.1007/s40072-019-00152-8

    Article  Google Scholar 

Download references

Acknowledgements

This author is indebted to his advisor N.V. Krylov for the statement of the problem, useful suggestions and attention to this work. The author is also grateful to the organizers of RISM school on “Developments in SPDEs in honour of G. Da Prato”, where he had an opportunity to present the results of this paper and discuss it with other participants. Finally, this author would like to thank the anonymous referees whose comments led to the improvement of the presentation of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Timur Yastrzhembskiy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

Definition 5.1

Denote \( h = 1/n, \) \( \varkappa (t) = -1 \vee t \wedge 1, t \in {\mathbb {R}}. \) We say that the process \( w^k_n (t), t \ge 0 \) is the polygonal approximation of \(w^k\) if

$$\begin{aligned} w^k_n (t) = w^k ( (l-1) h ) + 1/h\, ( t - l h ) \varkappa (w^k ( l h ) - w^k ( (l-1)h )), \end{aligned}$$
(5.1)

for \( t \in [l h, (l+1) h) \), and \( l \in \{0, 1, 2, \ldots \} \).

The following lemma is similar to Proposition 6.3.1 of [27].

Lemma 5.1

Let \(p > 0\), \(\varepsilon > 0\), \( \theta \in (0, 1/2), \) \(\theta ' \in (0, \theta )\) be numbers. Assume that \(w^i_n\) is given by (5.1). Then, for any ij, the following assertions hold.

$$\begin{aligned}&\mathrm{(i)} \, E \left| \left| \delta w^i_n \right| \right| ^p_{ C [0, T] } \le N (p, T, \varepsilon ) n^{- p/2 + \varepsilon }. \\&\mathrm{(ii)} \, E \left| \left| \delta w^i_n \right| \right| ^p_{C^{1/2 - \theta }[0, T]} \le N (p, T, \theta , \theta ') n^{- \theta ' p}. \end{aligned}$$

(iii) If, in addition, \(p \ge 1\), then,

$$\begin{aligned}&\, E \left| \left| s^{i j}_n \right| \right| ^p_{ C [0, T]} \le N (p, T, \varepsilon ) n^{- p/2 + \varepsilon }. \\&\mathrm{(iv)} \, E \int _0^T \left| D s^{i j }_n (t) \right| ^p \, dt \le N (p, T). \\&\mathrm{(v)} \, E \left| \left| s^{i j}_n \right| \right| ^p_{ C^{1/2 - \theta } [0, T]} \le N (p, T, \theta , \theta ') n^{- \theta ' p}. \end{aligned}$$

Proof

Denote \(h = 1/n\), \( t_k = k h, k \in \{ -1, 0, 1, \ldots \}. \) For any \(a > 0\), \( f : {\mathbb {R}}\rightarrow {\mathbb {R}}, \) denote

$$\begin{aligned} \varDelta _a f (x)= & {} f (x + a) - f(x), \\ \rho _{f} ( h, T)= & {} \sup _{t, s \in [0, T]: |t - s| \le h} |f (t) - f(s)|. \end{aligned}$$

For the sake of convenience, in the proofs (i), (ii) we denote \( w: = w^i, w_n := w^i_n. \)

(i):

For \( t \in [t_l, t_{l+1}), \) we have

$$\begin{aligned} |\delta w_n (t)| \le |w(t) - w (t_{l-1})| ~ +\mid \varkappa (\varDelta _{h} w (t_{l-1}))\mid , \end{aligned}$$
(5.2)

and

$$\begin{aligned} \mid \varkappa (\varDelta _{h} w (t_{l-1}))\mid \le \varDelta _{ h } (w (t_{ l - 1 })) + I_{A_{l, n}}, \end{aligned}$$
(5.3)

where

$$\begin{aligned} A_{l, n} =\mid \varDelta _{ h } w (t_{l-1})\mid > 1. \end{aligned}$$

