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We study the effect of additive noise to the inversion of FIOs associated to a diffeomorphic canonical relation. We use the microlocal defect measures to measure the power spectrum of the noise in the phase space and analyze how that power spectrum is transformed under the inversion. In general, white noise, for example, is mapped to noise depending on the position and on the direction. In particular, we compute the standard deviation, locally, of the noise added to the inversion as a function of the standard deviation of the noise added to the data. As an example, we study the Radon transform in the plane in parallel and fan-beam coordinates, and present numerical examples.

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums., This is a companion paper to arXiv:2305.10365. We apologize for the text overlap, due to some common preliminary notions

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability property is now well established. In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H>1/3$ and its Malliavin derivatives., There will be a companion paper to this contribution, establishing weak convergence results for the modified Euler scheme. We apologize in advance for the text overlaps

We prove the existence of density for the solution to the multiplicative semilinear stochastic heat equation on an unbounded spatial domain, with drift term satisfying a half-Lipschitz type condition. The methodology is based on a careful analysis of differentiability for a map defined on weighted functional spaces.

We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" developed in (Gubinelli, 2010) and (Hairer and Kelly, 2015) we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough volterra equations in driven by more irregular signals., Comment: 34 pages

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}\Delta u+u\dot{W}_\alpha$, where $\Delta$ is the hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_\alpha; \alpha>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-\Delta)^{-\alpha}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_\alpha$; (ii) Prove that one can solve the stochastic heat equation in the It\^{o} sense as soon as $\alpha>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.

We introduce a proper notion of 2-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a 2-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here we implement a simple procedure for texture classification. In this context, we show that a low dimensional set of features based on signatures produces an excellent accuracy.

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

We study infinite server queues driven by Cox processes in a fast oscillatory random environment. While exact performance analysis is difficult, we establish diffusion approximations to the (re-scaled) number-in-system process by proving functional central limit theorems (FCLTs) using a stochastic homogenization framework. This framework permits the establishment of quenched and annealed limits in a unified manner. At the quantitative level, we identity two parameter regimes, termed subcritical and supercritical indicating the relative dominance between the two underlying stochasticities driving our system: the randomness in the arrival intensity and that in the serivce times. We show that while quenched FCLTs can only be established in the subcritical regime, annealed FCLTs can be proved in both cases. Furthermore, the limiting diffusions in the annealed FCLTs display qualitatively different diffusivity properties in the two regimes, even though the stochastic primitives are identical. In particular, when the service time distribution is heavy-tailed, the diffusion is sub- and super-diffusive in the sub- and super-critical cases. The results illustrate intricate interactions between the underlying driving forces of our system.

This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform hypoellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution. Our methodology relies heavily on the rough paths structure of the equation.

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution., arXiv admin note: substantial text overlap with arXiv:1505.04924

We consider a stochastic differential equation with additive fractional noise of Hurst parameter H > 1/2, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate √ τ as τ → ∞, when the solution is observed continuously on the time interval [0, τ ]. The proof uses ergodic properties of the equation and a Poincaré type inequality.

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.

Dedicated to David Nualart on occasion of his 60th birthday; International audience; In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space.

International audience; For a support vector machine, model selection consists in selecting the kernel function, the values of its parameters, and the amount of regularization. To set the value of the regularization parameter, one can minimize an appropriate objective function over the regularization path. A priori, this requires the availability of two elements: the objective function and an algorithm computing the regularization path at a reduced cost. The literature provides us with several upper bounds and estimates for the leave-one-out cross-validation error of the ℓ2-SVM. However, no algorithm was available so far for fitting the entire regularization path of this machine. In this article, we introduce the first algorithm of this kind. It is involved in the specification of new methods to tune the corresponding penalization coefficient, whose objective function is a leave-one-out error bound or estimate. From a computational point of view, these methods appear especially appropriate when the Gram matrix is of low rank. A comparative study involving state-of-the-art alternatives provides us with an empirical confirmation of this advantage.

