For Itô stochastic equations in $\mathbb{R}^{d}$ with drift in $L_{d}$ several results are discussed such as the existence of weak solutions, the existence of the corresponding Markov process, Aleksandrov type estimates of their Green's functions, which yield their summability to the power of $d/(d-1)$, the Fabes-Stroock type estimates which show that Green's functions are summable to a higher degree, the Fanghua Lin type estimates, which are one of the main tools in the $W^{2}_{p}$-theory of fully nonlinear elliptic equations, the fact that Green's functions are in the class $A_{\infty}$ of Muckenhoupt and a few other results.
The results established by Flandoli, Gubinelli and Priola ({\it Invent. Math.} {\bf 180} (2010) 1--53) for stochastic transport equation with bounded and Hölder continuous drift are generalized to bounded and Dini continuous drift. The uniqueness of $L^\infty$-solutions is established by the Itô--Tanaka trick partially solving the uniqueness problem, which is still open, for stochastic transport equation with only bounded measurable drift. Moreover the existence and uniqueness of stochastic diffeomorphisms flows for a stochastic differential equation with bounded and Dini continuous drift is o…
Existence and uniqueness results of fully coupled forward stochastic differential equations without drifts and backward stochastic differential equations in a degenerate case are obtained for an arbitrarily large time duration.
In this paper we prove well-posedness and stabibility of a class of stochastic delay differential equations with singular drift. Moreover, we show local well-posedness under localized assumptions.
Via a special transform and by using the techniques of the Malliavin calculus, we analyze the density of the solution to a stochastic differential equation with unbounded drift.
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 prove that there exists a unique strong solution to reflecting stochastic differential equations with merely measurable drift giving an affirmative answer to the longstanding problem. This is done through Zvonkin transformation and a careful analysis of the transformed reflecting stochastic differential equations on non-smooth time-dependent domains.
We prove via a direct fixpoint argument the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite $p$-variation with $p \in [1,2)$. An application to the Feynman-Kac representation of semilinear rough partial differential equations is given.
This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $ν$-almost everywhere stochastic invertible flow, where $ν$ is the Riemannian measure, which is quasi-invariant with respect to $ν$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.
We consider both the effect of particle inertia on stochastic Stokes' drift, and also a related process which could be considered as a crude model of stochastic Stokes' drift driven by an eddy diffusivity. In the latter, the stochastic forcing is a stable OU process rather than Brownian motion. We show that the eddy Stokes' drift velocity has a peak at a non-zero value of the correlation time-scale for particles that have the same (limiting) diffusivity. For both of the models considered, this study shows that not only can stochastic Stokes' drift be used to sort particles with different diffu…
In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.
We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}Δu+b(u)u+σ(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and $σ:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function with suitable property. We will construct nonnegative solutions to such SPDEs.
We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.
For Itô stochastic processes in $\mathbb{R}^{d}$ with drift in $L_{d}$ Aleksandrov's type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding elliptic and parabolic operators in $L_{p}$ and $L_{p+1}$, respectively, where $p\geq d$.
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν. The generalization which we deal with can be interpreted as allowing more general set functions ν, for example signed measures which are only σ-finite. However, we use a different approach to describe the singular drift. For the considere…
We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.
We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.
We consider a one-dimensional stochastic differential equations (SDE) with irregular coefficients. The purpose of this paper is to estimate the $L^p(Ω)$-difference of SDEs using the norm of the difference of coefficients, where the discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. As an application, we consider the stability problem.
We consider the stochastic Stokes drift of a flexible dumbbell. The dumbbell consists of two isotropic Brownian particles connected by a linear spring with zero natural length, and is advected by a sinusoidal wave. We find an asymptotic approximation for the Stokes drift in the limit of a weak wave, and find good agreement with the results of a Monte Carlo simulation. We show that it is possible to use this effect to sort particles by their flexibility even when all the particles have the same diffusivity.
We prove the solvability of Itô stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A.V. Skorokhod. Weak uniqueness of solutions is an open problem even if the diffusion is constant.
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 $β>1/2-α/2$.
We derive the strong consistency of the least squares estimator for the drift coefficient of a fractional stochastic differential system. The drift coeffcient is one-sided dissipative Lipschitz and the driving noise is additive and fractional with Hurst parameter $H \in (\frac{1}{4}, 1)$. We assume that continuous observation is possible. The main tools are ergodic theorem and Malliavin calculus. As a by-product, we derive a maximum inequality for Skorohod integrals, which plays an important role to obtain the strong consistency of the least squares estimator.
