Uniform large deviation principles for the stochastic heat equation over unbounded sets of initial dataThanks: The first author was supported by Simons Foundation Grant 962543. The second author was supported by the VICI subsidy VI.C.212.027 of the Netherlands Organisation for Scientific Research (NWO)
Abstract.
We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and initial data. For these two classes, the ULDP holds uniformly over -bounded subsets, with and allowed respectively, so these sets are highly unbounded in the state space. The admissible range of is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to . This yields ULDPs over -bounded subsets, enabling the study of exit times in physical systems with mass conservation.
Key words and phrases:
Uniform large deviation principle, stochastic heat equation2020 Mathematics Subject Classification
Primary: 60H15, Secondary: 60F10, 35K58Introduction
We consider the following stochastic heat equation (SHE) on the one-dimensional torus , for small :
| (0.1) |
where is a space-time white noise. In this paper, we study uniform large deviation principles (ULDPs) for families of solutions to (0.1), as . We prove ULDPs in the topology for and the topology on the path space of solutions to (0.1). Unlike previous results for this problem, our main results prove that the ULDP holds uniformly over initial data that are bounded in norm, for some . In this regime, the set is unbounded in . Previous results mostly prove uniformity over bounded sets or compact sets of initial data [9, 5].
A large deviation principle formally quantifies the exponential decay rates for probabilities of rare events. In the context of the SHE, large deviation principles allow us to accurately study the asymptotic behavior of solutions , for the small-noise limit . Beyond guaranteeing almost sure convergence of to the deterministic solution , they provide the precise rates of decay for probabilities of relevant events. A powerful application of this theory is the Freidlin–Wentzell exit time problem, due to Freidlin and Wentzell [23]. For (or ), using large deviation principles, one can characterize the exponential growth rates of the corresponding exit times for the solutions , defined as
For these exit time applications, it is crucial to establish an LDP that holds uniformly with respect to the initial data in , i.e. a ULDP (see Definition 1.2). We refer the reader to [23, 18] for details on the general proof strategy used in such applications.
A highly influential development for proving ULDPs for small-noise SPDEs is the weak convergence approach by Budhiraja, Dupuis, and Maroulas [5], which yields uniformity over compact sets of initial data. This approach has been applied in a large body of literature on LDPs [4, 28, 27, 25, 46, 1, 2, 8, 16, 19, 29, 30, 31, 32, 34, 36, 37, 38, 44, 45, 48, 49, 35] and ULDPs [3, 6, 24, 21]. However, for the exit problem above, one typically wants to consider sets with non-empty interior. Because such sets are non-compact in infinite-dimensional Banach spaces and one needs uniformity over initial data in , the weak convergence approach cannot be applied directly. In [39], the first author showed that the variational principle underlying [5] can nevertheless be used to establish uniform large deviation principles on non-compact, and even unbounded, sets of initial data, provided that the controlled equations satisfy a stronger mode of convergence than weak convergence.
Concerning the SHE and more general reaction-diffusion equations, the earliest LDP results include [22, 43, 26] and treat Lipschitz coefficients and . In [9], Cerrai and Röckner relaxed the assumptions of those works by removing Lipschitz continuity from reaction terms and removing ellipticity conditions on the multiplicative noise terms, and they were the first to prove a ULDP over supremum-norm-bounded subsets of initial data. More recently, the work [40] by the first author further relaxed the assumptions of [9] by removing local Lipschitz and polynomial growth conditions for the reaction terms, while establishing a ULDP that still holds uniformly over supremum-norm-bounded subsets of initial data, and even uniformly over all initial data in case is bounded.
The purpose of the current work is to establish, for unbounded noise coefficients , a ULDP that holds uniformly over sets of initial data that are unbounded in , i.e. sets for which we do not have pointwise-in-space control. More precisely, we will prove large deviation principles that hold uniformly over -bounded subsets of initial data, where . We assume that and are Lipschitz and grow (sub)linearly:
Assumption 0.1.
