Intrinsic scaling properties for nonlocal operators
We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques.
Key words and phrases:integrodifferential operators, regularity, jump processes, intrinsic scaling
2010 Mathematics Subject Classification:Primary 35B65; Secondary: 35R09, 47G20, 60J75, 31B05
In recent years, regularity results for linear and nonlinear integrodifferential operators have been addressed by many research articles. Scaling properties are crucially used in these approaches. We reconsider these cases and, at the same time, include limit cases where standard scaling properties do not hold anymore. We study linear operators of the form
which, provided certain assumptions on are satisfied, are well defined for smooth and bounded functions . The quantity equals the jump intensity of jumps from to for the Markov process that is generated by the linear operator . If is independent of the first variable, then is a Lévy process. If for all and some appropriate function , then the increments of have stable distributions. Looking at the operator as an integrodifferential operator, this property is important because it allows to use scaling techniques.
Scaling techniques themselves are crucial when studying regularity properties of functions satisfying the equation in a domain for some function . In this work we study such properties with the emphasis on two features. We do not assume any regularity of the kernel function with respect to the first variable except for boundedness. Moreover, and this is the main new contribution, we systematically study classes of kernels that do not possess the aforementioned scaling property.
Our results include a growth lemma (expansion of positivity) and Hölder-type regularity estimates. Moreover, we provide several estimates on the corresponding Markov jump process. Recall that, in the case of an elliptic operator of second order , the standard growth lemma reads as follows:
There is a constant such that, if and satisfying
then in .
The above lemma holds true also for several nonlinear operators. Such lemmas are systematically studied and applied in [Lan71]. Their importance is underlined in the article [KrSa79], in which the authors establish a priori bounds for elliptic equations of second order with bounded measurable coefficients. Nowadays they form a standard tool for the study of various questions of nonlinear partial differential equations of second order, see [CaCa95] and [DBGV12]. Note that the property formulated in 1 is also referred to as expansion of positivity which describes the corresponding property for .
In this work we prove a similar growth lemma for integrodifferential operators, see 2 below. An important instance of an operator that we have in mind is
for some measurable function . Note that, in the last years, similar results have been studied for kernels of the form for some and we refer the reader to the short discussion below. The case is of particular interest because in this case the corresponding growth lemma fails. Our results apply to more general kernels than the one appearing in (1).
1.1. Main assumptions and results
Let be a measurable function such that for all and
where , are fixed constants and is a function satisfying and, for some , , and , the following:
The last two conditions often are referred to as weak scaling conditions.
Examples: The standard example is given by for some . Other examples satisfying the above conditions include for and a function that varies slowly at . More generally, the conditions () and () are satisfied if the function is regularly varying at zero of order and satisfies some weak bounds for large values of . The case is very interesting. The choice is possible if . In the case an interesting example is provided by for some .
We define an auxiliary function by , which is strictly decreasing. Note that, under our assumptions and . Furthermore, we define a measure on by
and, for , a scale function by .
Now we can formulate our first main result, a growth lemma for nonlocal operators. We state the result for functions which, together with their first and second derivatives, are continuous and bounded. It is an important feature of this result, that none of the constants depends on the regularity of the function under consideration. Thus, the result is tailored for later applications to viscosity solutions of fully nonlinear partial differential equations.
Remark: As the proof shows, the value of is a multiple of .
Note that, in the by now well-known case where and for , this result reduces to a growth lemma which is very similar to those given in [Sil06] and [CaSi09]. Let us now formulate the second main result.
In the case , we set , and the estimate (4) reduces to
which one would expect from standard scaling behavior of the integrodifferential operator.
In the case we obtain a Liouville theorem.
If (A) holds for , then every function satisfying on is a constant function.
