Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs

Hölder estimates for solutions of parabolic SPDEs

This paper considers second-order stochastic partial differential equations with additive noise given in a bounded domain of $\mathbb R^n$. We suppose that the coefficients of the noise are $L^p$-functions with sufficiently large $p$. We prove that the solutions are Hölder-continuous functions almost surely (a.s.) and that the respective Hölder norms have finite momenta of any order.

Elementary pathwise methods for nonlinear parabolic and transport type SPDE with fractal noise

We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.

Maximum Principle for Quasi-linear Reflected Backward SPDEs

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally,…

On pointwise Malliavin differentiability of solutions to semilinear parabolic SPDEs

We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

On forward and backward SPDEs with non-local boundary conditions

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.

一类带跳的倒向随机偏微分方程及其应用

A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications

提出一类带跳的高阶向量倒向随机偏微分方程，在Lipschitz和线性增长条件下证明适应解的存在唯一性，并应用于金融领域。

Schauder-type estimates for higher-order parabolic SPDEs

In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial d…

一类带分数噪声的抛物型SPDE光滑稳定流形的存在性

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。

Existence and regularity of the density for the solution to semilinear dissipative parabolic SPDEs

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal mono…

Degenerate backward SPDEs in domains: non-local boundary conditions and applications to finance

Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a standard Cauchy condition at the terminal time, problems with special non-local boundary conditions are considered. These non-local conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results …

带光滑输运噪声的SPDEs的De Giorgi-Nash-Moser估计的随机流方法

A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

针对带输运噪声的抛物型SPDEs正则性理论中Hölder连续性这一开放问题，本工作在噪声系数空间足够正则的假设下，基于Kunita随机特征线方法结合新先验估计与适配公式，解决该问题并确立拟线性SPDEs全局正则解的存在性。

$L^{p}$-估计、线性和半线性反向SPDEs的局部适定性与可控性

$L^{p}$-estimates, local well-posedness and controllability for linear and semilinear backward SPDEs

本文研究了线性和半线性反向SPDEs的$L^p$估计、局部适定性与可控性，提出新的Itô公式并解决了半线性方程的局部存在性问题。

Uniqueness in Law for the Allen-Cahn SPDE via Change of Measure

We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure $L^2$ condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function $a(t,x,u)=Cu^γ$, $1/2\leγ\le1$ and $C\ne0$. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.

Convergence rates in expectation for a nonlinear backward parabolic equation with Gaussian white noise

The main purpose of this paper is to study the problem of determining initial condition of nonlinear parabolic equation from noisy observations of the final condition. We introduce a regularized method to establish an approximate solution. We prove an upper bound on the rate of convergence of the mean integrated squared error.

On divergence form SPDEs with growing coefficients in $W^{1}_{2}$ spaces without weights

We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the second power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variable.

An order approach to SPDEs with antimonotone terms

We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.

非局部拟线性奇异SPDEs

Non-local quasilinear singular SPDEs

本文研究二维环面上系数为非局部泛函的拟线性广义抛物型安德森模型，在扩散系数正性假设下，利用抛物控制微分框架建立了局部时间解理论。

Rough path stability of (semi-)linear SPDEs

We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its Lévy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.

Backward parabolic Ito equations and second fundamental inequality

Regularity of solutions is studied for backward stochastic parabolic Ito equations. An analog of the second energy inequality and the related existence theorem are obtained for domains with boundary.

Local $L_\infty$-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.

Singular perturbations and asymptotic expansions for SPDEs with an application to term structure models

We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.

On the Dirichlet Problem for Backward Parabolic Stochastic Partial Differential Equations in General Smooth Domains

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.

Duality and semi-group property for backward parabolic Ito equations

We study existence, uniqueness, semi-group property, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution.

Second order backward stochastic differential equations and fully non-linear parabolic PDEs

We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte Carlo methods for their numerical treatment.

On the $l_p$ stability estimates for stochastic and deterministic difference equations and their application to SPDEs and PDEs

In this paper we develop the $l_p$-theory of space-time stochastic difference equations which can be considered as a discrete counterpart of N.V. Krylov's $L_p$-theory of stochastic partial differential equations. We also prove a Calderon-Zygmund type estimate for deterministic parabolic finite difference schemes with variable coefficients under relaxed assumptions on the coefficients, the initial data and the forcing term.

On a Small Elliptic Perturbation of a Backward-Forward Parabolic Problem, with Applications to Stochastic Models

We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markov-modulated queues. We employ singular perturbation methods to study the problem for small values of the parameter.