, are Lipschitz continuous, , and there is a constant such that for all :
| (0.2) | ||||
While we anticipate that our results generalize to multi-dimensional systems and non-Lipschitz dissipative drifts, our primary focus in this work is establishing uniformity with respect to unbounded sets.
Our first main result is the following. The precise notions of mild solution and ULDP will be discussed in Section 1.
Theorem 0.2 (ULDP on ).
Remark 0.3.
If , then Theorem 0.2 establishes the ULDP uniformly over -bounded subsets of initial data in , in the notably strong topology.
In many applications, the solutions to the SHE are nonnegative and the solutions represent mass densities of diffusing molecules. In this setting, the total mass of the system at time is the norm . Thus, taking , Theorem 0.2 shows that when , the solutions satisfy a ULDP in the uniform topology that is uniform over sets of initial data with bounded mass. This is a big improvement over previous results that could only prove a ULDP that was uniform over sets of densities bounded in the norm. Returning to the motivation discussed above, this improvement paves the way for deriving probabilistic exit time estimates for mass-conserving systems.
Besides initial data in , we will also consider initial data in for and study uniform large deviations in the topology. In this setting, one expects the ULDP to hold in the topology uniformly over -bounded sets of initial data, and consequently uniformly over -bounded sets for (since ). We cover this expected regime. The core novelty of the current work, however, lies in establishing the ULDP for the regime where and the balls are unbounded in the topology. Our result is as follows.
Theorem 0.4 (ULDP on ).
The lower bound for satisfies , thus indeed, Theorem 0.4 yields an admissible range of for which we have uniform large deviations over -bounded – but -unbounded – sets of initial data. In particular, we obtain uniformity over -bounded initial data for all if the growth rate is at most , and for a -dependent range of if :
Remark 0.5.
If , then the lower bound for in Theorem 0.4 is automatically satisfied for and any . Thus we have a ULDP on over -bounded subsets of initial data in for all .
If , then we also have the ULDP on with uniformity over -bounded subsets of , for all in the non-empty range . In particular, is always admissible.
We prove the ULDP theorems by studying the convergence of controlled mild solutions. We study
where
Here, is the heat semigroup and is the heat kernel defined in Subsection 1.1. A key ingredient in our analysis is the semigroup regularization estimate
The above time singularity shows that for all satisfying . We demonstrate that under the assumptions of Theorem 0.4 or 0.2, the integral terms , and all belong to (resp. ), and establish important bounds that depend on the norm of the initial data . These calculations enable us to prove the ULDP and also yield novel existence and uniqueness results for rough initial data. The latter are presented in Theorems 2.5 and 2.7. Smoothing properties of the SHE with rough initial data have previously been studied in [10, 11, 12, 13, 7, 14, 20, 42, 15]. Our focus differs from those results because we establish estimates that are uniform over -bounded initial data, and we show how the growth rate of affects the regularization.
Building on our well-posedness results and the proofs of the ULDP theorems above, we establish in Section 5 additional ULDPs that hold uniformly over arbitrary bounded sets of -initial data. That is, the sets of initial data are no longer assumed to be subsets of or . Because of this, the solutions no longer take values in or , so the ULDP in these topologies is inherently ill defined. However, if we subtract off the irregular semigroup term, we can prove that satisfies a ULDP in the (resp. ) topology, see Corollary 5.1. As a further consequence, we also establish ULDPs for in weaker topologies, such as -type topologies, while allowing for initial data belonging only to .
The organization of this paper is as follows. In Section 1, we review the relevant background theory, specify the solution notion that we adopt, and state sufficient conditions from [39] that we will use to prove our ULDP results. In Section 2, we establish deterministic and stochastic convolution estimates for the stochastic heat equation and for associated control problems that are relevant for the ULDP. Using these estimates, we establish new well-posedness and regularity results in Theorems 2.5 and 2.7. In Sections 3 and 4, we prove our main ULDP results (Theorems 0.2 and 0.4) using the estimates of Section 2 together with an efficient Grönwall scheme. In Section 5, we derive further ULDPs based on these results. In particular, we establish ULDPs for initial data belonging only to , using alternative topologies.
