Abstract
We prove a uniform large deviations principle for the law of the solutions to a class of Burgers-type stochastic partial differential equations in any space dimension. The equation has nonlinearities of polynomial growth of any order, the driving noise is a finite dimensional Wiener process, and the proof is based on variational principle methods. We prove the uniform large deviations principle for the law of the solutions in two different topologies. First, in the C([0, T] : Lρ(D)) topology where the uniformity is over Lρ(D)-bounded sets of initial conditions, and secondly in the \(C([0,T]\times \bar D)\) topology with uniformity being over bounded subsets in the \(C_{0}(\bar D)\) norm.
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Appendix A: Results from [10]
Appendix A: Results from [10]
The following results from [10] are used in proving the main theorems.
Proposition 2 (Green’s function estimates (Proposition 3.5 of 10, (3.15) in [10]))
For any multi-index δ, there exist Borel functions a, b, c such that for p ≥ 1 there exist some constants K, C > 0 such that for all 0 ≤ s < t ≤ T, ξ, η ∈ D
-
(E1)
\(\displaystyle |D^{\delta }_{\xi } G_{t-s}(\xi ,\eta )|\leq a(t-s, \xi -\eta ), ~~~~~~~~~~~~~|a(t, \cdot )|_{p}\leq K_{p}t^{-1+{\kappa }_{p}}\),
-
(E2)
\(\displaystyle |\frac {\partial }{\partial \xi _{i}} D^{\delta }_{\xi } G_{t-s}(\xi ,\eta )|\leq b(t-s, \xi -\eta ), ~~~~~~~~~~~~~|b(t, \cdot )|_{p}\leq K_{p}t^{-1-{\theta }_{p}+{\kappa }_{p}}\),
-
(E3)
\(\displaystyle |\frac {\partial }{\partial s} D^{\delta }_{\xi } G_{t-s}(\xi ,\eta )|\leq c(t-s, \xi -\eta ), ~~~~~~~~~~~~~|c(t, \cdot )|_{p}\leq K_{p}t^{-1-{\upsilon }_{p}+{\kappa }_{p}}, \)
In the above estimates
$$ \kappa_{p}: = \frac{1}{2}(d/p-d+2 - |\delta|), \ \ \ \theta_{p} :=(|\delta| + 1)/2, \ \ \ \ \upsilon_{p}:=(|\delta|+2)/2. $$(42) -
(E4)
\(\displaystyle |{D_{t}^{n}}D_{\xi }^{\gamma } G_{t-s}(\xi ,\eta )|\leq K t^{-(d+2n+|\gamma |)/2} \exp \bigg (-C\frac {|\xi -\eta |^{2}}{t-s}\bigg )\) for 2n + |γ| ≤ 3, where \({D_{t}^{n}} := \partial _{n}/\partial t^{n}, ~D_{\xi }^{\gamma } := \partial ^{\gamma }_{1}/\partial {\xi _{1}}^{\gamma _{1}}, \cdots \partial ^{\gamma }_{d}/\partial {\xi _{d}}^{\gamma _{d}}, \\~\gamma := (\gamma _{1}, {\cdots } \gamma _{d})\) is a multi-index, |γ| := γ1 + γ2 + ⋯ γd.
Lemma 3 ([10, Corollary 3.6])
Set
where \(\displaystyle \phi ^{\varepsilon } = \{\phi ^{\varepsilon }(t,\xi ) = (\phi ^{\varepsilon }_{1}(t,\xi ) , {\cdots } ,\phi ^{\varepsilon }_{k}(t,\xi ) ): t\in [0,T], \xi \in D\}\) is a sequence of \(\mathcal F_{t}\)-adapted random fields. If for ρ > d we have a constant C such that \(\displaystyle |\phi ^{\varepsilon }|_{\rho } \leqslant C\) for all t ∈ [0, T] and for all ε ∈ [0, 1], then Γ(ϕε) is tight in C([0, T] × D). In general, there is a number \(\bar \rho >d\) such that if for \(\rho >\bar \rho \) we have \(\displaystyle \sup _{\varepsilon }{\mathbb E} (\sup _{t\leqslant T} |\phi ^{\varepsilon }(t)|_{\rho }^{\rho })<\infty \), then Γ(ϕε) is tight in C([0, T] × D).
