Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line

Abstract

We prove a Stroock–Varadhan’s type support theorem for a stochastic partial differential equation on the real line with a noise term driven by a cylindrical Wiener process on $L_2 ({\mathbb {R}})$. The main ingredients of the proof are V. Mackevičius’s approach to support theorem for diffusion processes and N.V. Krylov’s $L_p$-theory of SPDEs.

Appendix

The following lemma is taken from Appendix A of [19].

Lemma 6.1

Let $p > 1$, $T > 0$, $h \in (0, 1 \wedge T)$, $\varepsilon > 0$, $\theta \in (0, 1/2)$, $\theta ' \in (0, \theta )$ be numbers. Assume that (A4) (h) holds. Then, for any $i, j \in {\mathbb {N}}$, the following assertions hold.

$$\begin{aligned}&(i)\, E || \delta w^i (\cdot , h) ||^p_{ C [0, T] } \le N (p, T, \varepsilon ) h^{ p/2 - \varepsilon }. \\&(ii)\, E ||\delta w^i (\cdot , h) ||^p_{ C^{1/2 - \theta } [0, T] } \le N (p, T, \theta , \theta ') h^{ \theta ' p}. \\&(iii)\, E || s^{i j} (\cdot , h) ||^p_{ C [0, T] } \le N (p, T, \varepsilon ) h^{p/2 - \varepsilon }. \\&(iv) \, E \int _0^T (|\delta w^i (t, h) |^p + |s^{i j} (t, h)|^p) \, dt \le N (p, T) h^{p/2},\\&\qquad E \int _0^T |D s^{i j} (t, h)|^p \, dt \le N (p, T). \\&(v) \, E || s^{i j} (\cdot , h) ||^p_{ C^{1/2 - \theta } [0, T]} \le N (p, T, \theta , \theta ') h^{ \theta ' p}. \end{aligned}$$

Proof

The proof of (i)–(iii), (v) and the proof of the second assertion of (iv) can be found in [19] (see Lemma 5.1). Here we show that the first estimate of (iv) holds. To do that, we only need to prove the following:

$$\begin{aligned} \sup _{t \le T} E (|\delta w^i (t, h)|^p + |s^{i j} (t, h)|^p) \le N (p, T) h^{p/2}. \end{aligned}$$

The above inequality for $\delta w^i (\cdot , h)$ follows from the formulas (5.2) and (5.7) of [19].

Next, we prove the claim for $s^{i j} (\cdot , h)$. First, consider the case $i = j$. By Itô’s formula

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

and, then, by Burkholder–Davis–Gundy inequality

$$\begin{aligned} \sup _{t \le T} E |s^{i i} (t, h)|^p \le N (p, T) \sup _{t \le T} E (|\delta w^i (t, h)|^p + |\delta w^i (t, h)|^{2p}) \le N (p, T) h^{p/2}. \end{aligned}$$

The proof in case $i \ne j$ was actually given in [19] (see formulas (5.14)–(5.16)). $\square$

Lemma 6.2

Assume that $(A4) (\gamma _n)$ holds for some sequence $\{\gamma _n, n \in {\mathbb {N}}\}$ of positive numbers. Let $\theta \in (0, 1), T > 0, p \ge 2$ be numbers, and let $\varDelta _n$ be any function of class ${{\varvec{\Delta }}_{\mathbf{n}}}$. Then, the following assertions hold.

(i):

$$\begin{aligned} J_n : = E \int _0^T ||\varDelta _n (t, \star ) ||^p_p \, dt \le N n^{N} \gamma _n^{p/2}, \end{aligned}$$

where $N = N (p, T)$.

(ii):

For any $\varepsilon > 0$, and any $\delta \in (0, 1)$,

$$\begin{aligned} E \sup _{t \le T} || \varDelta _{n} (t, \star ) ||^p_{C^{2 - \delta }} \le N n^{N} \gamma _n^{p/2 - \varepsilon }, \end{aligned}$$

where $N = N (p, T, \delta , \varepsilon )$.

(iii):

For any $\varepsilon > 0$, and $\varepsilon ' \in (0, \varepsilon )$,

$$\begin{aligned} E || \varDelta _n ||^p_{ C^{1/2 - \varepsilon } ([0, T], H^{\theta }_p ) } \le N n^{N} \gamma _n^{\varepsilon ' p}, \end{aligned}$$

where $N = N (p, T, \varepsilon , \varepsilon ')$.