Since is harmonic in every ball we can consider in Theorem 3 and use in order to prove that is a constant function. ∎
Our method to prove 2 and Theorem 3 is based on a purely analytic technique introduced in [Sil06]. As mentioned above, a second aim of this work is to explain a probabilistic approach to results like Theorem 3. The starting point for these observations is that, for several linear differential or integrodifferential operators , variants of 1 can be established with the help of corresponding Markov processes. Let be the strong Markov process associated with the operator , i.e. we assume that the martingale problem has a unique solution. Denote by the hitting resp. exit time for a measurable set and by the measure on the path space with . The following property then implies 1.
There is a constant such that for every and every measurable set with and
This result is established for non-degenerate diffusions in [KrSa79], thus leading to a result like Theorem 3 for elliptic differential operators of second order. The case of integrodifferential operators with fractional order of differentiability is treated in [BaLe02]. Therein it is shown that 5 holds true for jump processes generated by integral operators of the form (2) under the assumptions and for all and where is fixed. Note that this class includes the case and versions with bounded measurable coefficients.
5 fails to hold for several cases that we are interested in. One example is given by as in (2) with for and some appropriate condition for . For example, the geometric stable process with its generator , , can be represented by (2) with a kernel with such a behavior for close to zero. The operator resp. the corresponding stochastic process can be shown not to satisfy a uniformly hitting estimate like (5), see [Mim14].
This leads to the question whether a priori estimates can be obtained by the approach from [BaLe02] at all. In the second part of this work we address this question. It turns our that our main idea, i.e., to determine an new intrinsic scale can also be used to establish a modification of (5). As we did in the proof of 2, we choose a measure different from the Lebesgue measure for the assumption . We refer the reader to LABEL:sec:prob_estim for further details
Since we employ methods from two different fields - methods from partial differential operators as in [Sil06] and from stochastic analysis as in [BaLe02] - it is interesting to compare both approaches. In both approaches we need to make several assumptions, e.g., solvability of the equation and existence of the corresponding Markov jump process. The conditions in the analysis approach are slightly less restrictive than those imposed in the approach using stochastic analysis. Note that, although we assume the solutions to be twice differentiable in the first part, the assertions resp. the constants in our results do not depend on the regularity of the functions . Thus, the techniques and assertions presented here can be applied to nonlinear problems.
Let us look at different choices for the function used in condition (A). Note that () does not allow to be zero (different from ). Since the behavior of at zero is most important and characteristic, we provide examples of functions . For , set and .
1.3. Related results in the literature
Let us comment on related results in the literature. The probabilistic approach, which we explain in LABEL:sec:intro-prob, is based on the approach of [BaLe02]. The analytic method, which we employ in the first part of the article, is based on [Sil06]. Both approaches have been refined in many articles, allowing for more general kernels and treating fully nonlinear integrodifferential equations, but all these articles assume standard scaling properties, i.e., something like for some . We refer to the discussions in [KaSc14, SiSc14] for further references. Note that our regularity result Theorem 3 is stronger than LABEL:theo:main-prob because we can allow for right-hand sides in the integrodifferential equation and for more general kernels.
The current work comprises the two preprints [KaMi13] and [KaMi14] where the approaches by analytic and probabilistic methods are explained separately. After [KaMi13] had appeared, several articles have made use of the ideas therein. In [Bae14] nonlocal problems are studied where the kernels are supposed to satisfy certain upper and lower scaling conditions. These assumptions do not include limit cases like (1) since some comparability with kernels like for is still assumed. In [KKL14] the authors study fully nonlinear problems with similar assumptions on the kernels as in [Bae14]. In [ChZh14] the authors extend the regularity estimates of [KaMi13] to time-dependent equations with drifts. The article [JaWe14] is not directly related to [KaMi13] but mentions the need to consider . We solve this problem.