Acknowledgement
The authors are grateful to Mark Veraar for encouraging and supporting E.T.’s research visit to M.S. at Boston University, where this work was carried out. They also thank Boston University for hosting this visit.
1. Preliminaries
The following notation will be used throughout the paper.
- •
The notation ‘’ means ‘less than or equal to a constant multiple of’. Subscripts, as in ‘’, indicate the parameters on which the implicit constant may depend. In statements of the form ‘ … a.s.’, the constant is uniform with respect to a.e. .
Although the measure of the torus is fixed throughout the paper, we retain in such subscripts to emphasize when the finite measure of the domain is used.
- •
denotes the convolution operator over the domain for the spatial variable, or over for the time variable, as will be clear from the context.
- •
denotes the stochastic convolution with respect to space-time white noise.
In this section, we discuss the theoretical background that is relevant for our ULDP results. From now on, we fix an arbitrary filtered probability space satisfying the usual conditions.
1.1. Heat semigroup
Let be the periodic heat kernel defined by
and define through
The kernel satisfies the following (see [41, Lem. 2.1]):
| (1.1) | ||||
| (1.2) |
where is a constant independent of and .
As a consequence of (1.1) and Young’s convolution inequality, for all . Moreover, one can show that is a strongly continuous semigroup on , and on . For all these domains, we call the resulting semigroup the heat semigroup and denote it simply by .
The kernel bounds above and interpolation (or Hölder’s inequality) yield for all :
| (1.3) |
Combining (1.3) with Young’s convolution inequality, the following smoothing estimate holds for the heat semigroup, for any , , and :
| (1.4) |
where is such that .
Finally, using the semigroup property of and the fundamental lemma of the calculus of variations for instance, one can show that the kernel satisfies for all and :
| (1.5) |
1.2. Space-time white noise
We will use both Walsh’s definition and the functional analytic definition of space-time white noise.
Let , let be an orthonormal basis for and let be a sequence of independent standard -Brownian motions. For any progressively measurable process such that , we define
| (1.6) |
where the integral denotes the Itô integral. We call the space-time white noise measure.
We can identify with an -cylindrical Brownian motion , where the latter satisfies for all and .
Let and let be a progressively measurable process such that , identifying . We can define the Walsh integral pointwise in through (1.6) for . Then, the following identity holds a.s. in (with respect to ):
where the integral is the stochastic integral in the framework of UMD spaces (see [47]), and
By the -Fubini isomorphism (see [47, (3.1), (3.2)]), is a random operator such that
for a.e. . Moreover,
The BDG inequality in the UMD Banach space (see [47, (5.5)]) and the isomorphism above yield
| (1.7) |
1.3. Solution notion
For , we define
| (1.8) | ||||
For and , we consider the following implicit equation
| (1.9) |
where
| (1.10) | ||||
Here, the and integrals are Lebesgue integrals, and the integral is the stochastic integral with respect to the space-time white noise as defined in Subsection 1.2.
Observe that if , then the equation for corresponds to the mild formulation of the original stochastic heat equation (0.1) discussed in the introduction. Furthermore, if and (so is -independent), then the equation for corresponds to the so-called skeleton equation that appears in the rate functions in Theorems 0.2 and 0.4 (see (0.3)).
We define solutions in our framework as follows.
1.4. ULDP and sufficient criteria
The Freidlin–Wentzell uniform large deviation principle (with speed ) is defined as follows.
Definition 1.2 (ULDP).
Let be a Polish space, completely metrized by a metric . Let be a probability space. Let be any set, and let be a family of -valued random variables. Let be a collection of lower-semicontinuous maps . Let be a collection of subsets of . Then, the family is said to satisfy a Freidlin–Wentzell uniform large deviation principle (ULDP) with respect to the rate functions uniformly over , if
- •
For any , , and ,
- •
For any , , and ,
where .
