Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Introduction

Under (a1) and (a2), we first search for minimal conditions on $b$ ensuring the existence and pathwise uniqueness of mild solutions to (1.1), then study gradient estimates and Harnack inequalities of the associated semigroup.

In general, the mild solution (if exists) can be explosive. So, we consider mild solutions with life times.

then the mild solution is non-explosive and $P_{t}$ is strong Feller for $t>0$.

Without loss of generality, in Theorem 1.1 one may take $B=0$ in Theorem 1.1. But the situation is different in the next result (Theorem 1.2) where the singular part $b$ is bounded in the space variable, so that the appearance of $B$ allows the whole drift $B_{t}+b_{t}$ unbounded and singular.

Comparing with the above mentioned results of , Theorem 1.1 contains the following several new points: (1) It works for multiplicative noise; (2) It works for arbitrary starting points and non-Hölder continuous drift; (3) The assertion on the strong Feller property is new, see also Remark 4.1 for a discussion on Harnack inequalities. Moreover, condition (1.4) is more general than

for some constants $C,p>0$ which is used in [7, Theorem 16] to enure the non-explosion of the solution. See [7, Remark 17] for an explanation on the reasonability of such a condition in infinite dimensions.

The main difficulty in the proof of Theorem 1.1 comes from the singular drift $b$. To overcome this difficulty, a regularization argument has been introduced in and further developed in , to reformulate the mild solution by using a regular functional instead of $b$. This functional is constructed by solving an equation involving in the resolvent associated to the corresponding regular equation, i.e. the equation (1.1) without $b$. Based on such a regularization formulation, the uniqueness can be proved as in where the transport equation for Hölder continuous vector fields with a finite-dimensional multiplicative noise is concerned. See also and references therein for the study of singular SPDEs using regularization by the space-time white noise.

The key point in the proof of Theorem 1.1 is to realize the idea of for the present situation where $Q$ is non-constant and $b$ is non-Hölder continuous. This is done by establishing necessary derivative estimates using minimal continuity conditions on $b$.

holds for some $\Psi\in C([0,\infty))$ and all $T\in[0,\infty)$.

$Q$ satisfies (1.3). Moreover, for any $T>0$, there exists $\phi\in\mathscr{D}$ such that

For any $T>0$ there exists a constant $C(T)>0$ such that

(1) According to , the key point in the proof of Theorem 1.2 is the gradient estimate

As the regularization formula (2.30) still contains a non-Lipschitz term $Au_{s}$, the standard argument in the literature is invalid. Our proof is new in this singular setting (see the proof of Lemma 6.1(2)).

Then the assertions in Theorem 1.2 hold for $P_{s,s+t}$ in place of $P_{t}$ with $C(T)$ and $C$ depending also on $s$. If conditions in (a2’) and (a3’) are uniformly in $T$ (i.e. they hold with $T=\infty$ and $[0,\infty)$ in place of $[0,T]$), then these constants are independent of $s$ and, by the semigroup property, we may take $C(T)=c_{1}\text{\rm{e}}^{c_{2}T}$ for some constants $c_{1},c_{2}>0$.

The remainder of the paper is organized as follows. In Section 2, we present some gradient estimates on the semigroup for the corresponding O-U type equation, i.e. (1.1) with $B=b=0$. These gradient estimates enable us to prove the desired regularization representation of the mild solution to (1.1) with non-Hölder drift $b$. In Section 3, we prove the pathwise uniqueness using the regularization representation and, in Section 4, we investigate the strong Feller property and discuss Harnack inequalities for the semigroup. Results in Sections 3-4 are derived under some global conditions. Combining these results with a truncating argument, we prove Theorem 1.1 in Section 5. Finally, we prove Theorem 1.2 in Section 6 by using the regularization representation and finite-dimensional approximations.

Regularization representation of mild solutions

If $(X_{t})_{t\in[0,T]}$ solves (1.1), then $\{\theta_{t}(X_{t})\}_{t\in[0,T]}$ solves a regular equation having pathwise uniqueness.

