Bilateral Harnack Inequalities for Stochastic Differential Equation with Multiplicative Noise

An, Zihao; Zong, Gaofeng

doi:https://doi.org/10.1155/2022/5464688

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 5464688 | https://doi.org/10.1155/2022/5464688

Bilateral Harnack Inequalities for Stochastic Differential Equation with Multiplicative Noise

Zihao An¹and Gaofeng Zong²

Academic Editor: Salah Mahmoud Boulaaras

Received08 Mar 2022

Accepted11 Jul 2022

Published30 Jul 2022

Abstract

By constructing a coupling with unbounded time-dependent drift, a lower bound estimate of dimension-free Harnack inequality with power is obtained for a large class of stochastic differential equation with multiplicative noise. The key is an application of the inverse Hölder inequality. Combining this with the well-known upper bound, bilateral dimension-free Harnack inequality with power is established. As a dual inequality, the bilateral shift-Harnack inequalities with power are also investigated for stochastic differential equation with additive noise. Applications to the study of heat kernel inequalities are provided to illustrate the obtained inequalities.

1. Introduction

We consider the following stochastic differential equation (SDE for brief) on :

where is the -dimensional Brownian motion on a filtered complete probability space satisfying the usual hypotheses, and

are progressively measurable and locally bounded in the second variable and are continuous in the third variable. We will assume throughout this paper that, for any initial value , the SDE above has a unique strong solution which is nonexplosive and continuous in time . To establish the bilateral Harnack inequalities, we will introduce the following assumptions:

Assumption 1. For , there exists an increasing function so that for all , all and

Assumption 2. There exists a decreasing function s.t. for all

Assumption 3. For , there exists an increasing positive function s.t. for all , all and

Assumption 4. For , there exists a constant s.t. for any

It follows from the above assumptions that the existence and uniqueness of the strong solution of SDE (1) are ensured (see Protter [1]). For more results about existence and uniqueness of the solution of differential equations, we can see [2–4].

Let be the solution to the SDE (1) for . In this paper, we will establish the bilateral Harnack inequalities for the operator ; for any , here is the class of all bounded nonnegative measurable functions:

where is the expectation with respect to the probability measure . In Wang [5], under the above assumptions (Assumptions 1–4), the author has successfully established the dimension-free Harnack inequalities for a large class of SDEs with multiplicative noise by constructing a coupling.

The dimension-free Harnack inequality was firstly established by Wang [6] for diffusion semigroups on Riemannian manifolds under a curvature condition. As a useful tool in the study of diffusion semigroups, in particular, for the uniform integrability, contractivity properties, and estimates on heat kernels, Wang’s remarkable work has cause a lot of studies in the last two decades (see Aida and Zhang [7], Arnaudon et al. [8], and Da Prato [9]).

Wang’s Harnack inequality has been extended in a large number of papers. For instance, by using coupling method, the upper bounded of dimension-free Harnack inequality has been established in [10] for a delay SDE with additive noise, in Wang and Yuan [11] for a delay SDE with multiplicative noise, in Wang and Zhang [12] for SDE with non-Lipschitz drift and driven by additive anisotropic cylindrical -stable process, and in Wang et al. [13] for stochastic Burgers equation. Recently, using coupling by change of measures, the dimension-free Harnack inequality is investigated in Wang [14] for a distribution-dependent stochastic differential equation with regular coefficients and in Huang and Wang [15] for a distribution-dependent SDE with singular coefficients. For more details, we can refer to the book of Wang [16] for a deep analysis about dimension-free Harnack inequalities. All existing literatures focus on the upper bound of the dimension-free Harnack inequality. So far, we have not seen the lower bound estimate of dimension-free Harnack inequality with power in any previous literature. The aim of this paper is to establish the lower bound of the dimension-free Harnack inequality with power for SDE with multiplicative noise.

Theorem 1 in Wang [5] gives an upper bound of the dimension-free Harnack inequality with power for SDE with multiplicative noise. Hence, combining Theorem 1 in Wang [5] with our lower bound of the dimension-free Harnack inequality with power for SDE with multiplicative noise, actually, we have the following bilateral dimension-free Harnack inequality with power for SDE with multiplicative noise.

Theorem 5. Let where , and where .
For and , the bilateral dimension-free Harnack inequality with power holds for all and .

In this paper, we will give a direct proof about the lower bound estimate of dimension-free Harnack inequality with power for a large class of SDE with multiplicative noise, rather than the reciprocal transformation based on the upper bound of dimension-free Harnack inequality with power. Of course, based on the reciprocal transformation of our lower bound estimate, we can also get the upper bound of dimension-free Harnack inequality with power. Unfortunately, our method does not adapt to establish the lower bound estimate of dimension-free log-Harnack inequality. At the same time, there is no evidence that the lower bound estimate of dimension-free log-Harnack inequality can be easily obtained by reciprocal transformation. This also leaves an open problem of how to establish the lower bound estimate of dimension-free log-Harnack inequality for a large class of SDE with additive noise or multiplicative noise.