From now on, we specialize to being a solution to the stochastic heat equation (0.1), and we will let , , or , , respectively. In these settings, we will consider large deviation principles that hold uniformly over families of -bounded subsets of and . For and , we define
| (1.11) |
The next result follows from [39, Th. 2.13]. It provides sufficient criteria for the ULDPs stated in Theorems 0.2 and 0.4, and Corollary 5.4. We will use these criteria to prove the latter theorems. Here, we denote by the unique mild solution that will be provided by Corollary 2.8.
2. Convolution estimates, existence, uniqueness, and regularity
First, in Subsection 2.1, we prove deterministic and stochastic convolution results for the heat semigroup, which will then be used for our well-posedness results in Subsection 2.2 and are also essential for the ULDP proofs later on.
2.1. Deterministic and stochastic convolution estimates
The next lemma demonstrates a regularizing effect – specifically, an improvement in spatial integrability – for the mapping . This estimate will serve as a crucial tool for bounding terms and in our subsequent proofs.
Lemma 2.1.
Let and suppose that , , or , . Let be such that . Let
| (2.1) |
and let . Define
Then, it holds that , and for any ,
Moreover, if , then .
Proof.
First, by the Cauchy–Schwarz inequality in , we have for all and :
If , then we apply Minkowski’s inequality, followed by Young’s convolution inequality (with powers and ) and the bound (1.3) for , giving
Note that by the assumed ranges of and .
Consequently, putting and applying Hölder’s inequality with , we have for all :
| (2.2) |
where we used that if and only if
i.e. if and only if (2.1) holds.
If , then and the above still applies with and . Thus, in either case, we conclude that and the claimed bound holds.
Next, we prove that if , by approximating and . Assume that and . Then, and is a strongly continuous semigroup on , so by classical semigroup theory we have . Consequently, the following bilinear operator is well-defined:
By the bound (2.2) (take ), we have
| (2.3) |
Thus, by density of in and in , extends to a bounded bilinear operator on , proving the claimed continuity for the case .
It remains to prove that when . Since is bounded, we may assume without loss of generality that while (2.1) is still satisfied and . For , we have by classical strongly continuous semigroup theory. Then, from (2.2) and density of in and , it follows that extends uniquely to a bounded bilinear map , completing the proof. ∎
Next, we prove a -norm moment estimate for stochastic convolutions with the heat semigroup. Here, we denote by the stochastic convolution in time with respect to the space-time white noise , viewed as an -cylindrical Brownian motion (see Subsection 1.2).
Proposition 2.2.
Let , and . Let be a progressively measurable process such that . Let
where is the heat semigroup on . Then, , and for all ,
Proof.
It suffices to prove the bound with , since one can afterwards apply the bound to for . Furthermore, it suffices to prove the bound for pointwise bounded . Indeed, is a linear operator, and pointwise bounded, progressively measurable processes are dense in the class of progressively measurable processes in . Frow now on, we assume that is pointwise bounded.
We employ the factorization method from [17]. We fix an arbitrary , recalling that . By (1.1) and Young’s convolution inequality, we have Therefore, since and , [17, Prop. 5.9] yields that
| (2.4) |
where . Furthermore, if , then a.s.
where denotes the stochastic convolution with respect to the space-time white noise , and
Combining the formula for above with (2.4), we have
| (2.5) |
Now, we first apply the BDG inequality in the UMD space (see (1.7)) for fixed . After that, we apply the bound (1.2) for , followed by Minkowski’s inequality (using that ) and Young’s convolution inequality in space, giving
Applying Tonelli’s theorem twice, and applying the above, we obtain
In the second-last line, we applied Young’s convolution inequality in time with . In the last line, we used that . Combining the last estimate with (2.5), we conclude that
Choosing, for instance, , the implicit constant depends only on and . ∎
The following estimate for the stochastic convolution, uniform in both time and space, is due to [41, Th. 1.2]. The continuity of in time and space follows from the proof of [41, Th. 1.2] and the factorization method [17, Prop. 5.9, Th. 5.10] applied with , , , and .