1 figure; In this paper we obtain Gaussian type lower bounds for the density of solutions to stochastic differential equations (sde's) driven by a fractional Brownian motion with Hurst parameter H. In the one dimensional case with additive noise, our study encompasses all parameters 01/2. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.

27 pages; In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H> 1/3. We show that under some geometric conditions, in the regular case H>1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H>1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

15 pages; We study a least square-type estimator for an unknown parameter in the drift coefficient of a stochastic differential equation with additive fractional noise of Hurst parameter H>1/2. The estimator is based on discrete time observations of the stochastic differential equation, and using tools from ergodic theory and stochastic analysis we derive its strong consistency.

In this article, we analyze a system modeling bacteriophage treatments for infections in a noisy context. In the small noise regime, we show that after a reasonable amount of time the system is close to a sane equilibrium (which is a relevant biologic information) with high probability. Mathematically speaking, our study hinges on concentration techniques for delayed stochastic differential equations., 17 pages, 2 figures

In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space., Comment: Dedicated to David Nualart on occasion of his 60th birthday

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/3. We show that under some geometric conditions, in the regular case H > 1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H > 1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space., Dedicated to David Nualart on occasion of his 60th birthday

This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H��lder regularity greater than 1/4. After recalling how to treat the case of ordinary stochastic differential equations, we mainly focus on the case of delay equations. A careful analysis is then performed in order to show that a fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions of our abstract theorems., 32 pages

This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H\"older regularity greater than 1/4. After recalling how to treat the case of ordinary stochastic differential equations, we mainly focus on the case of delay equations. A careful analysis is then performed in order to show that a fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions of our abstract theorems., Comment: 32 pages

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional Brownian motion with Hurst coefficient greater than 1/3, 32 pages, 2 figures

In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author of the article, and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on the analytic fBm can be constructed for any value of the Hurst parameter in (0,1/2). This allows in particular to solve differential equations driven by this process in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli., 32 pages

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H��lder exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H��lder exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3., 31 pages

This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a space-time Gaussian field W assumed to be white noise in time and function-valued in space. According to the behavior of the spatial covariance W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any $\alpha, Comment: 31 pages; accepted for publication in Annals of Probability; Revised version of "Some scaling limits for a brownian polymer in a Gaussian medium "

In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an It\^{o}-type formula for the process X., Comment: 23 p

In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an It��-type formula for the process X., 23 p

In this paper we consider a n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H>1/3. After solving this equation in a rather elementary way, following the approach of Gubinelli, we show how to obtain an expansion for E[f(X\_t)] in terms of t, where X denotes the solution to the SDE and f:R^n->R is a regular function. With respect to the work by Baudoin and Coutin, where the same kind of problem is considered, we try an improvement in three different directions: we are able to take a drift into account in the equation, we parametrize our expansion with trees (which makes it easier to use), and we obtain a sharp control of the remainder., 46 pages

We compute the Onsager-Machlup functional for a stochastic evolution equation with additive noise, using L2-type techniques. The regularity that has to be imposed to the drift coefficient is a trace type condition on its derivative. The expression of the divergence part of the functional does not depend on the stochastic convolution related to our evolution system. Dans cet article, nous calculons la fonctionnelle d'Onsager-Machlup associée à une équation d'évolution stochastique avec bruit additif, en utilisant des techniques de type L2. Les hypothèses de régularité que l'on impose au coefficient de dérive sont des conditions de trace sur sa dérivée. Notons que l'expression de la partie divergence de la fonctionnelle ne dépend pas de la convolution stochastique associée au système d'évolution.

In this paper, we show that, in case of a perceptron model for which the number of outputs is a small proportion of the size of the system, the limiting behavior of the magnetization of a given spin, namely the random variable 〈 σ k 〉 , can be identified. In fact, we prove a L 2 convergence for a collection of those random variables.