1.4. Organization of the article
In Section 2 we review the relation between translation invariant nonlocal operators and semigroups/Lévy processes. Presumably, 6 is of some interest to many readers since it establishes a one-to-one relation between the behavior of a Lévy measure at zero and the multiplier of the corresponding generator for large values of . LABEL:sec:proof-expans1 and LABEL:sec:proof-theo-analysis contain the proof of 2 and Theorem 3 respectively. In LABEL:sec:intro-prob we explain the probabilistic approach to Theorem 3, which leads to LABEL:theo:main-prob. Note that we are slightly changing the assumptions there. The probabilistic approach is based on a Krylov-Safonov type hitting lemma, which is LABEL:prop:hitting_new. LABEL:sec:prob_estim contains the proof of this result and of LABEL:theo:main-prob. The last section is LABEL:sec:appendix in which we collect important properties of regularly resp. slowly varying functions.
The authors thank Tomasz Grzywny and Jongchun Bae for discussions about the assumptions used in this article. Further thanks are due to an anonymous referee for comments which helped us to improve the presentation.
2. Multipliers and Lévy measure: Analysis meets Probability
The aim of this section is to provide some background about translation invariant integrodifferential operators and related stochastic processes. The results explained here motivated the search for a new scale function which is a key element of the whole project. However, the material of this section is not needed for the proofs of the main results.
In this paper we provide two approaches to Theorem 3. One approach uses techniques from analysis, the other one uses stochastic processes. Note that the quantity in (2) has a clear interpretation in terms of probability. For fixed , the quantity describes the intensity with which the corresponding process performs jumps from some point to a point from the set . In this sense, the conditions ()-() say something about the behavior of the process. On the other hand, the conditions say something about mapping properties of the operator . In this section we explain the link between these two viewpoints. We restrict ourselves to the cases of translation invariant operators, i.e., we assume to be independent of the first variable. This allows us to give a focused presentation. Note that the results of the section are not used in the rest of the article.
In the translation invariant case, i.e. when does not depend on , there is a one-to-one correspondence between and multipliers, semigroups and stochastic processes. One aim is to prove how the behavior of for small values of translates into properties of the multiplier or characteristic exponent for large values of . This is achieved in 6. We add a subsection where we discuss which regularity results are known in critical cases of the (much simpler) translation invariant case. Note that our set-up, although allowing for an irregular dependence of on , leads to new results in these critical cases.
2.1. Semigroups, generators and Lévy processes
A stochastic process on a probability space is called a Lévy process if it has stationary and independent increments, and its paths are -a.s. right continuous with left limits. For we define a to be the law of the process . In particular, for and measurable sets .
Due to stationarity and independence of increments, the characteristic exponent of is given by
where is called characteristic exponent of . It has the following Lévy-Khintchine representation
where is a symmetric non-negative definite matrix, and is a measure on satisfying called the Lévy measure of .
The converse also holds; that is, given as in the Lévy-Khintchine representation (6), there exists a Lévy process with the characteristic exponent . The equality (6) provides a link to an analytic viewpoint on Lévy processes. If is a Lévy measure, i.e., a Borel measure on , then one can construct a convolution semigroup of probability measures such that the Fourier transform of equals with as in (6). This approach can be found in [BeFo75].
Let be a Lévy process corresponding to the characteristic exponent as in (6) with , and a Lévy measure . Then defines a strongly continuous contraction semigroup of operators on the space of bounded uniformly continuous functions on equipped with the supremum norm. Moreover, it is a convolution semigroup, since
with . The infinitesimal generator of the semigroup is given by
if is sufficiently regular, see the proof of [Sat99, Theorem 31.5]. Note that the process is a martingale (with respect to the natural filtration) for every , see the proof of [ReYo99, Proposition VII.1.6]. In this sense the process corresponds to the given Lévy measure and, in our set-up, to the kernel . For details about Lévy processes we refer to [Ber96, Sat99].
Let us now explain the connection between the characteristic exponent and the symbol of the operator . To be more precise, if denotes the Fourier transform of a function , then
for any , where is the Schwartz space, see [Ber96, Proposition I.2.9]. Hence, is the symbol (multiplier) of the operator . The following result explains how, in the case , the kernel is related to the characteristic exponent resp. the multiplier.
The assumptions of 6 allow to treat sophisticated examples. However, it is instructive to think about the simple examples