Theorem 2.3 ([41, Th. 1.2]).
Let . Let be a progressively measurable process such that . Let
where is the heat semigroup on . Then, , and for all ,
2.2. Existence, uniqueness, and regularity for solutions with initial data in
With the results of the previous subsection at hand, we are ready to begin our study of the controlled SHE for , i.e.
where and are as in (1.10). In the main results of this subsection, we allow for , thereby treating the skeleton equation (), the controlled SHE (), and the original SHE () simultaneously.
Throughout this subsection, we let
| (2.6) |
Our goal is to prove well-posedness and regularity results for initial data belonging only to , allowing and . In this case, solutions cannot be - or -valued because of the initial condition. However, we will establish well-posedness and regularity by studying , which exhibits greater regularity. First, in Theorem 2.5, we assume the parameter conditions of Theorem 0.4 and establish these properties using the space . Then, in Theorem 2.7, we prove an analogous result using the space under the conditions of Theorem 0.2.
For our first theorem, the following lemma will be essential to setting up a suitable Picard iteration.
Lemma 2.4.
Let the conditions of Theorem 0.4 be satisfied. Let
Let , let with , and let . Define mappings and by
| (2.7) | ||||
Proof.
Step 1a ( is well-defined): Let and . By contractiveness of on , (1.4), and the linear growth in (0.2) for , we have
using that . Moreover, by the estimates above, we have . By classical continuous semigroup theory, it follows that . The estimate above gives
proving that is well-defined.
Step 1b ( is well-defined): Let . First, we show that . If , then the semigroup bound (1.4) gives
If , then by Hölder’s inequality, and using the contractivity of on ,
Combining both cases, we have for all :
| (2.9) |
provided that
Assuming and , the latter holds if and only if
which coincides exactly with our assumptions for the case . Thus indeed, .
Let . To prove that , we can apply Lemma 2.1 with and , provided that . By the growth bound (0.2) for and by (2.9), we have
| (2.10) |
where we used that . Thus, Lemma 2.1 indeed gives
Step 1c ( is well-defined): Let . We apply Proposition 2.2 with and . Thanks to this result, it suffices to prove that . Applying (2.2) pointwise in and taking the expectation yields
| (2.11) |
thus is well-defined.
Step 2a ( is Lipschitz): Let . Since is contractive on and is Lipschitz continuous, we have for all :
Taking the supremum over , raising both sides to the -th power, and applying Hölder’s inequality (recall that ) yields
Step 2b ( is Lipschitz): Lemma 2.1 and the Lipschitz continuity of imply that for all and ,
Step 2c ( is Lipschitz): Proposition 2.2, the Lipschitz continuity of , and Tonelli’s theorem imply that for all and :
Step 3 ( and on ): Let and . Applying the estimates of Steps 1a and 1b pointwise in gives , and the Lipschitz estimates of (2.8) apply a.s. ∎
Next, we establish existence and uniqueness for the controlled stochastic heat eaquation (1.9).
Theorem 2.5.
Proof.
Define the space by (2.6). We use a Picard iteration scheme in the space . Let . Using the mappings from Lemma 2.4, define recursively for :
| (2.12) |
By the Lipschitz bounds of Lemma 2.4, we have for all and :
By induction, using the preceding estimate, we obtain for all and :
for a constant depending only on and . Consequently,
Since , it follows that is Cauchy in , and therefore, it has a limit in . Moreover, Tonelli’s theorem yields
Hence, a.s. By a telescoping sum, it follows that is Cauchy in a.s., hence converges also a.s. to in (by uniqueness of limits in probability).