Denote \( M = \lfloor Tn \rfloor . \) By Chebyshov’s inequality, for any \(q > 0\),

$$\begin{aligned} P \left( \cup _{l = 0}^M A_{l, n}\right) \le \sum _{ l = 0}^M P (A_{l, n}) \le N/h \, E |w (h)|^q \le N h^{ q/2 - 1}. \end{aligned}$$
(5.4)

Then,

$$\begin{aligned} \begin{aligned}&E \max _{l = 0, \ldots , M} |\varkappa (\varDelta _{h} w (t_{l-1}))|^p \\&\quad \le N \left( E \rho ^p_{w} (h, T) + P \left( \cup _{l = 0}^M A_{l, n}\right) \right) \le N h^{p/2 - \varepsilon }, \end{aligned} \end{aligned}$$
(5.5)

where in the second inequality we used the estimate of \(\rho _w\) which we state below. By Theorem 2.3.2 of [17], for any \(\alpha > 0\), there exists a positive random variable \( N_{\alpha , T} \) such that, for any \( r > 0, \, E N_{\alpha , T}^r < \infty , \) and

$$\begin{aligned} \rho _{w} (\lambda , T) \le N_{\alpha , T} \, \lambda ^{ 1/2 - \alpha }, \, \forall \omega \in \varOmega , \lambda \in [0, T]. \end{aligned}$$
(5.6)

Then, the claim follows from (5.2), (5.5) and (5.6). Similarly, for all l, we have

$$\begin{aligned} E | \varkappa (\varDelta _{h} w (t_l))|^p \le N h^{p/2}. \end{aligned}$$
(5.7)
(ii):

Fix any \( \alpha \in (0, \theta ). \) First, we consider the case when \( |t - s| \ge h, \) \( t, s \in [0, T]. \) We have

$$\begin{aligned} 1/(t-s)^{1/2 - \theta } | \delta w_n (t) - \delta w_n (s)| \le 2 h^{ \theta - 1/2 } ||\delta w_n ||_{ C [0, T] }, \end{aligned}$$

and this combined with (i) yields the claim.

Next, we take any \(t, s \in [0, T]\) such that \(|t-s| < h\). There are two subcases: either

$$\begin{aligned} (t, s) \in B_1 = \cup _{l = 0}^M \{ (t, s) \in [0, T]^2: t, s \in [t_l, t_{ l+1 }] \} \end{aligned}$$
(5.8)

or

$$\begin{aligned} \begin{aligned} (t, s) \in B_2&= \cup _{l = 0}^{M-1} \{ (t, s) \in [0, T]^2: |t - s|< h, \\&\qquad t_l< s \le t_{l+1} \le t < t_{ l + 2} \}. \end{aligned} \end{aligned}$$
(5.9)

To handle (5.8) we write

$$\begin{aligned} |\delta w_n (t) - \delta w_n (s)| \le | w_n (t) - w_n (s)| + |w (t) - w (s)|. \end{aligned}$$
(5.10)

Using (5.6) and the fact that \( |t - s| \le h, \) we get

$$\begin{aligned} |w (t) - w (s)| \le N_{\alpha , T} h^{ \theta - \alpha } | t - s|^{1/2 - \theta }. \end{aligned}$$
(5.11)

Next, by (5.1), (5.5) we obtain

$$\begin{aligned} \begin{aligned} E \sup _{ (t, s) \in B_1 } | w_n (t) - w_n (s) |^p&\le |t - s|^p/h^p \, E \max _{l = 0, \ldots , M} |\varkappa (\varDelta _{h} w (t_{l - 1}))|^p \\&\le N h^{ (\theta - \alpha ) p} \, |t - s|^{(1/2 - \theta )p}. \end{aligned} \end{aligned}$$
(5.12)

Then, the claim in this subcase follows from (5.10) - (5.12).

We move to (5.9). Observe that

$$\begin{aligned}&| w_n (t) - w_n (s)| \\&\quad \le |w_n (t) - w_n (t_{l+1})| + |w_n (s) - w_n (t_{l+1})|, \end{aligned}$$

and \( (t, t_{l+1}), (s, t_{l+1}) \in B_1. \) This combined with (5.12) and (5.11) proves the assertion for the second subcase.