In this way we prove the pathwise uniqueness of (1.1). For readers’ convenience, we briefly explain the idea of the construction of $\theta$ (see also ).

To ensure that coefficients in (2.1) are regular as required by point (b), we set

i.e. $\partial_{t}u_{t}=-L_{t}u_{t}-b_{t}$. In particular, with $u_{T}=0$ we have

where $\{P_{s,t}^{0}\}_{0\leq s\leq t}$ is the semigroup associated to the O-U type equation

It is well known that under assumptions (a1) and (a2’), the equation (2.3) has a unique mild solution which is non-explosive (see ). We have

To ensure $\|\nabla u_{s}\|_{\infty}<1$ as required by point (a), instead of (2.2) we consider

for large enough $\lambda>0$, which also ensures the desired regularity of the equation (2.1), see (6.3) and (6.4) below for details.

To verify the regularity properties of $u_{s}$ solving (2.4) for large $\lambda>0$, we first consider derivative estimates on $P_{s,t}^{0}$. In the following result, (2.5) is more or less standard, but (2.6) is new.

Assume (a1) and (a2’) with $b=0$. Let $T>0$ be fixed.

There exists two constants $c_{1},c_{2}>0$ such that for any increasing $\phi:[0,\infty)\to[0,\infty)$ with concave $\phi^{2}$,

(1) We shall make use of the following Bismut formula

holds for some constant $c>0$; see, for instance, [9, Remark 9.5].

In the same manner of [9, Remark 9.5], but using the Malliavin derivative $D_{h}$ to replace the directional derivative $\nabla_{\eta}$, we see that the Malliavin derivative process $(D_{h}Z_{s,r}^{x})_{r\in[s,t]}$ is the unique mild solution to the equation

Combining this with (2.9) and the definition of $h$, we see that both $(\frac{r-s}{t-s}\nabla_{\eta}Z_{s,r}^{x})_{r\in[s,t]}$ and $(D_{h}Z_{s,r}^{x})_{r\in[s,t]}$ solve the equation

By the uniqueness of the mild solution to this equation, we obtain $D_{h}Z_{s,t}^{x}=\frac{t-s}{t-s}\nabla_{\eta}Z_{s,t}^{x}=\nabla_{\eta}Z_{s,t}^{x}.$ So, by the chain rule and the integration by parts formula in the Malliavin calculus, we arrive at

Now, according to (2.8), (2.10) and (a2’), there exists a constant $c>0$ such that

Next, writing $P_{s,t}^{0}=P_{s,\frac{t+s}{2}}^{0}P_{\frac{t+s}{2},t}^{0}$ by the Markov property, and applying (2.8) to $\frac{t+s}{2}$ and $P_{\frac{t+s}{2},t}^{0}f$ instead of $t$ and $f$, we obtain

To verify this formula, we need to apply the dominated convergence theorem. In the spirit of [9, Remark 9.5], (a1), (a2’) and (2.9) imply that $\gamma_{r}:=\nabla_{\eta^{\prime}}\nabla_{\eta}Z_{s,r}^{x}$ is the unique mild solution to the equation

holds for some constant $c>0$. Combining this with (2.12), (2.10) and (a2’), we derive (2.14) from (2.13) by using the dominated convergence theorem. Moreover, (2.14) implies

holds for some constant $c>0$. Combining this with (2.12) we prove (2.5).

Since $Z_{s,t}^{x}-\text{\rm{e}}^{(t-s)A}x=\int_{s}^{t}\text{\rm{e}}^{(t-r)A}Q_{r}(Z_{s,r}^{x})\text{\rm{d}}W_{r}=:\beta_{s,t},$ by (2.7) and noting that $\phi^{2}$ is concave and increasing, we obtain

holds for $c:=\|Q\|^{2}_{T,\infty}2^{\varepsilon}\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{1-\varepsilon}}<\infty,$ (2.6) follows from (2.17). ∎

As a straightforward consequence of Lemma 2.1, we have the following result on the resolvent

Assume (a1) and (a2’) with $b=0$. Let $T>0$ be fixed.