SPDE limit of the global fluctuations in rank-based models

We consider systems of diffusion processes ("particles") interacting through their ranks (also referred to as "rank-based models" in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of la…

Well-posedness of Backward Stochastic Partial Differential Equations with Lyapunov Condition

In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

抛物型随机偏微分方程的瞬时处处爆破

Instantaneous everywhere-blowup of parabolic SPDEs

本文证明随机热方程在Osgood条件下解几乎必然瞬时处处爆破，方法结合微分不等式击中时间界与Malliavin微积分分析随机卷积空间增长。

Continuous Dependence for Backward Parabolic Operators with Log-Lipschitz Coefficients

We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.

Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $μ$, solvability of the associated Kolmogorov equation in $L^1(μ)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variati…

Forward-backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary…

Strong solutions of semilinear parabolic equations with measure data and generalized backward stochastic differential equations

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem be means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation…

On backward uniqueness for parabolic equations when Osgood continuity of the coefficients fails at one point

We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.

Strong Backward uniqueness for sublinear parabolic equations

In this paper, we establish strong backward uniqueness for solutions to sublinear parabolic equations of the type (1.1). The proof of our main result Theorem 1.1 is achieved by means of a new Carleman estimate and a Weiss type monotonicity that are tailored for such parabolic sublinear operators.

On the backward uniqueness property for a class of parabolic operators

We give sharp regularity conditions, ensuring the backward uniquess property to a class of parabolic operators.

Stochastic partial differential equations with singular terminal condition

In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +$\infty$ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.

Backward uniqueness for parabolic operators with non-Lipscitz coefficients

We investigate the relation between the backward uniqueness and the regularity of the coefficients for a parabolic operator. A necessary and sufficient condition for uniqueness is given in terms of the modulus of continuity of the coefficients.

Conditional stability for backward parabolic operators with Osgood continuous coefficients

We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgodd continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.

On Lipschitz solutions for some forward-backward parabolic equations. II: The case against Fourier

As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.

Backward uniqueness for general parabolic operators in the whole space

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. This result extends in some ways a classical result of Lions and Malgrange [12] and a recent result of the authors [10].

Global in Time Asymptotic Solutions to Kolmogorov--Feller-Type Parabolic Pseudodifferential Equations with a Small Parameter. Forward and Backward in Time Motion

The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics.

时间反向的半线性耦合抛物型系统的稳定性

Stability of backward-in-time semilinear coupled parabolic systems

本文研究有界区域中半线性耦合抛物型系统的反向问题，利用带权函数$e^{λt}$的修正Carleman估计方法，证明了线性与半线性强耦合系统的条件稳定性估计。

Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization

This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth, and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Si…

A Semi-Linear Backward Parabolic cauchy Problem with Unbounded Coefficients of Hamilton-Jacobi-Bellman Type and Applications to optimal control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the Value Function of the controlled equation and that the feedback law is verified.

Backward uniqueness for parabolic operators with non-Lipschitz coefficients

In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [Math. Scand. ${\bf 8}$ (1960), 277--286] and some more recent results of Del Santo and Prizzi [J. Math. Pures Appl. ${\bf 84}$ (2005), 471--491; Ann. Mat. Pura Appl., to appear].

Two-phase entropy solutions of forward-backward parabolic problems with unstable phase

This paper study the two--phase problem for the forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are both stable. Here we consider the situation in which the unstable phase is taken in account, obtaining not trivial solution of the problem. It is interesting to note that such solutions are given by solving generalized Abel's equations.

Short time existence for a general backward-forward parabolic system arising from Mean-Field Games

We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.

A new result on backward uniqueness for parabolic operators

Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.

Conditional stability up to the final time for backward-parabolic equations with Log-Lipschitz coefficients

We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.

Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase

This paper study the well--posedness of the entropy formulation given by Plotnikov in [{Differential Equations}, 30 (1994), pp. 614--622] for forward-backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in [C. Mascia, A. Terracina, and A. Tesei, {Arch.\ Ration.\ Mech.\ Anal.}, 194 (2009), pp. 887--925] that such formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold.

Remarks on backward uniqueness of parabolic equations and incompressible Navier-Stokes well-posedness

We explain why the theory of Escauriaza, Seregin, and Sverak (Russian Math. Surveys, 2003) on potential finite time singularity in Navier-Stokes solutions must be largely misapprehended. It is found that the proofs of the backward uniqueness theorem for parabolic equations contain technical errors. The stated validity of a theorem for vorticity is established on ill-informed analyses as the solenoidal constraint is not taken into account. There are many cases where parabolic scalings are erroneously applied. We briefly discuss a number of related issues.