Proof

(i):

Due to Definition 3.1 we have

$$\begin{aligned} \begin{aligned} J_n&\le N n^{2p - 2} \sum _{i, j = 1}^{ n} \sum _{ k, l, m = 0}^2 (||{D^{k}} \phi _i||_p^p + || \phi _{i, j}^{l, m} ||_p^p) \\&\quad \times E \int _0^T (|s^{i j}_n (t)|^p + |\delta w^i_n (t)|^p) \, dt. \end{aligned} \end{aligned}$$

(6.1)

Next, it is well-known (see Lemma 1.5.2 of [16]) that, for any $\rho \in [2, \infty ]$, and $k \in {\mathbb {N}}$,

$$\begin{aligned} || \phi _k ||_{\rho } \le N (\rho ). \end{aligned}$$

(6.2)

In addition (see Sect. 1.1 of [16]),

$$\begin{aligned} D H_k (x) = 2k H_{k-1} (x). \end{aligned}$$

Then, by formula (2.5) and what was just said we have

$$\begin{aligned} || D^k \phi _j ||_{\rho } \le N (k, \rho ) j^{k/2}, \, \, k \in {\mathbb {N}}\cup \{0\}, j \in {\mathbb {N}}. \end{aligned}$$

(6.3)

Combining (6.1) and (6.3) with Lemma 6.1 (iv), we prove the assertion.

(ii):

The proof is similar the one above. First, note that by the interpolation inequality for Hölder spaces (see Theorem 3.2.1 in [8]), we may replace $2 - \delta$ by 3. Second, by the product rule

$$\begin{aligned} || \phi ^{l, m}_{i, j} ||_{ C^3 } \le N ||\phi _i||^l_{C^3} ||\phi _j||^m_{C^3}. \end{aligned}$$

(6.4)

Third, by Lemma 6.1 (i), (iii), for any $i, j \in {\mathbb {N}}$,

$$\begin{aligned} E ||q_{i j}||_{C[0, T]}^p \le N (p, T, \varepsilon ) \gamma _n^{p/2 - \varepsilon }, \end{aligned}$$

(6.5)

where $q_{i j} \in \{ \delta w^i_n, s^{i j}_n \}.$ Now the assertion follows from (6.4), (6.3), and (6.5).

(iii):

First, by the properties of $H^{\theta }_p$ spaces, for $k, l, m \ge 0,$

$$\begin{aligned} \begin{aligned} ||D^{k} \phi _i||_{\theta , p} + ||\phi _{i, j}^{ l, m}||_{\theta , p}&\le || \phi _i ||_{ k + 1, p } + || \phi _{i, j}^{ l, m} ||_{ 1, p}\\&\le N (p, k) ||\phi _i||_{W^{k+1}_p} + N (p) || \phi _{i, j}^{ l, m} ||_{ W^1_p }. \end{aligned} \end{aligned}$$

(6.6)

Clearly, we may assume that $m \ne 0$. Note that by (6.3) the right hand side of (6.6) is less than

$$\begin{aligned}&N (p, k) i^{(k+1)/2} + N (p) || \phi _{i }||^l_{C^1} || \phi _j ||^{m-1}_{ C^1 } (|| \phi _j ||_p + ||D \phi _j||_p) \\&\quad \le N (p, k, l, m) (i^{(k+1)/2} + i^{l/2} j^{m/2}). \end{aligned}$$

This combined with Lemma 6.1 (ii) and (v) proves the claim.

$\square$

Lemma 6.3

Let $\varepsilon \in (0, 1)$ be a number. Let $\rho \in C^{\infty }_0$ be a function supported on (0, 1), and $h \in {\mathbb {L}}_p (T)$. Denote

$$\begin{aligned} {\tilde{h}} (t, x) = \varepsilon ^{-2} \iint _{{\mathbb {R}}^2} h (t - s, x - y) I_{t - s > 0} \rho ( s/\varepsilon ) \rho ( y/\varepsilon ) \, ds dy. \end{aligned}$$

Then, the following assertions hold.

(i):

For any $\gamma > 0$, ${\tilde{h}} (t, \star ), t \ge 0$ is a continuous ${\mathcal {F}}_t$-adapted $H^{\gamma }_p$-valued process, and, hence, it is a predictable $H^{\gamma }_p$-valued process.