Define and note that by (2.9). We prove that is a mild solution to (1.9). Since is deterministic and Borel measurable, it is progressively measurable. Moreover, since a.s. in and each is adapted, completeness of the filtration implies that has an indistinguishable adapted continuous version, which is progressively measurable. Hence is indistinguishable from a progressively measurable process. Next, we verify that satisfies the mild solution formula. By (2.8) and Tonelli’s theorem,
In particular, we conclude that the following convergences hold in , a.s. for all :
Consequently, we have a.s. for all , a.e. on :
Recalling the definition of and , this proves that is a mild solution.
For the regularity claims, note that by construction, . Moreover,
Since the mappings were proved to map , we conclude that and belong to .
Finally, we prove uniqueness within . If are mild solutions and belong to , then for : and
Hence, (2.8) applied with and implies that for all :
and Grönwall’s inequality yields a.s., proving uniqueness. ∎
Next, we work toward our second well-posedness and regularity result, Theorem 2.7, which treats initial data in under the parameter conditions of Theorem 0.2. As a preliminary step, we derive a suitable analog of Lemma 2.4 using the space .
Lemma 2.6.
Proof.
The proof of Lemma 2.4 can be copied with the small adjustments indicated below. Let and be as in the statement.
Step 1a: We have by the linear growth of and (1.4):
Consequently, and since is a contractive semigroup on ,
For continuity of in space and time, note that by the smoothing properties of the heat semigroup and Lipschitz continuity of . Moreover, is strongly continuous on , so standard semigroup theory yields .
Step 1b: We derive an bound for the term . When , (1.4) guarantees that
When , we have and
Combining both cases, we obtain
| (2.14) |
If , we have if , or equivalently, Our assumption that thus guarantees that .
Consequently, for , we have by (2.2) applied with and noting that . Hence, for well-definedness of , we can apply Lemma 2.1 with and . Note that for these and , (2.1) holds if and only if , which we assumed.
Step 1c: For well-definedness of , we apply Theorem 2.3 instead of Proposition 2.2. Note that for , we have by (2.11) applied with and since .
Steps 2a–2c: In the proofs of these steps, we can replace every by and by , applying Lemma 2.1 with in Step 2b and applying Theorem 2.3 in Step 2c.
Step 3: In the proof we can replace by and , by , . ∎
Theorem 2.7.
Proof.
We can copy the Picard iteration scheme from the proof of Theorem 2.5 with replaced by . Since Lemma 2.6 fully replaces Lemma 2.4, all arguments remain valid and yield existence and uniqueness of mild solutions belonging to . Moreover, the regularity claims can be proved analogously, using the splitting with . The latter then implies that , , and . ∎
As a consequence of Theorems 2.5 and 2.7, we obtain the following results for classical initial data and solution spaces. For the uncontrolled case (), corresponding results are classical, see e.g. [17]. For the controlled problem, part (2) below also follows from [40, Th. 7.1], noting that the boundary conditions do not affect that result.
Corollary 2.8 (Well-posedness, classical cases ()).
3. ULDP on over -bounded subsets of
In this section we prove Theorem 0.2, using the sufficient condition stated in Theorem 1.3(1). To verify that condition, we will establish uniform estimates for , for a suitable . Note that we can write
As we will show in the proof of Theorem 0.2, the differences and can be estimated suitably in terms of , and will disappear after an application of Grönwall’s inequality, thus it does not lead to any -dependence. In contrast, does depend on , and we need to derive estimates for this term that are uniform with respect to -bounded initial data .
Let us outline how we will deal with the term . Using Theorem 2.3, it will be shown that for a suitable , we have
Since we study the limit , we can assume that , and the second term in the sum above will be absorbed through Grönwall’s inequality. Due to the prefactor above, it then remains to prove that
In Subsection 3.1, we will deal with , and establish an even stronger, almost sure bound, which is uniform over . In Subsection 3.2, we provide the proof of Theorem 0.2.