(iii):

We follow the proof of Proposition 6.3.1 of [27]. First, we consider the case \(i = j\). By Itô’s formula, for any t, a.s.

$$\begin{aligned} \left| w^i (t) - w^i_n (t)\right| ^2 = 2 \int _0^t \left( w^i (s) - w^i_n (s)\right) \, d\left( w^i (s) - w^i_n (s)\right) + t, \end{aligned}$$

and, then,

$$\begin{aligned} s^{i i}_n (t) = \int _0^t \delta w^i_n (s) \, dw^i (s) - 1/2 \, \left| w^i (t) - w^i_n (t)\right| ^2. \end{aligned}$$

Using Burkholder–Davis–Gundy inequality (see, for example, Theorem iv.4.1 [17])) and assertion (i), we get

$$\begin{aligned} E \left| \left| s^{i i}_n (t) \right| \right| ^p_{ C [0, T] } \le N E \left| \left| \delta w_n^i \right| \right| ^p_{ C [0, T] } \le N h^{p/2 - \varepsilon }. \end{aligned}$$
(5.13)

Now we assume \(i \ne j\). Note that, for \(t \in [t_k, t_{k+1}]\), we have

$$\begin{aligned} w^i_n (t) = -\varDelta _h w^i (t_{k - 1}) + w^i (t_k) + 1/h (t - t_k) \varkappa \left( \varDelta _h w^i (t_{k-1})\right) . \end{aligned}$$

Then, for each \(\omega , t\) we may write

$$\begin{aligned} s^{i j}_n (t) = I_1 (t) + I_2 (t) + I_3 (t), \end{aligned}$$
(5.14)

where

$$\begin{aligned} I_{1} (t)= & {} \sum _{ l = 0}^{ \lfloor tn \rfloor } \varkappa \left( \varDelta _{h} w^j (t_{l - 1})\right) \int _{t_l}^{t_{l+1}} \left( w^i (s) - w^i (t_l)\right) /h \, I_{s \le t} \, ds, \\ I_2 (t )= & {} \sum _{ l = 0}^{ \lfloor tn \rfloor } \varkappa \left( \varDelta _{h} w^j (t_{l - 1})\right) \varDelta _h w^i (t_{l - 1}), \\ I_3 (t)= & {} - h^{-2} \sum _{ l = 0}^{ \lfloor tn \rfloor } \int _{t_l}^{t_{l+1}} (s - t_l) I_{s \le t } \, ds \, \varkappa \left( \varDelta _{h} w^j (t_{l - 1})\right) \varkappa \left( \varDelta _{h} w^i (t_{l - 1})\right) . \end{aligned}$$

Observe that \( \varkappa (\varDelta _{h} w^i (t_{l - 1})) \) is a symmetric random variable as a composition of an odd function with a symmetric random variable. It follows from the Markov property of Wiener process that \( I_1 (t) \) is a sum of independent centered random variables. Since \(p \ge 1\), Burkholder–Davis–Gundy inequality is applicable [5]. By this and (5.7) we get

$$\begin{aligned} \begin{aligned}&E \sup _{t \in [0, T]} |I_1 (t) |^p \le N \left( \sum _{ l = 0}^{ \lfloor tn \rfloor } E \left| \varkappa \left( \varDelta _{h} w^j (t_{l - 1})\right) \right| ^2 \right. \\&\quad \left. \times E \left| \int _{t_l}^{t_{l+1}} \left( w^i (s) - w^i (t_l)\right) /h \, I_{s \le t} \, ds\right| ^2\right) ^{p/2} \le N h^{p/2}. \end{aligned} \end{aligned}$$
(5.15)

Further, the same argument yields

$$\begin{aligned} E \sup _{t \le T} \left( | I_{2} (t) |^p + |I_3 (t)|^p\right) \le N h^{p/2}. \end{aligned}$$
(5.16)

The claim follows from (5.13)–(5.16).