Let q ≥ 1 and Rt(ξ, η) := ∂ηGt(ξ, η) or Gt(ξ, η) for r, t ∈ [0, T] and ξ, η ∈ D. For \(v\in L^{\infty }([0,T];L^{q}(D))\) define the linear operator J by
\(\displaystyle J(v)(t,\xi ) := {{\int \limits }_{0}^{t}}{\int \limits }_{D} R_{t-r}(\xi ,\eta ) v(r,\eta ) d\eta dr, ~~~~~~~~~~t\in [0,T], ~~~\xi \in D,\)
J(v)(t, ξ) := 0 if ξ ∉ D.
provided the integral exists.
Lemma 4 (Lemma 3.1 of 10)
Let \(\rho \in [1,\infty ]\), q ∈ [1, ρ], and p := (1 + 1/p − 1/q)− 1. Let κp be as in Eq. 42. Let \(\alpha _{0} := \min \limits \{\kappa _{p}/{\upsilon }_{p}, \kappa _{p}\}\), \(\beta _{0}:= \min \limits \{\kappa _{p}/\theta _{p}, 1/\rho \}\). The following statements hold.
-
(i)
J is a bounded linear operator from Lγ([0, T] : Lq(D)) → C([0, T] : Lρ(D)) for every γ > 1/κp and there exists C > 0 such that
$$ |J(v)(t,\cdot)|_{\rho} \leq C{{\int}_{0}^{t}}(t-s)^{\kappa_{p}-1}|v(s)|_{q}ds \leq C t^{\kappa_{\rho} - \frac{1}{\gamma}} \left( {{\int}_{0}^{t}} |v(s)|_{q}^{\gamma} ds \right). $$(43) -
(ii)
For every α ∈ (0, α0), γ > (α0 − α)− 1 there exists C > 0 such that
$$ |J(v)(t,\cdot) - J(v)(s,\cdot)|_{\rho} \leq C|t-s|^{\alpha} \left( {\int}_{0}^{t\vee s} |v(r)|_{q}^{\gamma} \right)^{\frac{1}{\gamma}}. $$(44) -
(iii)
Let β ∈ (0, β0). Assume that \(\kappa _{p^{\prime }}>0\) for \(p^{\prime }:=(1/p - \beta )^{-1}\). Then for every γ > (β0 − β)− 1, there exists C > 0 such that
$$ |J(v)(t,\cdot) - J(v)(t,\cdot + \zeta)|_{\rho} \leq C|\zeta|^{\beta} \left( {{\int}_{0}^{t}} |v(r)|_{q}^{\gamma} dr \right)^{\frac{1}{\gamma}}. $$(45) -
(iv)
If \(\rho =\infty \) then for \(0< \beta < \min \limits \{\kappa _{p}/\theta _{p},1\}=:\beta ^{\prime }\) and \(\gamma >(\beta ^{\prime }-\beta )^{-1}\), there is a constant C > 0 such that
$$ |J(v)(t,\xi) - J(v)(t,\xi+\zeta)| \leq C|\zeta|^{\beta} \left( {{\int}_{0}^{t}} |v(r)|^{\gamma}_{q} \right)^{\frac{1}{q}}. $$(46)
There is a slight error in the statement of Corollary 3.2 in [10]. The statement assumes |ζn(t,⋅)|p ≤ ξn almost surely. This should say |ζn(t,⋅)|q ≤ ξn. We present the correct version below.
Lemma 5 ([10, Corollary 3.2])
Let ζn(t, ξ) be a sequence of random fields on [0, T] × D such that almost surely
where 𝜗n is a finite random variable for every n. Assume that the sequence 𝜗n is bounded in probability, i.e.
Let ρ ≥ 1 and p := (1 + 1/ρ − 1/q)− 1. Then for \(0\leq \alpha <\min \limits ({\kappa _{p}}/{\upsilon }_{p}, \kappa _{p})\), the sequence J(ζn) is tight in Cα([0, T] : Lρ(D)). In the case \(\rho =\infty \) the sequence J(ζn) is tight in the \(C^{\alpha , \beta }([0,T]{\times \bar D})\) for \(0\leq \alpha < \min \limits ({\kappa _{p}}/{\upsilon }_{p},{\kappa }_{p})\), \(0\leq \beta <\min \limits (\kappa _{p}/{\theta }_{p}, 1)\).
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Salins, M., Setayeshgar , L. Uniform Large Deviations for a Class of Burgers-Type Stochastic Partial Differential Equations in any Space Dimension. Potential Anal 58, 181–201 (2023). https://doi.org/10.1007/s11118-021-09936-x
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DOI: https://doi.org/10.1007/s11118-021-09936-x
Keywords
- Large deviations
- Weak convergence method
- Stochastic partial differential equations
- Infinite dimensional dynamical systems