3.1. Uniform -bound for
In the next lemma, we prove an almost sure, uniform bound for solutions to equation (1.9) with . This bound is uniform with respect to -bounded (forcing) functions , and uniform with respect to -bounded initial data .
Lemma 3.1.
Let Assumption 0.1 hold. Let and . Suppose that and . Then, there exists a constant such that for all and :
| (3.1) |
The value of depends only on and .
Proof of Lemma 3.1.
The condition is there only to ensure that the range for is non-empty. Note that for a.e. , i.e. stochasticity only enters into through . Therefore, recalling the definition of , it suffices to prove that for all with , the bound in (3.1) holds.
Let with . We will apply Grönwall’s inequality to . Using the notation of (1.10), we have
By (2.14), we have for any :
| (3.2) |
For , assuming for the moment that , we have by Minkowki’s inequality, Hölder’s inequality and the linear growth of :
For the convolution term , we apply Lemma 2.1 with and , and use the growth bound (0.2) for and the fact that , giving for all :
Raising this estimate to the power gives
For all , we conclude that
Hence, using also (3.2) and integrating over , we find for all :
Grönwall’s inequality yields
where is a constant depending only on and . Setting , this concludes the proof if .
If , then we put . Note that and since and . Thus the proof applies with and gives for all :
Combined with Jensen’s inequality (for the concave function ), we obtain a.s.
Redefining , we again obtain (3.1). ∎
3.2. Conclusion of the ULDP
Proof of Theorem 0.2.
Fix any and . For all and , Theorem 2.7 provides a unique mild solution to (1.9). Using this regularity and the growth condition in (0.2), we have . Therefore, we may apply Theorem 2.3 with , giving together with (0.2):
In the third line we used that , and we used that since . For the last term in the upper bound above, we have by Lemma 3.1 (see (1.11)):
Furthermore, by (2.13) of Lemma 2.6 (, for ) and the Tonelli’s theorem, we have for all and :
Combining all estimates above, we find that for all , , , , and , it holds that
Grönwall’s inequality gives that
where is a constant that depends only on , and . Finally, Chebychev’s inequality provides the convergence in probability as that allows us to apply Theorem 1.3(1), completing the proof of the ULDP. ∎
4. ULDP on over -bounded subsets of
In this section, we prove Theorem 0.4. We will verify the sufficient criterion from Theorem 1.3(2), using an analogous series of estimates as in Section 3, but with replacing . Eventually, we will prove uniform bounds for for a suitable .
Recall the splitting (see (1.9)). Again, we will absorb the differences and in our Grönwall estimate, and are left with bounding . As in Section 3, we will reduce this problem to bounding uniformly in , , and . The latter will be achieved in Subsection 4.1.
4.1. Uniform -bound for
The following result will be crucial for controlling the term in the proof of our main result. It can be viewed as an analogue of Lemma 3.1. We note that the stated range for is always non-empty: this follows from (4.1) when , and holds trivially when .
Lemma 4.1.
Then, there exists a constant such that for all and :
The constant depends only on and .
Proof.
Let and . Since for a.e. , and by the definition of , it suffices to prove the bound for with . We will apply Grönwall’s inequality to . Recall that . We estimate these terms separately.
Thanks to the parameter assumptions, we can apply (2.9), giving for all :
Moreover, assuming for the moment that and , we have by Minkowki’s inequality, Hölder’s inequality and the linear growth of :
Integrating over , we conclude that for all :
By the last part of Lemma 2.1 applied with , , and , the growth bound (0.2) for , and the fact that , we have for all :
Raising the estimate to the power and integrating over , we conclude that for all :
Combining the estimates above, if and , we obtain for all :
Grönwall’s inequality thus gives
concluding the proof if and .
If or , then we define and . We verify that the parameter conditions also hold with . Clearly, and . Moreover, if , then (since ), hence and . Also, (2.1) is satisfied for : if , then by assumption, and if , then . Consequently, our result can be applied with and , giving a.s.