Let be the density of the operator with respect to a Radon measure . It follows from Theorem 5 that the following corollary on bilateral heat kernel inequalities is a direct consequence.

Corollary 6. Assume that Assumptions 1–4 hold and the operator have a strictly positive density with respect to . Then, for , the upper bound of heat kernel inequality holds, and for , the lower bound of heat kernel inequality, and for all , holds.

The main aim of this paper is to establish lower bound of dimension-free Harnack inequality with power for SDE with multiplicative noise. As a dual inequality, the shift-Harnack inequality has been developed and applied in Wang [17]. Obviously, it is relatively easy to establish lower bound of shift-Harnack inequality by the method used in this paper.

Next, we establish the bilateral dimension-free Wang’s shift-Harnack inequalities with power for introduced in Wang [17]. We now only consider the additive noise for which the SDE (1) reduces to

Theorem 7. Let from to and from to be measurable and satisfy the assumptions in Section 1. Furthermore, the function is invertible and satisfies the assumption.

Assumption 8. There exists a decreasing function from to s.t. for all , Then, (1)for any and for any ,(2)for any and for any ,

It is quite remarkable that the lower bound of dimension-free Harnack inequality with power in Theorem 5 is essentially equivalent to the upper bounded of dimension-free Harnack inequality with power through appropriate transformation. The same remark as Theorem 5 is also true for bilateral dimension-free shift-Harnack inequalities with power in Theorem 7. In the future, we want to establish bilateral Harnack inequalities for various models, for example, functional SDEs with additive/multiplication noise, distribution-dependent SDEs, distribution-dependent SDEs with singular coefficients, and SDEs driven by cylindrical -stable processes. However, the method we use here is not suitable for the lower bounded dimension-free log-Harnack inequality. How to establish the lower bounded dimension-free log-Harnack inequality is a very interesting and meaningful question.

In Section 2, we construct a coupling and prove several auxiliary results which will be needed for the proof of theorem. The dimension-free bilateral Harnack inequality with power and dimension-free bilateral shift-Harnack inequality with power are given in Sections 3–4, respectively. As applications, bilateral heat kernel estimates are derived in Section 5.

2. Auxiliary Results

Let and

be fixed such that . Due to , we have

For any , write

Then, function is smooth and strictly positive on such that

We now construct the following coupling:

We set by convention. Let

and is the explosion time of , and then, the coupling is a well-defined continuous process for . For , let

If and

is a uniformly integrable martingale for , then by the martingale convergence theorem, exists and is a martingale. In this case, by the Girsanov theorem, is a -dimensional Brownian motion under the probability . We can rewrite (21) as

Let

We have as because of the nonexplosibility of .

By the Lemma 2.1 in Wang [5], we know that for any , the following limits

exist such that with respect to is a uniformly integrable martingale, and

so that .

Lemma 9. Assume Assumptions 1–4. Then, for , where the expectation is under the probability measure .

Proof. Fixed . Using the Itô’s formula for and Assumption 1, we have, for , Combining this with the fact , we obtain for . Multiplying this inequality by and then integrating in , we get By the Girsanov theorem, ia the -dimensional Brownian motion under the probability measure . Taking expectation with respect to , hence we obtain, for any , By the dominated convergence theorem, we have where the expectation is under the probability measure ; hence, we obtain the desired result.

Lemma 10. Assume Assumptions 1–4. Let Then,

Proof. By the definition of , we have Observing that for any exponential integrable martingale with respect to , one has where . It follows that Noting that take i.e.,
which minimizes (see Figure 1), such that Hence, noting that we can write (39) as Furthermore, we see from (40) that Noting that it follows from the above relation (42) that Due to the choice of and the definition of (35) and (43), we can compute For any it follows from (32) that Taking we have Now, returning to (41), and noticing that , then we obtain Therefore, by letting , the Fatou lemma implies that This proves the assertion.

3. Proof of Lower Bounded of Harnack Inequality

We will only give the proof of the lower bounded of the dimension-free Harnack inequality with power, since the upper bounded of the dimension-free Harnack inequality with power has been established in Wang [5]. We will first introduce an inverse Hölder inequality, which is the key building stone for this paper to obtain successfully lower bound of the dimension-free Harnack inequality with power. In the literature, we can find the earlier version of inverse Hölder inequality with integral form in Hardy et al. [18], P23-24, and more inverse Hölder inequalities can be found in Nehari [19] and Lin and Bai [20] for the following type and the references there cited.

