Nonlinear Young integrals and differential systems in Hölder media
For Hölder continuous random field and stochastic process , we define nonlinear integral in various senses, including pathwise and Itô-Skorohod. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with is also studied and its applications to the transport equation in rough media is given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients are given, where is a second order elliptic differential operator with random coefficients (dependent on ). To establish such formula the main difficulty is the exponential integrability of some nonlinear integrals, which is proved to be true under some mild conditions on the covariance of and on the coefficients of . Along the way, we also obtain an upper bound for increments of stochastic processes on multidimensional rectangles by majorizing measures.
Key words and phrases:Gaussian random field; sample path property; majorizing measure; nonlinear Young integral; nonlinear Itô-Skorohod integral; transport equation; stochastic parabolic equation; multiplicative noise; Feynman-Kac formula; Malliavin calculus; diffusion process; exponential integrability of the Hölder norm of diffusion process.
2000 Mathematics Subject Classification:Primary 60H30; Secondary 60H10, 60H15, 60H07, 60G17
- 1 Introduction
- 2 Nonlinear Young integral
- 3 Differential equations
- 4 Feynman-Kac formula - A pathwise approach
- 5 Asymptotic growth of Gaussian sample paths
- A Other types of nonlinear stochastic integral
- B Estimates for diffusion process
- C Schauder estimates
Feynman integral is an important tool in quantum physics. The Feynman-Kac formula is a variant of Feynman integral and plays very important role in the study of (parabolic) partial differential equations (see  and ). Recently, there have been several successes in extending the Feynman-Kac formula to the following stochastic partial differential equations with noisy (random) potentials on (see e.g. , , and ): , where is the Laplacian with respect to spatial variable and is a Gaussian noise (the derivatives in the sense of Schwartz distribution of a Gaussian field). As indicated in the aforementioned papers, there are three tasks to accomplish for establishing the Feynman-Kac formula. The first one is to give a meaning to the nonlinear stochastic integral for a -dimensional Brownian motion (whose generator is ), independent of . The second one is to establish the exponential integrability of and hence the Feynman-Kac expression (which we may call the Feynman-Kac solution) has a rigorous meaning. The final task is to show that the Feynman-Kac expression is indeed a solution to the equation in certain sense. It should be emphasized that the independence between and plays crucial role in previous studies.
In many applications, one needs to study more general stochastic partial differential equations. For example, in modeling of the pressure in an oil reservoir in the Norwegian sea with a log normal stochastic permeability one was led to study the stochastic partial differential equation on some bounded domain in of the form , where the permeability is the (Wick) exponential of white noise, is the divergence operator, and is the gradient operator, see  and in particular the references therein. Recently, there have been a great amount of research on uncertainty quantification. Among the huge literature on this topic let us just mention the books , , and the references therein. Many different types of stochastic partial differential equations with random coefficients have been studied.
This motivates us to study the Feynman-Kac formula for general stochastic partial differential equations with random coefficients, namely,
and for notational simplicity and up to a time change we assume that the terminal condition is given. The product in (1.1) is the ordinary product. If satisfies and if is the solution of the following stochastic differential equation
then should be the Feynman-Kac solution to (1.1) with . As indicated above, there are three tasks to complete to justify the above claim. The first task to give a meaning to the nonlinear stochastic integral is much more challenging than what has been accomplished before (see for instance , , and ). Although the major focus of the work  is to give a meaning to the nonlinear integral . However, in that paper is a Brownian motion independent of and then we can consider as “deterministic”. In our current situation since and are correlated, the nonlinear integral is a true stochastic one. In addition, the noise may enter to in an anticipative way. Thus, the general stochastic calculus for semimartingales cannot be applied in a straightforward way due to the lack of adaptedness.
If is only continuous in (without any Hölder continuity in ) but has certain differentiability on , then we can use semimartingale structure of plus some new techniques developed in Section 4 to define and study the corresponding Feynman-Kac solution to (1.1). This result extends the work of  in two aspects. One is that the Laplacian is replaced by general second order elliptic operator with general and in particular random coefficients. The other one is that in , the Hurst parameter in time is assumed to be greater than , while the result of this paper is applicable to fractional Brownian field whose Hurst parameter in time can be any number between and .
When has certain (Hölder) regularity in time variable, it is natural to see whether one can reduce its regularity in spatial variable to define . Having in mind the recent development on rough path analysis and encouraged by the previous success in the case when is the Brownian motion (, , and ), we dedicate ourselves to a systematic study of the nonlinear integral , where is a Hölder continuous function on and and is also a Hölder continuous function. Some elementary properties of the integral are obtained as well. These results are presented in Section 2. Let us emphasize that this nonlinear integral is defined in a purely deterministic way. In fact, it is an extension of integration of Young type ().