Applying Jensen’s inequality twice (for on and for on ), we find
Redefining , and noting that depend only on and , this proves the result for the general case. ∎
4.2. Conclusion of the ULDP
Proof of Theorem 0.4.
First, we observe that it suffices to prove the theorem for . Indeed, , so if , then we can pick and note that -bounded sets are also bounded (since is bounded). Thus the ULDP follows by applying the stated ULDP with .
From now on, let and . We will apply Grönwall’s inequality to the mapping . For all , we have
Note that the difference equals , thus the last part of Lemma 2.4 and Hölder’s inequality yield for all :
Similarly, , so the last part of Lemma 2.4 and Hölder’s inequality yield
For , we can apply Proposition 2.2 with (so ). The latter, combined with the growth bound (0.2) for and the fact that , gives
Combining the estimates above, we find that for any fixed , for all , and for all ,
Now, Grönwall’s inequality gives
| (4.2) |
for a constant that depends on and . Moreover, Lemma 4.1 gives for all (see (1.11)):
| (4.3) |
Fixing any in the (non-empty) range above, and combining (4.2) and (4.3) with Chebychev’s inequality proves that the condition of Theorem 1.3(2) is satisfied, thus the latter yields the ULDP. ∎
5. Further ULDPs
In this section, we derive further ULDPs using our previous results and the arguments developed in their proofs. Our next corollary establishes ULDPs for initial data that need only belong to , without the additional requirement of belonging to , as in Theorem 0.4.
Corollary 5.1 (ULDPs with initial data in ).
Let the conditions of Theorem 0.4 hold. For and , let be the mild solution to (0.1) provided by Theorem 2.5. Then:
- (1)
The family satisfies the ULDP on uniformly over bounded subsets of , with rate functions given by
where and is the mild solution to
- (2)
If is any subset of , and is any Polish topological vector space such that
(5.1) and there exists a constant such that for all with ,
(5.2) then the family satisfies the ULDP on uniformly over -bounded subsets of , with rate functions defined by the formula (0.3).
Proof.
Proof of (1): We apply [39, Th. 2.13] with the Polish space . We have existence of measurable solution maps such that a.s., thanks to Theorem 2.5 and the Yamada–Watanabe theorem. Indeed, for each fixed , is a mild solution to
| (5.3) |
Because is a mild solution to (5.3) if and only if is a mild solution to (0.1), the pathwise uniqueness stated in Theorem 2.5 implies that is the unique mild solution to (5.3) within . The Yamada–Watanabe theorem (see [33]) yields measurable mappings such that a.s., where we identify as in (1.6).
Taking into account Remark 4.2, the proof of Theorem 0.4 gives immediately
| (5.4) |
Since , the ULDP now follows by [39, Th. 2.13].
Proof of (2): One can combine part (1) either with a contraction argument and the definition of the ULDP, or with another application of [39, Th. 3.12]. Here, we use the latter approach.
Let . The proof of (1) yielded measurable maps such that a.s. Now, for any , we have by assumption, and is continuous since and since is a topological vector space (so translations are continuous). Thus, the mappings are Borel measurable and satisfy a.s.
Analogous ULDP results can also be established under the conditions of Theorem 0.2.
Remark 5.2.
The following can be derived as a special case of Corollary 5.1(2).
Corollary 5.3 (ULDP in ).
Let the conditions of Theorem 0.4 hold. For and , let be the mild solution to (0.1) provided by Theorem 2.5. Let be such that .
Then, the family satisfies the ULDP on uniformly over bounded subsets of , with rate functions defined by the formula (0.3).
Proof.
Our final result shows that a ULDP over -bounded sets of initial data always holds in the topology for , whenever the noise has sublinear growth (). If the noise has linear growth (), we obtain the ULDP over -bounded sets of initial data for any .
Corollary 5.4 (ULDP on for ).
Let Assumption 0.1 hold. Let and let if , and if .
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