(iv):

Observe that

$$\begin{aligned} D s^{i j}_n (t) = \delta w^i_n (t) \varkappa \left( \varDelta _h w^j (t_{k-1})\right) - \delta _{i j}/2, \end{aligned}$$
(5.17)

for \(t \in [t_k, t_{k+1}]\). Then, by this and Cauchy–Schwartz inequality we get

$$\begin{aligned}&E \int _0^T \left| D s^{i j}_n (t)\right| ^p \, dt \le N (p, T) \\&\quad +\, h^{-p} \sum _{k = 0}^{ \lfloor T n \rfloor } \left( E \left| \varkappa \left( \varDelta _h w^j_n (t_{k-1})\right) \right| ^{2p}\right) ^{1/2} \int _{t_k}^{t_{k+1}} \left( E \left| \delta w^i_n (t)\right| ^{2p}\right) ^{1/2} \, dt. \end{aligned}$$

This combined with (5.2) and (5.3) proves the claim.

(v):

By (5.17) and Cauchy–Schwartz inequality we have

$$\begin{aligned} E \left| \left| D s^{i j }_n \right| \right| _{ L_{ \infty }[0, T] }^p \le h^{-p} M_{1, n} M_{2, n}, \end{aligned}$$

where

$$\begin{aligned} M_{1, n} = \left( E \left| \left| \delta w^i_n \right| \right| _{ C [0, T] }^{2p}\right) ^{1/2}, \quad M_{2, n} = \left( E \max _{l = 0, \ldots , \lfloor T n \rfloor } \left| \varkappa \left( \varDelta _{h} w (t_{l-1})\right) \right| ^{2p} \right) ^{1/2}. \end{aligned}$$

By (i) and (5.5)

$$\begin{aligned} M_{1, n}, M_{2, n} \le N (p, \varepsilon , T) h^{ p/2 - \varepsilon }, \end{aligned}$$

Then, by the above

$$\begin{aligned} E \left| \left| D s^{ i j }_n \right| \right| ^p_{ L_{\infty } [0, T] } \le N (p, \varepsilon , T) h^{- 2\varepsilon }. \end{aligned}$$
(5.18)

By the interpolation inequality (see, for example, Theorem 3.2.1 in [19]), for any \(\lambda > 0\),

$$\begin{aligned} \left| \left| s^{ i j }_n \right| \right| _{ C^{1/2 - \theta } [0, T]} \le N (\theta , T) \left( h^{1/2 + \theta } \left| \left| D s^{i j}_n\right| \right| _{ L_{\infty } [0, T]} + h^{ \theta - 1/2 } \left| \left| s^{i j}_n \right| \right| _{ C[0, T]}\right) . \end{aligned}$$

We finish the proof by combining this with (iii) and (5.18).

\(\square \)

Appendix B

Lemma 6.1

Let \( g \in C^1_{loc} ({\mathbb {R}}), \) \( D g \in L_{\infty } ({\mathbb {R}}), \) \(g(0) = 0\), and \(u \in H^1_p ({\mathbb {R}}^d)\). Then,

$$\begin{aligned} || g (u (\cdot ) ) ||_{ 1, p } \le N ( d, p) || D g ||_{\infty } || u ||_{ 1, p }. \end{aligned}$$

Proof

Recall that the spaces \(W^1_p ({\mathbb {R}}^d)\) and \(H^1_p ({\mathbb {R}}^d)\) coincide as sets and have equivalent norms. By this and the chain rule in \(W^1_p ({\mathbb {R}}^d)\) we may write

$$\begin{aligned} || g (u (\cdot )) ||_{ 1, p }\le & {} N (d, p) ( || g ( u (\cdot ) ) ||_p + || D_i u (\cdot ) D g (u (\cdot ))||_p) \\\le & {} N || D g ||_{\infty } (|| u ||_p + || D_i u ||_{p}) \\= & {} N ||D g ||_{\infty } ||u ||_{W^1_p ({\mathbb {R}}^d)} \le N || D g ||_{\infty } || u ||_{1, p}. \end{aligned}$$