For any $\phi\in\mathscr{D}$, there exists a decreasing function $\delta_{\phi}:[0,\infty)\to(0,\infty)$ such that

Next, since for $\phi\in\mathscr{D}$ we have $\int_{0}^{T}\frac{\phi(c_{2}s^{\varepsilon/2})}{s}\text{\rm{d}}s<\infty,$ Lemma 2.1(2) implies the second assertion for

In the next result, we characterize the solution $u_{s}$ to (2.4) which will be used to formulate the mild solution to (1.1) (see Proposition 2.5 below). To prove the formulation in infinite-dimensions, we shall adopt an approximation argument based on the second assertion of the following result.

Assume (a1) and (a2’), and let $T>0$ be fixed. Then there exists a constant $\lambda(T)>0$ such that the following assertions hold.

Let $P_{s,t}^{\{n\}}$ be defined as $P_{s,t}^{0}$ for $Q\circ\pi_{n}$ in place of $Q$, and let $b^{\{n\}}=b\circ\pi_{n}$. Then for any $\lambda\geq\lambda(T)$ and $n\geq 1$, the equation

where $\delta$ is a function such that $\delta(\lambda)\to 0$ as $\lambda\to\infty$. If, moreover, $b$ satisfies $\eqref{BB}$ for some $\phi\in\mathscr{D},$ then

holds for some positive function $\delta_{\phi}$ such that $\lim_{\lambda\to\infty}\delta_{\phi}(\lambda)=0$.

Next, by (2.12) and the definition of $\Gamma$, we have

for some constants $C_{1},C_{2}>0$. Combining this with (2.23) we may find $\lambda_{0}(T)>0$ such that the operator $\Gamma$ is a contraction operator on $\mathscr{H}$ when $\lambda\geq\lambda_{0}(T).$

(2) Obviously, if $(B,b,Q)$ satisfies (a2’), so does $(B\circ\pi_{n},b\circ\pi_{n},Q\circ\pi_{n})$ uniformly in $n\geq 1.$ By (1), $(u^{\{n\}})_{n\geq 1}$ are well defined for $\lambda\geq\lambda_{0}(T)$. Due to (2.12), there exists a constant $C>0$ such that

Taking $\lambda_{1}(T)\geq\lambda_{0}(T)$ such that $C\int_{0}^{T}\frac{\text{\rm{e}}^{-\lambda_{1}(T)t}}{\sqrt{t}}\text{\rm{d}}t\leq\frac{1}{2}$, we obtain

Combining this with the definition of $u^{\{n\}}$ we prove

for some function $\delta$ with $\delta(\lambda)\to 0$ as $\lambda\to\infty$. Moreover, we have

for some constant $C_{1}>0$. Obviously, by the dominated convergence theorem we have

Moreover, by (2.19) and Lemma 2.2(1), for every $t\in[0,T]$, the following Lemma 2.4 applies to

for some constant $C>0$ according to (2.12) which also holds for $P_{s,t}^{\{n\}}$ in place of $P_{s,t}^{0}$, we can apply the dominated convergence theorem to obtain

holds for some constant $C_{2}>0.$ Taking $\lambda_{2}(T)=\lambda_{1}(T)\lor(4C_{2}^{2})$, we obtain $\|g\|_{T,\infty}\leq\frac{1}{2}\|g\|_{T,\infty}$ for $\lambda\geq\lambda_{2}(T).$ Since $g$ is bounded, this implies

provided $\lambda\geq\lambda_{2}(T)$. Combining this with (2.24) and Lemma 2.4 below, and using again the dominated convergence theorem, we obtain $\limsup_{n\to\infty}|u_{s}-u_{s}^{\{n\}}|=0.$ Therefore, (2.21) holds for $\lambda\geq\lambda_{2}(T).$

Finally, let $b$ satisfy $\eqref{BB}$ for some $\phi\in\mathscr{D}$. Then, by Lemma 2.2(1), there exists a constant $c>0$ such that the functions

