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Mild Solutions and Harnack Inequality for Functional Stochastic Partial Differential Equations with Dini Drift

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Abstract

The existence and uniqueness of a mild solution for a class of functional stochastic partial differential equations with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.

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References

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Acknowledgements

The authors would like to thank Professor Feng-Yu Wang and Jianhai Bao for corrections and helpful comments.

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Correspondence to Xing Huang.

Additional information

Supported in part by NNSFC (11431014, 11626237).

Appendix

Appendix

In this section, we give [6, Theorem 4.3.1 and Theorem 4.3.2] in detail as follows.

Fix a constant \(r_0>0\). Let \({\mathscr {C}}(\mathbb {R}^d):=C([-\,r_0,0],\mathbb {R}^d)\). For simplicity, we denote \({\mathscr {C}}^d={\mathscr {C}}(\mathbb {R}^d)\). Consider the functional SDE on \(\mathbb {R}^d\):

$$\begin{aligned} \begin{aligned} \text {d}Z(t)=\{a(t,Z(t))+c(t,Z_t)\}\text {d}t+\sigma (t,Z(t))\text {d}w(t), \end{aligned} \end{aligned}$$
(6.1)

where w is a standard m-dimensional Brownian motion, \(a: [0,\infty )\times \mathbb {R}^d\rightarrow \mathbb {R}^d\), \(c: [0,\infty )\times {\mathscr {C}}^d\rightarrow \mathbb {R}^d\), and \(\sigma : [0,\infty )\times \mathbb {R}^d\rightarrow \mathbb {R}^d\otimes \mathbb {R}^m\) are measurable and locally bounded (i.e., bounded on bounded sets).

To establish the Harnack inequality, we shall need the following assumption:

(A) :

For any \(T>r_0\), there exist constants \(K_{1}, K_{2}\ge 0\), \(K_{3}>0\) and \(K_{4}\in \mathbb {R}\) (\(K_{1}\), \(K_{2}\), \(K_{3}\), \(K_{4}\) only depend on T) such that

$$\begin{aligned}&\left| (\sigma ^{*}(\sigma \sigma ^{*})^{-1})(t,\eta (0))\{c(t,\xi )-c(t,\eta )\}\right| \le K_{1}\Vert \xi -\eta \Vert _{\infty }; \end{aligned}$$
(6.2)
$$\begin{aligned}&\left\| \sigma (t,x)-\sigma (t,y)\right\| \le K_{2}(1\wedge |(x-y)|; \end{aligned}$$
(6.3)
$$\begin{aligned}&\left\| (\sigma ^{*}(\sigma \sigma ^{*})^{-1})(t,x)\right\| \le K_{3}; \end{aligned}$$
(6.4)
$$\begin{aligned}&\left\| \sigma (t,x)-\sigma (t,y)\right\| _{HS}^{2}+2\left\langle x-y, a(t,x)-a(t,y)\right\rangle \le K_{4}|x-y|^{2} \end{aligned}$$
(6.5)

hold for \(t\in [0,T]\), \(\xi ,\eta \in {\mathscr {C}}^d\), and \(x,y\in \mathbb {R}^d\).

(A) implies [6, (A4.1)], so by [6, Corollary 4.1.2], for any \(\xi \in {\mathscr {C}}^d\), (6.1) has a unique strong solution \(Z_t^\xi \) with \(Z_0=\xi \). Let \(P_T\) be the associated Markov semigroup defined as

$$\begin{aligned} P_T f(\xi )=\mathbb {E}f(Z_T^\xi ), \quad f\in {\mathscr {B}}_b({\mathscr {C}}^d),\xi \in {\mathscr {C}}^d. \end{aligned}$$

Lemma 6.1

Assume (A). Then for any \(T>r_0\), every positive function \(f\in {\mathscr {B}}_{b}({\mathscr {C}})\),

  1. (1)

    the \(\log \)-Harnack inequality holds, i.e.,

    $$\begin{aligned} P_T\log f(\eta )\le \log P_T f(\xi )+H(T,\xi ,\eta ),~\xi ,\eta \in {\mathscr {C}}^d \end{aligned}$$
    (6.6)

    with

    $$\begin{aligned} H(T,\xi ,\eta )=C\left( \frac{|\xi (0)-\eta (0)|^2}{T-r}+\Vert \xi -\eta \Vert ^2_\infty \right) \end{aligned}$$

    for some dimension-free constant \(C>0\).

  2. (2)

    For any \(p>(1+K_2K_3)^{2}\), the Harnack inequality with power

    $$\begin{aligned} P_{T}f(\eta )\le (P_{T} f^{p}(\xi ))^{\frac{1}{p}}\exp {\Psi _{p}(T;\xi ,\eta )},\quad \xi ,\eta \in {\mathscr {C}}^d \end{aligned}$$
    (6.7)

    holds, where

    $$\begin{aligned} \Psi _{p}(T;\xi ,\eta )=C(p)\left\{ 1+\frac{|\xi (0)-\eta (0)|^2}{T-r} +\Vert \xi -\eta \Vert _{\infty }^{2}\right\} \end{aligned}$$

    for a dimension-free decreasing function \(C:\left( (1+K_2K_3)^{2},\infty \right) \rightarrow (0,\infty )\).

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Huang, X., Zhang, SQ. Mild Solutions and Harnack Inequality for Functional Stochastic Partial Differential Equations with Dini Drift. J Theor Probab 32, 303–329 (2019). https://doi.org/10.1007/s10959-018-0830-4

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  • DOI: https://doi.org/10.1007/s10959-018-0830-4

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