\(\square \)

Lemma 6.2

Let \( u, v \in H^1_p ({\mathbb {R}}^d) \) and assume that

$$\begin{aligned} || D g||_{\infty } + || D^2 g ||_{\infty } \le K, \quad || D_i u ||_{\infty } \le R. \end{aligned}$$
(6.1)

Then,

$$\begin{aligned} || g(u (\cdot ) ) - g(v (\cdot ) ) ||_{1, p} \le N (d, p) K ( 1 + R) || u - v ||_{1, p}. \end{aligned}$$

Proof

By the argument of the proof of Lemma 6.1 we have

$$\begin{aligned}&|| g(u (\cdot ) ) - g(v (\cdot ) ) ||_{ 1, p } \\&\quad \le N (d, p) (|| g (u (\cdot ) ) - g (v (\cdot ) ) ||_p + || D_i u (\cdot ) D g (u (\cdot ) ) - D_i v (\cdot ) D g (v (\cdot ) ) ||_p) \\&\quad \le N K ||u - v||_p + N || D_i u (\cdot ) [ D g (u (\cdot ) ) - D g (v (\cdot ) ) ] ||_p \\&\qquad + N || D g(v (\cdot ) ) [ D_i u (\cdot ) - D_i v (\cdot ) ]||_p. \end{aligned}$$

By this and (6.1) we obtain the assertion of the lemma. \(\square \)

Lemma 6.3

Let \( p > d\), \(\delta \in (0,1 )\), \(\gamma \in (d/p, 1), \tau > 0\) be numbers. Let \(g: {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a function such that \(D g \in C^{1 + \theta } ({\mathbb {R}})\), and \(g(0) = 0\). Denote

$$\begin{aligned} {\mathcal {B}}: = C^{\delta } \left( [0, \tau ], H^{1 + \gamma }_p ({\mathbb {R}}^d) \right) \end{aligned}$$

and take any \(u \in {\mathcal {B}}\). Then, there exists a constant N independent of g and u such that

$$\begin{aligned} ||g(u)||_{{\mathcal {B}}} \le N || D g ||_{ C^{1 + \theta } } \left( ||u||_{ {\mathcal {B}}} + ||u||^{2 + \theta }_{ {\mathcal {B}}} \right) . \end{aligned}$$

Proof

For the sake of convenience, we denote \( u_t = u (t, \cdot )\), and we omit the dependence of u on the spatial variable x. We set \( N_g = || D g ||_{ C^{1 + \theta } }. \)

First, we prove the supremum norm estimate. Note that \( H^{1 + \gamma }_p ({\mathbb {R}}^d) \) is embedded in \( C ({\mathbb {R}}^d) \) since \( 1 + \gamma > d/p\) (see, for instance, Theorem 13.8.1 of [20]). Then, by this and Corollary 3 combined with Remark 3 of Section 5.3.7 of [26] we have

$$\begin{aligned} E \sup _{t \le T} || g (u_{t}) ||_{ 1 + \gamma , p } \le N_g \left( || u ||_{ {\mathcal {B}}} + || u ||^{1 + \gamma }_{ {\mathcal {B}}}\right) . \end{aligned}$$
(6.2)

Next, take any \( s, t \in [0, \tau ] \) such that \(s \ne t\). By the fact that \((1 - D_i)\) is a strongly elliptic differential operator of order 1 we obtain (see Theorem 13.3.10 of [20])

$$\begin{aligned} || g ( u_t ) - g ( u_s) ||_{1+ \gamma , p } \le N \left( J^{(1)} + J^{(2)}\right) , \end{aligned}$$
(6.3)

where

$$\begin{aligned} J^{(1)}= & {} || g ( u_t ) - g( u_s ) ||_{ \gamma , p }, \\ J^{(2)}= & {} || D g ( u_t ) D_i u_t - D g( u_s ) D_i u_s ||_{ \gamma , p }. \end{aligned}$$