For Gaussian noise a very important (linear) stochastic integral is the Itô (or Itô-Skorohod) integral. It is also called divergence integral. In probability theory, this integral is a central concept in stochastic analysis. For our stochastic partial differential equation (1.1) it is needed if the product there is Wick product. We shall introduce the nonlinear Itô-Skorohod integral ( depends on ) by using Malliavin calculus. This is done in Appendix A. The relation of this integral with other types of integrals is also discussed in this section. Naturally, readers may ask the question to study the Itô-Skorohod type stochastic differential equation , where denotes the Wick product between and . However, this seems to be very complex since depends on in a sophisticated way and will not be considered in this work.
When is a semimartingale in for any fixed and is smooth in for any fixed , there has been many studies on stochastic flows which contributes significantly to the study of stochastic partial differential equations (see  and the references therein). The important tool there is the nonlinear stochastic integral (with respect to semimartingale) and the corresponding flow. After defining the nonlinear Young integral and motivated by this aspect, we study the pathwise flow associated with time dependent rough vector field . That is, we study the differential equation under joint Hölder continuity assumptions of . We shall study the flow and other properties of the solution . This is presented in Section 3. The applications to the transport equation in rough media of the form are also investigated in Subsection 3.4.
After completion of the first task of defining the nonlinear integral another major difficulty (the above mentioned second task) to overcome in the construction of the Feynman-Kac solution is the exponential integrability of . In the previous work of , , and , this is achieved by showing is finite. If we continue to follow the idea in aforementioned papers, then we are led to show
is finite, where is the solution to the equation (1.2) with a Brownian motion , independent of and . It seems to us that in our situation, due to the dependence of on , it is hard to show the above quantity is finite. To get around this difficulty, our strategy is then to show that is finite for every fixed path of , assuming some mild pathwise conditions on (see for instance (4.30)). The third (and the last) task to show that the Feynman-Kac solution is indeed a solution to (1.1) is relatively easier and will be completed by using approximation technique. All these will be done Section 4.
Intentionally, the paper is divided into two parts. The first three chapters can be read without knowledge of probability theory. A single (rough) sample satisfying some joint Hölder continuity and growth conditions is considered. For instance, the (stochastic) partial differential equation (1.1), the nonlinear Young integral (Definition 2.1), and the transport equation (3.30) are considered for every fixed sample path . Since is fixed, we also drop the dependence of and on throughout the paper. So, the integrals and equations are defined and studied for a (fixed) rough function. The stochastic partial differential equation considered in Section 4 is for a single rough sample path. But Brownian motion is used to represent the solution.
As a probabilist, one may ask whether a stochastic process satisfies the joint Hölder continuity conditions together with the growth conditions assumed throughout the paper. For instance, condition (4.30) in Section 4 requires the paths of to satisfy
for all and . We give a partial answer for this problem in Section 5, where an extra assumption for a fixed constant is imposed. Pathwise boundedness and pathwise regularity (Hölder continuity) have been extensively studied in the literature (see Section 5 for more detailed discussions.) However, estimates similar to (1.3) has not been studied thoroughly. Comparing with the existing literature (e.g. , ), where estimates for increments over one parameter interval are obtained, the left side of (1.3) is an increment over two parameter rectangle. Difficulties arise because the increments behave differently when the number of parameters get large. For instance, the corresponding entropic volumetric to the left side of (1.3), , does not satisfy the triangular inequality. Therefore, classical estimates (such as those appear in ) are no longer applicable, new tools are needed to prove (1.3). If in (1.3), are restricted in a compact set, a similar problem has been considered by the authors by extending the Garsia-Rodemich-Rumsey inequality (). Nevertheless, the exact growth rate when get large is not discussed in that paper. Motivated by this requirement, we extend and sharpen our previous work in  so that it is applicable to our current situation. Since in many applications, will be a Gaussian noise, we focus on the case satisfies normal concentration inequalities to obtain the desirable pathwise property from the covariance structure of the process. As is well-known it is usually hard to obtain properties for each sample path in the theory of stochastic processes. We hope this work will shed some light along this direction.
Notations: We collect here some notations that we will use throughout the entire paper. means there is a constant such . We represent a vector in as a matrix of dimension , represents the transpose of a matrix . Sometimes we write for column vector and for the row vector . We use the Einstein convention on summation over repeated indices. For instance, abbreviates for
2. Nonlinear Young integral
Let and be -valued functions defined on and respectively. We define in the current section the nonlinear Young integration .
We make the following assumption on the regularity of
There are constants , such that for all , the seminorm
About the function , we assume
is locally Hölder continuous of order . That is the seminorm
is finite for every .
Throughout the current section, we assume that . Among three terms appearing in (2.1), we will pay special attention to the first term. Thus, we denote
When , then we denote . If are clear in the context, we frequently omit the dependence on . For instance, is an abbreviation for , is an abbreviation for and so on. We shall assume that and are finite. It is easy to see that for any
Thus assumption 1 also implies that
For the results presented in this section, the condition 1 can be relaxed to
There are constants , such that for all and compact set in , the seminorm
However, the polynomial growth rate is needed in the following sections to solve differential equations.