Let $(Z_{s,t}^{\{n,x\}})_{t\geq s}$ solve the equation

By letting $r\downarrow 0$ we prove $\lim_{n\to\infty}|P_{s,t}^{0}f_{n}-P_{s,t}^{\{n\}}f_{n}|=0.$

Similarly to (2.28), we can prove $\lim_{n\to\infty}J_{n}=0$ uniformly in $|\eta|\leq 1.$ Moreover, since

from (2.27) we see that $\lim_{n\to\infty}J_{n}^{\prime}=0$ uniformly in $|\eta|\leq 1.$ Therefore,

Let $G^{(n)}_{r}=\nabla_{b_{r}^{(n)}}u^{(n)}_{r}+b_{r}^{(n)},r\geq 0.$ To regularize this functional, we fix $\delta>0$ and let

So, by (2.31) and Itô’s formula, for any $0\leq s\leq r\leq T$, we have

Then, by $u^{(n,\delta)}=u^{(n,\delta)}\circ\pi_{n}$ and Itô’s formula,

Combining this with (2.38) and noting that $(\nabla_{b_{s}^{(n)}}u_{s}^{(n)})\circ\pi_{n}=(\nabla_{b_{s}}u_{s}^{(n)})\circ\pi_{n},$ we obtain

Finally, we complete proof by using (a3’) and (a3”) respectively.

Moreover, combining (2.42) with (2.12) which also holds for $P_{s,t}^{(n)}$ in place of $P_{s,t}^{0}$, we obtain

So, to deduce (2.41) from (2.40) with $\delta\downarrow 0$, it remains to prove

This follows since by the boundedness of $b$, the uniform boundedness and continuous of $\nabla_{b_{s}}u_{s}^{(n)},G_{s}^{(n)},$ and (2.43), we have

Since $b_{s}$ and $Q_{s}$ are bounded and continuous, and $\|u^{(n)}\|_{\infty}+\|\nabla u^{(n)}\|_{\infty}$ is bounded in $n$ by Lemma 2.3(2), with $n\to\infty$ this implies (2.30) provided

Combining this with (2.33) and $b^{(n)}\circ\pi_{n}=b^{(n)}$, we conclude that $u^{\{n\}}:=u^{(n)}\circ\pi_{n}$ solves (2.20). Therefore, (2.44) follows from Lemma 2.3(2).

(ii) Assume (a3’). Then (1.3) and (1.5) hold for some $\phi\in\mathscr{D}$. By Lemma 2.3(2), $\|\nabla u^{(n)}\|_{\infty}+\|\nabla^{2}u^{(n)}\|_{\infty}$ is bounded in $n\geq 1$. Since

Therefore, repeating the argument in case (i) we prove (2.30). ∎

Pathwise uniqueness

In this section, we prove the pathwise uniqueness of mild solutions under (a1), (a2’), and either (a3’) or the following stronger version of (a3).

It suffices to prove that for any $T>0$ and $m\geq 1$,

Moreover, by (a1) there exists some function $\varepsilon(\lambda)\downarrow 0$ as $\lambda\uparrow\infty$ such that

Similarly, since (3.3) and (a2’) imply $\|\nabla u\|_{T,\infty}+\|\nabla Q\|_{T,\infty}<\infty$,

holds for the same type $\varepsilon(\lambda)$. Combining this with (3.4) and (3.5), and using (a2’) and (3.3), we may find a constant $C_{0}>0$ such that for large enough $\lambda>0$

Next, by the boundedness of $b$ and Lemma 2.3(1), we have $\|\nabla_{b}u+b\|_{T,\infty}<\infty.$ So, according to Lemma 2.2(1) and (2.4), there exists a constant $C_{1}>0$ such that

If (1.5) holds for some $\phi\in\mathscr{D}$, then by Lemma 2.2(1) and $\|b\|_{T,\infty}+\|\nabla u\|_{T,\infty}<\infty,$ we conclude that

Substituting this and (3.7) into (3.6), we arrive at

By the Gronwall inequality we obtain $\eta_{T}=0,$ which is equivalent to the desired (3.1). ∎

Strong Feller property and Harnack inequality

In this section, we investigate the strong Feller property and discuss Harnack inequalities of the semigroup associated to the equation (1.1).