Recall that, by the elementary embedding (see Sect.  2) we may replace \(\gamma \) by 1 in the expression for \(J^{(1)}\). Since \(\gamma > d/p\), by the embedding theorem for \(H^{\mu }_p ({\mathbb {R}}^d)\) spaces we have

$$\begin{aligned} \sup _{t \le T} ||D_i u_t ||_{\infty } \le N || u ||_{ {\mathcal {B}}}. \end{aligned}$$

Then, by Lemma 6.2 and what was just said we obtain

$$\begin{aligned} J^{(1)} \le N N_g (1 + || u ||_{ {\mathcal {B}}}) || u_t - u_s||_{1, p}. \end{aligned}$$
(6.4)

Next, by the triangle inequality

$$\begin{aligned} J^{(2)} \le J^{ (2, 1) } + J^{ (2, 2)} + J^{ (2, 3) }, \end{aligned}$$
(6.5)

where

$$\begin{aligned} J^{(2, 1)}= & {} || (D g ( u_t ) - D g (0)) ( D_i u_t - D_i u_s) ||_{ \gamma , p }, \\ J^{(2, 2)}= & {} || D_i u_s (D g( u_t ) - D g( u_s )) ||_{\gamma , p}, \\ J^{(2, 3)}= & {} |D g (0)| || u_t - u_s ||_{1 + \gamma , p}. \end{aligned}$$

It is well-known that \( H^{\gamma }_p ({\mathbb {R}}^d) \) is a multiplication algebra because \(\gamma > d/p\) (see, for example, Theorem 1 in Section 4.6.1 of [26]). Then, we get

$$\begin{aligned} J^{(2, 1)}\le & {} N || D g (u_t) - D g(0) ||_{\gamma , p} || u_t - u_s ||_{1 + \gamma , p}, \\ J^{(2, 2)}\le & {} N || u_s ||_{1 + \gamma , p} || D g (u_t) - D g(u_s) ||_{\gamma , p}. \end{aligned}$$

To handle \(J^{(2, 1)}\) we estimate \(|| D g (u_t) - D g(0) ||_{1, p}\) via Lemma 6.1. By this we have

$$\begin{aligned} J^{(2, 1)} \le N N_g || u_t ||_{ 1, p} || u_t - u_s ||_{1 + \gamma , p}. \end{aligned}$$
(6.6)

Next, using the fact that \(D g^2 \in C^{\theta }\), we obtain

$$\begin{aligned}&|| D g (u_t) - D g (u_s)||_{\gamma , p} \le || D g (u_t) - D g(u_s)||_{1, p} \\&\quad \le N || D g (u_t) - D g (u_s)||_{ p} \\&\qquad + N \left| \left| D^2 g (u_s) (D_i u_t - D_i u_s) \right| \right| _p \\&\qquad + N \left| \left| D_i u_t (D^2 g (u_t) - D^2 g (u_s) ) \right| \right| _p \\&\quad \le N N_g ||u_t - u_s||_{1 + \gamma , p} + N N_g || u_t ||_{ 1, p } || u_t - u_s||^{ \theta }_{ \infty }. \end{aligned}$$

Again, by the embedding theorem we may replace \(|| u_t - u_s||_{ \infty }\) by \(|| u_t - u_s ||_{ 1 + \gamma , p}\). Then, by the above we have

$$\begin{aligned} J^{(2,2)} \le N N_g ||u_s||_{ 1 + \gamma , p} \left( ||u_t - u_s||_{1 + \gamma , p} + ||u_t||_{1 + \gamma , p} || u_t - u_s||^{\theta }_{ 1 + \gamma , p}\right) \end{aligned}$$
(6.7)

The assertion follows from (6.3)–(6.7). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yastrzhembskiy, T. Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space. Stoch PDE: Anal Comp 9, 71–104 (2021). https://doi.org/10.1007/s40072-020-00168-5

Download citation

Keywords

  • SPDE
  • Wong–Zakai approximation
  • Stroock–Varadhan’s support theorem
  • Krylov’s \(L_p\)-theory of SPDEs

Mathematics Subject Classification

  • 35R60
  • 60H15