arXiv subjects
Publications and source records attributed to Antoine Hocquet.
We establish a simultaneous generalization of Itô's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering, - control and the conditional analysis of stochastic systems with common noise.
We show well-posedness for McKean--Vlasov equations with rough common noise and progressively measurable coefficients. Our results are valid under natural regularity assumptions on the coefficients, in agreement with the respective requirements of Ito and rough path theory. To achieve these goals, we work in the framework of rough stochastic differential equations recently developed by the authors of this article.
A summary of recent contributions in the field of rough partial differential equations is given. For that purpose we rely on the formalism of ``unbounded rough driver''. We present applications to concrete models including Landau-Lifshitz-Gilbert, Navier-Stokes and Euler equations.
We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form \( du_t -L_t(u_t)u_t dt = N_t(u_t)dt + F(u_t)\cdot d\mathbf X_t \), where \( \mathbf X\) is a \( γ\)-Hölder rough path for \( γ\in(1/3,1/2)\). We explore the mild formulation that combines functional analysis techniques and controlled rough paths theory which entail the local well-posedness of such equations. We apply our results to the stochastic Landau-Lifshitz-Gilbert and Shigesada-Kawasaki-Teramoto equation. In this framework we obtain a random dynamical system associated to the Landau-Lifshitz-Gilbert equation.
We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarily geometric) rough path of Hölder regularity $γ>1/3.$ Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path $\mathbf X$). As a technical tool, we introduce a version of the multiplicative sewing lemma, which allows to construct the so-called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.
Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called Itô-Lyons map in the energy spaces $L^\infty(0,T;H^k)\cap L^2(0,T;H^{k+1})$ for any $k\ge1$. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the second step where the continuity in the optimal norm is established through an application of the rough Gronwall lemma. Our approach is direct and does not rely on any transformation formula, which permits to treat multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type result, a large deviation principle for the solution and a support theorem.
We investigate the Cauchy problem for a quasilinear equation with transport rough input of the form $\mathrm{d} u-\partial_i(a^{ij}(u)\partial_j u)\mathrm{d} t =\mathrm{d} \mathbf{X}_t^i(x)\partial_i u_t,$ $u_0\in L^2$ on the torus $\mathbb T^d$, where $\mathbf{X}$ is two-step enhancement of a family of coefficients $(X^i_t(x))_{i=1,\dots d}$, akin to a geometric rough path with Hölder regularity $α>1/3.$ Using energy estimates, we provide sufficient conditions that guarantee existence in any dimension, and uniqueness in the case when $X$ is divergence-free. We then focus on the one-dimensional scenario, with slightly more regular coefficients. Improving the a priori estimates of the first results, we prove existence of a class of solutions whose spatial derivatives satisfy a Ladyzhenskaya-Prodi-Serrin type condition. Uniqueness is shown in the same class, by obtaining an $L^\infty(L^1)$ estimate on the difference of two solutions. The latter is obtained by establishing a link with a certain backward dual equation combined with a (rough) iteration lemma à la Moser.
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla + \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces $W^{k,p}.$ We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e.\ when $Y=0$), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form $u\mapsto F(u),$ where $F$ is $C^2$ and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for $F(u)=|u|^p$ with $p\geq 2,$ but also when $Y\neq 0$ and $p\geq 4.$ As an application of these results, we prove existence and uniqueness of a suitable class of $L^p$-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.
We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^α$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),α_t)dt+σ(t,X_t,\mathcal L(X_t),α_t)dW_t,$ $X_0\sim μ$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $α_t$ is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form $π(X_t)\leq0$. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control.
We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory.
We investigate existence and uniqueness for the stochastic liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in $L^p$-based spaces, for every $p>2.$ Thanks to a bootstrap principle together with a Gyöngy-Krylov-type compactness argument, this will ultimately lead us to prove the existence of a particular class of global solutions which are partially regular, strong in the probabilistic sense, and taking values in the "critical space" $L^2\times H^1.$
We give a new result on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau-Lifshitz-Gilbert model for thermal fluctuations in micromagnetics. We construct strong solutions that belong locally to the spaces $C([s,t);H^1)\cap L^2([s,t);H^2)$, $0\leq s<t\leq T$. It that sense, these maps are a counterpart of the so-called "Struwe solutions" of the deterministic model. We also give a natural criterion of uniqueness that extends A.\ Freire's Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.
We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow $u(t,\cdot ):\mathbb{D}^2\to\mathbb{S}^2$, from the two-dimensional unit disc onto the sphere. The diffusion term is assumed to have range one pointwisely in the tangent space $T_{u(t,x)}\mathbb{S}^2$, so that the noise preserves the 1-corotational symmetry of solutions. Under the assumption that its space-correlation is of trace class (in some appropriate hilbert space), we prove that the noise generates blow-up with positive probability. This scenario happens no matter how we choose the initial data, provided it fulfills the latter symmetry assumption.
We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.
We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence.