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.

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 i, j, 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). $\square$