(ii):

Let $T > 0$, $\gamma \in (0, 2)$, $\delta > 0$, $k \in {\mathbb {N}}\cup \{0\}$ be numbers, and $\tau \le T$ be a stopping time. Let $\{\gamma _n, n \in {\mathbb {N}}\}$ be sequence of positive numbers, and $\varDelta _n$ be any function of class ${\varvec{\Delta }}_{\mathbf{n}}$. Then, there exists a constant $N ( p, T, \gamma , k, \delta ) > 0$ such that

$$\begin{aligned} \begin{aligned} I: =&E \int _0^{\tau } || \varDelta _n (t, \star ) \partial ^k_t {\tilde{h}} (t, \star )||^p_{\gamma , p} \, dt \\ \le&N n^{N} \gamma ^{p/2 - \delta }_n \varepsilon ^{ -(\lceil \gamma \rceil + k) p} E \int _0^{\tau } || h (t, \star ) ||^p_{ p} \, dt. \end{aligned} \end{aligned}$$

(6.7)

Proof

(i):

Note that, for any $m \in {\mathbb {N}}\cup \{0\}$,

$$\begin{aligned}&D_x^m \partial ^k_t {\tilde{h}} (t, x) \nonumber \\&\quad = \varepsilon ^{-2 - m - k} \int _0^{ t} \int _{ {\mathbb {R}}} h (t - s, x - y) (\partial _s^k \rho ) ( s/\varepsilon ) (D_x^m \rho ) ( y/\varepsilon ) I_{t> s > 0} \, ds dy.\nonumber \\ \end{aligned}$$

(6.8)

Then, by the properties of $H^{\gamma }_p$ spaces (see Sect. 2), (6.8) and Minkowski inequality, and Hölder’s inequality, for any $t \ge 0$,

$$\begin{aligned} \begin{aligned}&|| \partial ^k_t {\tilde{h}} (t, \star ) ||_{\gamma , p} \le || \partial ^k_t {\tilde{h}} (t, \star ) ||_{\lceil \gamma \rceil , p} \le N (\gamma , p) || \partial ^k_t {\tilde{h}} (t, \star ) ||_{ W^{\lceil \gamma \rceil }_p} \\&\quad \le N (\gamma , p, \rho ) \varepsilon ^{ -1 - \lceil \gamma \rceil - k} \int _0^{ t} || h (t - s, \star ) ||_p |(\partial _t^k \rho ) ( s/\varepsilon )| \, I_{t> s > 0} ds \\&\quad \le N (\gamma , p, k, \rho ) t^{(p-1)/p} \varepsilon ^{ - 1 - \lceil \gamma \rceil - k} \left( \int _0^{ t} || h (s, \star ) ||^p_p \, ds\right) ^{1/p}. \end{aligned} \end{aligned}$$

(6.9)

Hence, $\partial ^k_t {\tilde{h}} (t, \star ), t \ge 0$ is an $H^{\gamma }_p$-valued function. By a similar argument combined with the dominated convergence theorem, it is also a continuous process taking values in the same space. Finally, by a standard approximation argument combined with (6.9) we conclude that $\partial ^k_t {\tilde{h}} (t, \star )$ is an ${\mathcal {F}}_t$-adapted $H^{\gamma }_p$-valued process.

(ii):

First, note that by (i) the integral on the left-hand side of (6.7) is a well-defined random variable. Next, by Lemma 5.2 (i) of [7]

$$\begin{aligned} I \le E \int _0^{\tau } || \varDelta _n (t, \star ) ||^p_{ C^{\gamma + \eta } } ||\partial ^k_t {\tilde{h}} (t, \star )||^p_{\gamma , p} \, dt, \end{aligned}$$

where $\eta \in (0, 2 - \gamma )$ is a number. Using Lemma 6.2 (ii),

$$\begin{aligned} I \le N n^N \gamma ^{p/2 - \delta }_n E \int _0^{\tau } ||\partial ^k_t {\tilde{h}} (t, \star )||^p_{\gamma , p} \, dt. \end{aligned}$$

To estimate the last integral we use the third inequality in (6.9). By a standard argument that uses Minkowski inequality we get

$$\begin{aligned} I \le N n^N \gamma ^{p/2 - \delta }_n \varepsilon ^{ -(\lceil \gamma \rceil + k) p} E \int _0^{\tau } || h (t, \star ) ||^p_p \, dt. \end{aligned}$$

$\square$