We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^β_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $β-1/q\ge-1$, $β>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=δ_0$ which corresponds to the skewed stochastic heat equation. For $β-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê~(2020).
In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributionsof a continuous function.
This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution.
In this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.
In this article, we derive the explicit transition density functions of skew Brownian motion (SBM in abbreviation) with two-valued drift for all $t>0$. As an important step of this result, it is also shown in this paper that SBM with two-valued drift is a strong Markov process by finding its symmetrizing measure and canonical scale function, from which one can tell what values of the drift make such a process transient or recurrent.
We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $α$-stable Lévy processes, $α\in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space $\mathcal{C}^β$ for $β>1/2-α/2$.
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{ö}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes th…
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
Motivated by recent developments on random polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. It was shown by Harrison and Williams (1987) that RBM in a polyhe- dral domain has an invariant measure in product form if a certain skew-symmetry condition holds. We show that (modulo technical assumptions) the generalised RBM has an invariant measure in product form if (…
We study the weak approximation error of a skew diffusion with bounded measurable drift and Hölder diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first establish two sided Gaussian bounds for the density of this approximation scheme. Then, a bound for the difference between the densities of the skew diffusion and its Euler approximation is obtained. Notably, the weak approximation error is shown to be of order $h^{η/2}$, where $h$ is the time step of the scheme, $η$ being the Hölder exponent of the diffusion…
We consider a stochastic heat equation driven by a space-time white noise and with a singular drift, where a local-time in space appears. The process we study has an explicit invariant measure of Gibbs type, with a non-convex potential. We obtain existence of a Markov solution, which is associated with an explicit Dirichlet form. Moreover we study approximations of the stationary solution by means of a regularization of the singular drift or by a finite-dimensional projection.
The law of a positive infinitely divisible process with no drift is characterized by its Lévy measure on the paths space. Based on recent results of the two authors, it is shown that even for simple examples of such processes, the knowledge of their Lévy measures allows to obtain remarkable distributional identities.
In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a rate of convergence in a suitable $L^1$-norm and we implement the scheme numerically. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift lives in a space of distributions. As a byproduct we also obtain an estimate of the convergence rate for a numerical scheme applied to SDEs with drift…
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of p…
We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions. Due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, that is, pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. The paper contains qualitative and…
We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of [Gubinelli, Perkowski 2016], but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of [Gubinelli, Perkowski 2016] to a non-stationary regime.
The Bessel process in low dimension (0 $\le$ $δ$ $\le$ 1) is not an It{ô} process and it is a semimartingale only in the cases $δ$ = 1 and $δ$ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift.
We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.
In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a "local time variational calculus" argument.
We study the distribution of the exponential functional $I(ξ,η)=\int_0^{\infty} \exp(ξ_{t-}) \d η_t$, where $ξ$ and $η$ are independent Lévy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $η$ is a Brownian motion with drift we show that this integral equation leads to an important functional e…
Perturbing Walsh's spider process by infinitesimal drifts
Lublin University of Technology（卢布林理工大学）；University of Geneva, Section de mathématiques（日内瓦大学数学系）；Institute of Mathematics of the Ukrainian National Academy of Sciences（乌克兰国家科学院数学研究所）
Multidimensional Stable driven McKean-Vlasov SDEs with distributional interaction kernel -- a regularization by noise perspective
University of Nantes（南特大学）；HSE University（高等经济大学）；Université d’Evry Val d’Essonne（埃夫里-瓦勒德埃松大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS (UMR 8071)（法国国家科学研究中心）
Based on an integration by parts formula for closed and convex subsets $Γ$ of a separable real Hilbert space $H$ with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein-Uhlenbeck process (perturbed by a bounded drift $B$) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary $\partial Γ$ is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vecto…
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.
We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem's data.…
In this note we propose an exact simulation algorithm for the solution of dX_t=dW_t+b(X_t)dt (1) where b is a smooth real function except at point 0 where b(0+)\neq b(0-). The main idea is to sample an exact skeleton of X using an algorithm deduced from the convergence of the solutions of the skew perturbed equation dX^β_t=dW_t+b(X^β_t)dt + βdL^0_t {X^β} (2) towards X solution of (1) as βtends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in \cite{etoremartinez1} for the solutions of (2) towards a limit algorith…