Lemma 11. For any , , then and for any positive random variables , the inverse Hölder inequality holds:

For our purpose to establish the dimension-free Harnack inequality with power and for simplicity, we only consider , although the inverse Hölder inequality also holds for . With the inverse Hölder inequality in hand, we now can start to prove the lower bounded of dimension-free Harnack inequality with power.

Theorem 12. Let where . If Assumptions 1–4 hold, then for , the dimension-free Harnack inequality with power holds for all and .

Proof. Since with respect to is a uniformly integrable martingale and is the -dimensional Brownian motion under new probability measure , thus can be solved up to time . Let and set by convention. We claim and thus, -a.s. Indeed, if for some such that , by the continuity of the processes, we have So, on the set , we have But according to Lemma 9, we obtain Hence, we conclude that . Therefore, -a.s.
Now, since and is the -dimensional Brownian motion under new probability measure , we have Using the inverse Hölder inequality, for , hence , we have Hence, we arrive at Here, .
Let . Noting the choice of the (18), we have that is, In fact, in (18), we choose which solves the equation .
Observing that the function is a decreasing function with respect to in if , it follows from Lemma 10 that Replacing with in (63), we can rewrite (63) and then obtain This completes the proof of this theorem.

Remark 13. In this theorem, we give a direct proof about the lower bound estimate of dimension-free Harnack inequality with power for a large class of stochastic differential equation with multiplicative noise, rather than the reciprocal transformation based on the upper bound of dimension-free Harnack inequality with power. Of course, based on the reciprocal transformation of our lower bound estimate, we can also get the upper bound of dimension-free Harnack inequality with power. Unfortunately, our method does not adapt to establish the lower bound estimate of dimension-free log-Harnack inequality. At the same time, there is no evidence that the lower bound estimate of dimension-free log-Harnack inequality can be easily obtained by reciprocal transformation. This also leaves an open problem of how to establish the lower bound estimate of dimension-free log-Harnack inequality for a large class of stochastic differential equation with additive noise or multiplicative noise.

4. Bilateral Shift-Harnack Inequalities

In this section, we establish the bilateral dimension-free Wang’s shift-Harnack inequalities with power for , and the upper bounded of this inequality has been introduced in Wang (2014). As seen in the previous section, the study for the multiplicative noise case is very complicated; we hence only consider the additive noise for which the SDE (1) reduces to

Proposition 14. Let function from to and from to be two progressively measurable process and satisfy assumptions in Section 1. Furthermore, the function is invertible and satisfies the following assumption.

Assumption 15. There exists a decreasing function from to s.t. for all , Then, (1)for any , and any ,(2)for any , and any ,

Proof. Let . Then, we have where Let By the Girsanov theorem, we have Similar to (58), by inverse Hölder inequality, we obtain Due to (71), we have Here, the last inequality holds according to Assumption 15, and Now, noticing from Assumption 1 that we see that for all , Thus, in view of the above assertion (77) and , estimate (74) becomes Using again the fact that the function is a decreasing function with respect to in if , it follows from (73) that This completes the proof of the first part of this theorem. Noticing that (78) holds for all , the second part of this theorem can be obtained.

In fact, we can get a better conclusion than Proposition 14.

Theorem 16. Let function from to and function from to be measurable and satisfy assumptions in Section 1. Furthermore, the function is invertible and satisfies Assumption 15. Then, (1)for any , and ,(2)for any , and ,

Proof. Similar to (39), we have Due to (77), we have (1)For that satisfies , using the fact that the function is a decreasing function with respect to in , it follows from (73) based on the inverse Hölder inequality thatThis completes the proof of the lower bounded (2)For , by Hölder inequality, we haveNotice that (83) holds for all ; hence, we get This completes the proof of this theorem.

Remark 17. From this simplified proof of the bilateral dimension-free shift-Harnack inequality with power for additive case, we can easily see that the moment estimation (83) is the key to establishing the bilateral dimension-free shift-Harnack inequality with power. We need to specifically point out that the moment estimation (83) holds for all . To establish the upper bound, we need to use the moment estimation (83) and Hölder inequality, but to establish the lower bound, we need to use the moment estimation (83) and the inverse Hölder inequality.

5. Proof of the Corollary 6

Let be the density of with respect to which is a Radon measure. Then, it follows from Corollary 1.2 in Wang [5] that for , the Harnack inequality with power in Wang [5] is equivalent to the following heat kernel inequalities:

For all , hence, (11) holds. Due to this, then we will complete the proof of the Corollary 6, if we prove the following proposition:

Proposition 18. Assume that Assumptions 1–4 hold. Let be a strictly positive density of the operator with respect to a Radon measure . Then, for , the following heat kernel inequality holds for all .

Proof. By Theorem 12, we have Here, Let be a transition probability such that Obviously, the above inequality implies that and are equivalent to each other. Indeed, if , then applying the above inequality to , we conclude that .
Let be the Radon-Nikodym derivative of with respect to .
Applying (89) to , we have So we obtain that is, Thus, Therefore, the desired result in (88) holds.

Data Availability

All data, models, and code generated or used during the study appear in the submitted article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Science Foundation of China (Grant Nos. 11501325 and 71871129) and the China Postdoctoral Science Foundation (Grant Nos. 2018T110706 and 2018M642641). We express special thanks to Professor Feng Zhang (SDUFE) for the several helpful comments.

References

P. Protter, Stochastic Integration and Differential Equations, Springer, 2005.
A. Benramdane, N. Mezouar, M. Alqawba, S. Boulaaras, and B. Cherif, “Blow-up for a stochastic viscoelastic Lamé equation with logarithmic nonlinearity,” Journal of Function Spaces, vol. 2021, Article ID 9943969, 10 pages, 2021.
View at: Publisher Site | Google Scholar
A. Ben Makhlouf, E. El-hady, S. Boulaaras, and L. Mchiri, “Stability results of some fractional neutral integrodifferential equations with delay,” Journal of Function Spaces, vol. 2022, Article ID 8211420, 7 pages, 2022.
View at: Publisher Site | Google Scholar
T. Bouali, R. Guefaifia, A. Choucha, S. Boulaaras, and M. Abdalla, “The existence of one non-trivial weak solution of generalized Yamabe equations,” Miskolc Mathematical Notes, vol. 23, no. 1, pp. 117–129, 2022.
View at: Publisher Site | Google Scholar
F.-Y. Wang, “Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds,” Annals of Probability, vol. 39, no. 4, pp. 1449–1467, 2011.
View at: Publisher Site | Google Scholar
F.-Y. Wang, “Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,” Probability Theory and Related Fields, vol. 109, no. 3, pp. 417–424, 1997.
View at: Publisher Site | Google Scholar
S. Aida and Z. Tusheng, “On the small time asymptotics of diffusion processes on path groups,” Potential Analysis, vol. 16, no. 1, pp. 67–78, 2002.
View at: Publisher Site | Google Scholar
M. Arnaudon, A. Thalmaier, and F. Wang, “Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below,” Bulletin des Sciences Mathematiques, vol. 130, no. 3, pp. 223–233, 2006.
View at: Publisher Site | Google Scholar
G. D. Da Prato, M. Röckner, and F. Y. Wang, “Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups,” Journal of Functional Analysis, vol. 257, no. 4, pp. 992–1017, 2009.
View at: Publisher Site | Google Scholar
A. Es-Sarhir, M.-K. von Renesse, and M. Scheutzow, “Harnack inequality for functional SDEs with bounded memory,” Electronic Communications in Probability, vol. 14, pp. 560–565, 2009.
View at: Google Scholar
F.-Y. Wang and Y. Chenggui, “Harnack inequalities for functional SDEs with multiplicative noise and applications,” Stochastic Processes and their Applications, vol. 121, no. 11, pp. 2692–2710, 2011.
View at: Publisher Site | Google Scholar
L. Wang and Z. Xicheng, “Harnack inequalities for SDEs driven by cylindrical -stable processes,” Potential Analysis, vol. 42, no. 3, pp. 657–669, 2015.
View at: Publisher Site | Google Scholar
F.-Y. Wang, J.-l. Wu, and L. Xu, “Log-Harnack inequality for stochastic burgers equations and applications,” Journal of Mathematical Analysis and Applications, vol. 384, no. 1, pp. 151–159, 2011.
View at: Publisher Site | Google Scholar
F.-Y. Wang, “Distribution dependent SDEs for Landau type equations,” Stochastic Processes and their Applications, vol. 128, no. 2, pp. 595–621, 2018.
View at: Publisher Site | Google Scholar
X. Huang and F.-Y. Wang, “Distribution dependent SDEs with singular coefficients,” Stochastic Processes and their Applications, vol. 129, no. 11, pp. 4747–4770, 2019.
View at: Publisher Site | Google Scholar
F.-Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer, New York, 2013.
F.-Y. Wang, “Integration by parts formula and shift Harnack inequality for stochastic equations,” Annals of Probability, vol. 42, no. 3, pp. 994–1019, 2014.
View at: Publisher Site | Google Scholar
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, 1934.
Z. Nehari, “Inverse Hölder inequalities,” Journal of Mathematical Analysis and Applications, vol. 21, no. 2, pp. 405–420, 1968.
View at: Publisher Site | Google Scholar
Z. Lin and B. Zhidong, Probability Inequalities, Springer, 2010.

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Copyright © 2022 Zihao An and Gaofeng Zong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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