A stochastic reconstruction theorem

Kern, Hannes

doi:10.48550/arxiv.2107.03867

Cited by 3 publications

(4 citation statements)

References 17 publications

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“…Indeed, note that already the above proof is exploiting self-similarity properties to transfer the one parameter setting to the present two parameter setting. As already in [25], a crucial role in the regularity estimates for local times was played by the Stochastic Sewing Lemma, we expect that in our setting, a "2D Stochastic Sewing Lemma" not yet available in the literature (see however [27] for a stochastic reconstruction theorem very close in spirit) might prove instrumental in establishing regularization from "both directions".…”

Section: Remark 32mentioning

confidence: 81%

“…The concept of regularization in this article has been presented in view of the local time associated with the field w. While we present here the case of regularization when the field w is given by a fractional Brownian sheet or as a sum of two independent fractional Brownian motions, further systematic investigations of the space-time regularity of the local time associated to various stochastic fields appears in order. In particular refined estimates for the local time of the fractional Brownian sheet using a "multiparameter Stochastic Sewing Lemma" or the application of a stochastic reconstruction theorem as recently provided in [27] The equation above could be reformulated by similar principles as in the current article, although certain extensions must be made with respect to the construction of a non-linear Young integral to account for the possibly singular nature of the Volterra operator S. The stochastic process obtained in t 0 R S t−s (x − y) ẇ(s, y) dy ds might indeed provide a regularizing effect in this equation, but it is then also needed to investigate the space-time regularity of the local time associated with this field. The authors of [2] have recently made certain progress in this direction the case where L is the heat operator.…”

Section: Further Challenges Open Problems and Concluding Remarksmentioning

confidence: 99%

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Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Bechtold

Harang

Rana

2023

Stoch PDE: Anal Comp

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We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

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Section: Remark 32mentioning

confidence: 81%

Section: Further Challenges Open Problems and Concluding Remarksmentioning

confidence: 99%

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Bechtold

Harang

Rana

2023

Stoch PDE: Anal Comp

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“…Let us point out that joint space-time estimates on the regularity of local times of stochastic processes are typically obtained by an application of the Stochastic Sewing Lemma [23] (see for example [21]). Consequently, a possible approach to the study of joint regularity results for local times associated with stochastic fields might consist in the employment of a "multiparameter Stochastic Sewing Lemma" or the application of a stochastic reconstruction theorem as provided in [22].…”

Section: Further Challenges Open Problems and Concluding Remarksmentioning

confidence: 99%

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Bechtold¹,

Harang²,

Rana³

2022

Preprint

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We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli in [6]. To this end we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of regularity of the local time associated to the perturbing stochastic field. As an illustration we prove wellposedness of a 1D non-linear wave equation with a noisy boundary given by two independent fractional Brownian motions. A discussion of several open problems and further investigations is provided.

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“…A multidimensional version of the sewing lemma is the reconstruction theorem [18,Theorem 3.10] of Hairer. A stochastic version of the reconstruction theorem was obtained by Kern [21]. It could be possible to extend Theorem 1.1 in the multidimensional setting, but we will not pursue it in this paper.…”

Section: Introduction and The Main Theoremmentioning

confidence: 99%

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

Matsuda,

Perkowski

2024

Forum of Mathematics, Sigma

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We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$ . Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v<s$ , and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$ . We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

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A stochastic reconstruction theorem

Cited by 3 publications

References 17 publications

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

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