For later purpose, we denote (respectively ) the collection of all functions satisfying condition 1 (respectively 1). denotes a universal generic constant depending only on and independent of , and . The value of may vary from one occurrence to another.
We define the nonlinear integral as follows.
Let be two fixed real numbers, . Let be a partition of with mesh size . The Riemann sum corresponding to is
If the sequence of Riemann sums ’s is convergent when shrinks to 0, we denote the limit as the nonlinear integral .
We observe that in the particular case when for some functions , the nonlinear integral defined above, if exists, coincides with the Riemann-Stieltjes integral . It is well known that if and are Hölder continuous with exponents respectively and , then the Riemann-Stieltjes integral exists and is called Young integral ().
More generally, for each partition of an interval , one can consider the (abstract) Riemann sum
where is a function defined on with values in a Banach space. A sufficient condition for convergence of the limit is obtained by Gubinelli in  via the so-called sewing map. This point of view has important contributions to Lyons’ theory of rough paths ([39, 40]). Since we will apply Gubinelli’s sewing lemma, we restate the result as follows.
Lemma 2.2 (Sewing lemma).
Let be a continuous function on with values in a Banach space and let . Suppose that satisfies
Then there exists a function unique up to an additive constant such that
In addition, when shrinks to 0, the Riemann sums (2.3) converge to .
In what follows, we adopt the notation . The map is called the sewing map. The setting of Lemma 2.2 is adopted from . In several occasions, one needs to prove a relation between two or more integrals. The following result provides a simple method for this problem.
Suppose and are two functions as in Lemma 2.2. In addition, assume that
for some positive constant . Then and are different by an absolute constant. That is for all .
From Lemma 2.2, and
for all . This implies for all . ∎
Returning to our main objective of the current section, we consider
Hence, by combining the sewing lemma and the previous estimate, we obtain
In addition, the following estimate holds
for all .
After the completion of this work, we are brought to the attention of the work  (and also [7, 8, 25]), where a similar nonlinear Young integral is studied. The objective of that paper is to define the averaging of the form for some process and for some irregular function . The sewing lemma that we follow is from  , which is after the work of .
(i) In the particular case when , Proposition 2.4 reduces to the existence of the Young integral . Hence, from now on we refer the integral as nonlinear Young integral.
It is evident that
This together with (2.5) imply easily the following.
Fractional calculus is very useful in the study of (linear) Young integral. It leads to some detailed properties of the integral and solution of a differential equation (see , , and the references therein). It is interesting to extend this approach to nonlinear Young integral. In fact, the authors obtain in  the following presentation for the nonlinear Young integral by using fractional calculus. Since this method is not pursued in the current paper, we refer the readers to  for further details.
2.2. Mapping properties
Let be a function as in Lemma 2.2. Let us define the quality
In several occasions, given two functions and such that and are finite, one would like to compare the integrals and . The following result answers this question.
Let and be two continuous functions on such that and are finite for some . Then for every
As an application, we study the dependence of the nonlinear Young integration with respect to the medium and the integrand .
Let and be real valued functions on satisfying the condition 1. Let be a function in and let . Then
Let be a function on satisfying the condition 1. Let and be two functions in and let . Let such that . Then for any
where and .
We put , and . Applying Lemma 2.9, we obtain, for any such that
It remains to estimate . It is obvious that for
On the other hand
Combining the two bounds for we get for any such that ,
This completes the proof. ∎
Let be a nonempty closed, bounded and connected interval. Let be in . Assuming condition 1 with . Then the map
is continuous and compact.
Continuity follows immediately from Proposition 2.11. For compactness, suppose is a bounded subset of . The estimate in Proposition 2.11 implies that is bounded in . By the Arzelà-Ascoli theorem, the set is relatively compact in for every . We show that is indeed relatively compact in . More precisely, suppose is a convergent sequence in in the norm of , by taking further subsequence, we can assume that the sequence converges to in , for some (this is possible since is bounded). It is sufficient to show that converges to in . To prove this, we choose and such that , and then we apply Proposition 2.11 to obtain
The constant depends only on and which is uniformly bounded with respect to . This shows converges to in and completes the proof. ∎
3. Differential equations
We are concerned with the existence, uniqueness, boundedness and the flow property of the solution. We shall also study the dependence of the solution on the initial conditions. Some related results on this direction are also obtained independently by Catellier and Gubinelli . Applications of the results obtained are represented in Subsections 3.3 and 3.4 where we consider a transport equation of the type
Literature on transport equations is vast and mostly focuses on irregularity of the spatial variables of the vector field (see for instance  for Sobolev vector fields,  for BV vector fields and  for Besov vector fields). In the case being a semi-martingale, the above equation is treated in . It appears to be new in the context of nonlinear Young integration.