We consider a linear stochastic differential equation with stochastic drift and multiplicative noise. We study the problem of approximating its solution with the process that solves the equation where the possibly stochastic drift is replaced by a deterministic function. To do this, we use a combination of deterministic Pontryagin's maximum principle approach and direct methods of calculus of variations. We find necessary and sufficient conditions for a function $u \in L^1(0,T)$ to be a minimizer of a certain cost functional. To overcome the problem of the existence of such minimizer, we also …
We obtain $T_2(C)$ for stochastic differential equations with Dini continuous drift and $T_1(C)$ stochastic differential equations with singular coefficients.
We examine existence and uniqueness of strong solutions of multi-dimensional mean-field stochastic differential equations with irregular drift coefficients. Furthermore, we establish Malliavin differentiability of the solution and show regularity properties such as Sobolev differentiability in the initial data as well as Hölder continuity in time and the initial data. Using the Malliavin and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula for mean-field stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectatio…
We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the correspon…
In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of $0$. We show the existence, uniqueness and positivity of the solutions. The estimations of moments, including the negative power moments, are given. Based on these estimations, strong convergence of the positivity preserving drift-implicit Euler-type scheme is proved, and optimal convergence rate is obtained. By using Lamperti transformation, we show th…
We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get…
The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.
We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein-Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by expl…
Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.
We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.
We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=Δu(t,x)+u^{-α}(t,x)+g(u(t,x))\dot{W}(t,x) \] where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $0<α<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$. (ii) If $α>3$ then with probability one, $u(t,x)\ne0$ for all $(t,x)$.
In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points, and the Malliavin differentiability with respect to sample paths. We also establish Bismut-Elworthy-Li's formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier-Stokes equations given by Constantin-Iyer \cite{Co-Iy} to prove the local well-posedness of NSEs in $\m…
In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of…
In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift $b$ and the weak gradient of Sobolev diffusion $σ$ are supposed to satisfy $||b(x){1}_{|x|\le R}||_{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})$ and $||\nabla σ(x)1_{|x|\le R} ||_{p_1} \le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$ respectively. The main tools for these results are the decomposition of global two-point motions, Krylov's estimate, Kh…
We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along with additional restrictions. For example, consider the forcing $f(u) = u \log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.
In this paper, we prove weak uniqueness of hypoelliptic stochastic differential equation with H{ö}lder drift, with H{ö}lder exponent strictly greater than 1/3. We then extend to a weak framework the previous work [CdR12] where strong uniqueness was proved when the H{ö}lder exponent is strictly greater than 2/3. We also show that this result is sharp, by giving a counter example to weak uniqueness when the H{ö}lder exponent is just below 1/3. Our approach is based on martingale problem formulation of Stroock and Varadhan and is based on smoothing properties of the associated PDE.
We generalize the results of Ambrosio [Invent. Math. 158 (2004), 227--260] on the existence, uniqueness and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. Then we construct an explicit solution to the corresponding stochastic transport equation in terms of the stochastic flow. The approximate differentiability of the flow is also studied when the drift is a Sobolev vector field.
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.
This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the discrete or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stocha…
When expanding a filtration with a stochastic process it is easily possible for semimartingale no longer to remain semimartingales in the enlarged filtration. Y. Kchia and P. Protter indicated a way to avoid this pitfall in 2015, but they were unable to give the semimartingale decomposition in the enlarged filtration except for special cases. We provide a way to compute such a decomposition, and moreover we provide a sufficient condition for Itô processes to remain Itô processes in the enlarged filtration. This has significance in applications to Mathematical Finance.
In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.
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 …
In this paper, using Zvonkin type transform, the large deviation principle is proved for stochastic differential equations with Dini continuous drifts, where the existed methods for large deviation principle are unavailable. The method and result are new in related fields. Moreover, the result is also extended to a class of degenerate stochastic differential equations with Dini continuous drifts.
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…
In this paper we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler-Maruyama method. The existence of the random periodic solution is shown as the limits of the pull-back flows of the SDE and discretized SDE respectively. We establish a convergence rate of the strong error for the backward Euler-Maruyama method and obtain the weak convergence result for the approximation of the periodic measu…
We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators, where the nonlinear drift includes a mean field term of McKean type and the driving noise is Gaussian. Uniform convergence in L^2 sense is achieved by applying L^2-type estimates and the Gronwall Theorem. The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by stochastic nonlinear oscillators with diffe…
We consider the problem of constructing weak solutions to the Itô and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.
In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a H{ö}lder drift, for a H{ö}lder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin, Veretennikov, Krylov and R{ö}ckner from non-degenerate to degenerate cases. The existence of a threshold for the H{ö}lder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is b…
In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin's transformation \cite{zvonkin\_transformation\_1974} and so on the regularization properties of the associated PDE, which is stated on the space $[0,T]\times \R^d\times \mathcal{P}\_2(\R^d)$, where $T$ is a positive number, $d$ denotes the dimension equation and $\mathcal{P}\_2(\R^d)$ is the space of probability measures on $\R^d$ with finite second order moment. Especi…