To this end, we formulate $P_{T}$ using the mild solution to the regular equation

Then $(Z_{t}^{x},W_{t}^{x})_{t\in[0,T]}$ is a weak mild solution to (1.1), so that

By the boundedness and continuity of $Q^{*}_{s}(QQ^{*})^{-1}_{s}b_{s},$ and noting that $Z_{t}^{x}$ is continuous in $x$, we conclude that

It is easy to see that $R_{T,n}^{z}$ is Malliavin differentiable and

Using coupling by change of measures as in [20, Chapter 4], in the situation of Proposition 4.1 we may derive the dimension-free Harnack inequality in the sense of . Here, instead of repeating the coupling arguments therein, we intend to show that (4.3) together with known Harnack inequalities of $P_{T}^{0}$ implies the corresponding inequalities for $P_{T}$. For instance, when $Q_{t}(x)=Q_{t}$ does not depend on $x$, by [20, Theorem 3.2.1] for $K=0$ and $\lambda_{T}:=\sup_{t\in[0,T]}\|Q_{t}^{*}(Q_{t}Q_{t}^{*})^{-1}\|^{2}$, the Harnack inequality

Comparing with (4.6), the Harnack inequality included in (4.7) is worse for short distance since

In particular, it does not imply the strong Feller property as (4.6) does. See Section 6 for the study of the log-Harnack inequality which is sharp for short distance as in the regular case.

Proof of Theorem 1.1

Throughout this section, we assume (a1), (a2) and either (a3). Using $b$ to replace $b+B$, we may and do assume that $B=0.$

By (a2) we see that (a2’) holds for $B=0$ and $Q^{[m]}$ in place of $Q$. Moreover, (a3) implies that $(Q^{[m]},b^{[m]})$ satisfies (a3’). Then by (a), (1.1) for $B=0$ and $(b^{[m]},Q^{[m]})$ in place of $(b,Q)$ has a unique mild solution $X_{t}^{(m)}$ starting at $X_{0}$ which is non-explosive. Let

Since $b_{s}^{[m]}(z)=b_{s}(z)$ and $Q_{s}^{[m]}(z)=Q_{s}(z)$ hold for $s\leq m$ and $|z|\leq m,$ by Proposition 3.1, for any $n,m\geq 1$ we have $X_{t}^{(n)}=X_{t}^{(m)}$ for $t\in[0,\tau_{n}\land\tau_{m}]$. In particular, $\tau_{m}$ is increasing in $m$. Let $\zeta=\lim_{m\to\infty}\tau_{m}$ and

Then it is easy to see that $(X_{t}^{x})_{t\in[0,\zeta)}$ is a mild solution to (1.1) for $B=0$ with life time $\zeta$ and, due to Proposition 3.1, the mild solution is unique. We prove Theorem 1.1(1) for $B=0$.

Due to (1.4) for $B=0$, the increasing property of $h,\Phi$, and $A\leq 0$, this implies that for any $T>0$,

Moreover, (a1) and $\|Q\|_{T,\infty}<\infty$ yield

where $X_{t}^{y}$ is the mild solution for $X_{0}=y$. Let $X_{t}^{(n,z)}$ solve (1.1) with $X_{0}=z$ for $B=0$ and $(b^{[n]},Q^{[n]})$ in place of $(b,Q)$, and let

By Proposition 4.1, $P_{T}^{(n)}$ is strong Feller. Since $X_{t}^{(n,z)}=X_{t}^{z}$ for $t\leq\tau_{n}^{z}$, where $\tau_{n}^{z}:=n\land\inf\{t\geq 1:|X_{t}^{(n,z)}|\geq n\},$ it follows that

Since $P_{T}^{(n)}$ is strong Feller, this implies

Proof of Theorem 1.2

Throughout this section, we assume (a1), (a2’) and (a3’). The idea of the proof is to transform (1.1) into an equation with regular coefficients, so that gradient estimates for the solution of the new equation can be derived. To this end, we use the regularization representation (2.30). Let us fix $T>0$. By Lemma 2.3, we take large enough $\lambda(T)>0$ such that for any $\lambda\geq\lambda(T)$ the unique solution $u$ to (2.4) satisfies

Now, let $X_{t}^{x}$ solve (1.1) for $X_{0}=x$. By (2.30), $Y_{t}^{x}:=\theta_{t}(X_{t}^{x})$ satisfies

$\bar{X}_{t}^{x}:=Y_{t}^{\theta_{0}^{-1}(x)}$ is a mild solution to the equation

We first study gradient estimates and the log-Harnack inequality for $\bar{P}_{t}$. To this end, one may wish to apply the corresponding results derived recently in . However, in the present case the assumption (A1) in is not available, i.e. our conditions do not imply the existence of $K\in L^{2}([0,T];\text{\rm{d}}t)$ such that

Hence, we are not at the position to apply results in .

But this is already enough for our purpose.

Assume (a1), (a2’) and (a3’). For any $T>0$ and large enough $\lambda\geq\lambda(T)$, there exists a constant $C>0$ such that the following assertions hold.

If in addition $\|B\|_{T,\infty}<\infty$, then $\eqref{FF2}$ holds.

Thus, by the dominated convergence theorem, (6.8) implies

Taking large enough $p>1$ such that $c(p)\leq\frac{1}{2},$ and substituting this into (6.10), we arrive at

Noting that $\langle\bar{b}_{s},e_{i}\rangle=(\lambda+\lambda_{i})\langle u_{s}\circ\theta_{s}^{-1},e_{i}\rangle+\langle(B_{s}+\nabla_{B_{s}}u_{s})\circ\theta_{s}^{-1},e_{i}\rangle,$ for $p\geq\lambda$ this implies

for large enough $p\geq\lambda$, since due to (6.1) and (6.2) we have $\|\nabla(u\circ\theta^{-1})\|_{T,\infty}\leq\frac{1}{7}.$ Combining this with (6.9), we prove $I=0$ which is equivalent to (6.7).

for some constant $C_{2}>0$. On the other hand, similarly to (6.12) and (6.13), for large enough $p>0$, there holds

Taking large enough $p_{0}=2C_{2}$, such that $\frac{C_{2}}{p}\leq\frac{1}{2}$ for $p\geq p_{0}$, we obtain

Combining this with (6.15) and (2), we prove (3). For instance, regarding $s$ as the starting time, we see that (2) also holds for $\bar{P}_{s,t+s}^{(n)}$ in place of $\bar{P}_{t}^{(n)}$, so that

for some constant $C_{3}>0$. Integrating over $[0,t]$ we prove (4) for some constant $C>0$. ∎

So, by letting $n\to\infty$ (up to a subsequence) in (6.17) we prove (6.16) for a.e. $t\in[0,T]$ with respect to the Lebesgue measure. By the continuity of $\bar{X}_{t}$, $\bar{P}_{t}f$ is continuous in $t$. Therefore, (6.16) holds for all $t\in[0,T].$

Now, according to (6.5) and (6.2), we only need to prove Theorem 1.2 for $\bar{P}_{t}f$ in place of $P_{t}$. By the above observation, Theorem 1.2(1) as well as (1.6) and (1.7) with $t\in(0,1]$ hold for $\bar{P}_{t}$ in place of $P_{t}$. Then the proof is complete by the following two facts: (a) Due to the semigroup property and Jensen’s inequality, if (1.6) and (1.7) hold for $t\in(0,1]$, then they also hold for all $t>0$; (b) According to [22, Proposition 1.3], (1.6) is equivalent to (1.8).

Therefore, as explained in (a) that these assertions also hold for $P_{t}$. ∎

The author would like to thank Jian Wang, X. Huang and X. Zhang for helpful comments and corrections.

References