In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens. In particular, we deal with the one-dimensional stochastic heat equation in $[0,1]$ driven by the space-time white noise, and the stochastic heat and wave equations in $\mathbb{R}^d$ ($d\geq 1$ and $d\leq 3$, respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.
We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.
We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associate…
This appendix provides a short proof for sample path continuity of the Brownian motion induced by an arbitrary centered Gaussian measure on a separable Banach space, and also some perturbation results for the spectrum of compact self-adjoint operators on a Hilbert space.
In this paper, we generalize to Gaussian Volterra processes the existence and uniqueness of solutions for a class of non linear backward stochastic differential equations (BSDE) and we establish the relation between the non linear BSDE and the partial differential equation (PDE). A comparison theorem for the solution of the BSDE is proved and the continuity of its law is studied.
In this paper, we establish the Freidlin-Wentzell's large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The proof is based on the weak convergence approach.
In this paper, we establish a small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone.
We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon$ and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation.
We prove an existence and uniqueness result for the obstacle problem for quasilinear stochastic integral-partial differential equations. Our method is based on the probabilistic interpretation of the solution using backward doubly SDEs with jumps.
Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the partial stochastic dominance result that the cumulative distribution function of the maximum of a multivariate normal random vector, with positive intraclass correlation coefficient, intersects the cumulative distribution function of a standard normal random variable at most once. This result can be applied to the Bayesian design of a clinical trial in which se…
The main goal of this article is to study the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle (LDP) based on exponential equivalence arguments) for a class of stochastic partial differential equations (SPDEs) with fully monotone coefficients driven by multiplicative Gaussian noise. The small time LDP obtained in this paper is applicable for various quasi-linear and semilinear SPDEs such as porous medium equations, Cahn-Hilliard equation, 2D Navier-Stokes equations, convection-diffusion equation, 2D liquid crystal model, power law fluids, Ladyzhenskaya …
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of the solution in terms of Malliavin calculus. We prove that when the drift is bounded and measurable the solution is directional Malliavin differentiable.
We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.
In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.
We prove the quasi-invariance of gaussian measures (supported by functions of increasing Sobolev regularity) under the flow of one dimensional Hamiltonian PDE's such as the regularized long wave (BBM) equation.
We consider the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.
Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasiline…
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures…
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this goal by introducing a simultaneous renormalization on the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow. In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed.
We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We disc…
We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted G…
Obstruction to quasi-invariance of Gaussian measures under transport flows on Riemannian Manifolds
Université Paris-Saclay（巴黎萨克雷大学）；Laboratoire de Mathématiques et de Modélisation d’Évry (LaMME) Université d’Évry（埃夫里大学数学与建模实验室）；Institut universitaire de France (IUF)（法国高等研究院）
The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance $[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter $s$.
We prove quasi-invariance of Gaussian measures $μ_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov [15] and Gunaratnam-Oh-Tzvetkov-Weber [10] who proved this result this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted adapted to higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introdu…
In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $δ$, $m\sim δ^{-4} s\log(N/sδ)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $δ$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $δ$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.
We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $α> 1$, subject to a Gaussian random initial data of negative Sobolev regularity $σ<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(α) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $α< \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the ini…
This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^2 u}{\partial x^2}+u\dot{W}$ on $[0, \infty)\times {\bf R}^d $ with $d\geq 1$ has a unique random field solution, where $W(t, x)$ is a fractional Brownian sheet on $[0, \infty)\times {\bf R}^d$ and formally $\dot W =\frac{\partial^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise $W(t, x)$ is white in time, our condition is both necessary and sufficient when the initial dat…
In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.
