arXiv subjects
Publications and source records attributed to Michael Röckner.
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorov's backward equations in some second order Sobolev spaces, which is analytically interesting in itself.
In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the averaging principe, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations between the original system and its averaged equation. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Furthermore, rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergence are shown not to depend on the regularity of the coefficients in the equation for the fast variable, which coincides with the intuition, since in the limit systems the fast component has been totally averaged or homogenized out.
One studies here, via the La Salle invariance principle for nonlinear semigroups in Banach spaces, the properties of the $ω$-limit set $ω(u_0)$ corresponding to the orbit $γ(u_0)=\{u(t,u_0);\ t\ge0\}$, where $u=u(t,u_0)$ is the solution to the nonlinear Fokker-Planck equation $$\begin{array}{l} u_t-Δβ(u)+{\rm div}(Db(u)u)=0\ \mbox{ in }(0,\infty)\times\mathbb{R}^d,\\ u(0,x)=u_0(x),\ \ x\in\mathbb{R}^d,\quad u_0\in L^1(\mathbb{R}^d),\ d\ge3.\end{array}$$ Here, $β\in C^1(\mathbb{R})$ and $β'(r)>0$, $\forall r\ne0$. Moreover, $β$ is a sublinear function, possibly degenerate in the origin, $b\in C^1(\mathbb{R})$, $b$ bounded, $b\ge b_0\in(0,\infty),$ $D$ is bounded such that $D=-\nablaΦ$, where $Φ\in C(\mathbb{R}^d)$ is such that $Φ\ge1,$ $Φ(x)\to\infty$ as $|x|\to\infty$ and satisfies a condition of the form $ΔΦ-α|\nablaΦ|^2\le0$, a.e. on $\mathbb{R}^d$. The main conclusion is that the equation has an equilibrium state and the set $ω(u_0)$ is a non-empty, compact subset of $L^1(\mathbb{R}^d)$ while, for each $t\ge0$, the operator $u_0\to u(t,u_0)$ is an isometry on $ω(u_0)$. In the nondegenerate case $0<γ_0\leβ'\leγ_1$ studied in [Barbu, Röckner: arXiv:1808.10706], it follows that $\lim\limits_{t\to\infty}S(t)u_0=u_\infty$ in $L^1(\mathbb{R}^d)$, where $u_\infty$ is the unique bounded stationary solution to the equation.
One proves the uniqueness of distributional solutions to nonlinear Fokker--Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean--Vlasov stochastic differential equation (SDE).
An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of $\mathds{R}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures $Γ$ is equipped with the vague topology and the corresponding Borel $σ$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $Γ$, we prove the existence of a unique family $\{P_{t,μ}: t\geq 0, \ μ\in \mathcal{P}_{\rm exp}\}$ of probability measures on the space of cadlag paths with values in $Γ$ that solves a restricted initial-value martingale problem for the mentioned system. Thereby, a Markov process with cadlag paths is specified which describes the stochastic dynamics of this particle system.
In this paper we present a unified approach to establish gradient type formulas and Bismut type formulas for backward stochastic differential equations (BSDEs). This approach relies on a mix of derivative formulas with respect to the conditional probability of forward SDEs and the expression of the solution of BSDEs. Some concrete examples are given to illustrate the results. As applications, we provide representation formulas for the control solutions to McKean-Vlasov BSDEs and derive gradient estimates for related PDEs.
We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with non-smooth coefficients. Depending on the averaging regime and the homogenization regime, two strong convergences in the averaging principle of functional law of large numbers type are established. Then we consider the small fluctuations of the system around its average. Nine cases of functional central limit type theorems are obtained. In particular, even though the averaged equation for the original system is the same, the corresponding homogenization limit for the normal deviation can be quite different due to the difference in the interactions between the fast scales and the deviation scales. We provide quite intuitive explanations for each case. Furthermore, sharp rates both for the strong convergences and the functional central limit theorems are obtained, and these convergences are shown to rely only on the regularity of the coefficients of the system with respect to the slow variable, and do not depend on their regularity with respect to the fast variable, which coincide with the intuition since in the limit equations the fast component has been totally averaged or homogenized out.
In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted $L^p$-path spaces is proved. In particular, the classical Caputo derivative is included as a special case. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type \begin{equation*} \frac{d}{dt} (k * u)(t) + A(t, u(t)) = f(t), \quad 0<t<T, \end{equation*} with (in general nonlinear) operators $A(t,\cdot)$ satisfying general weak monotonicity conditions. Here $k$ is a non-increasing locally Lebesgue-integrable nonnegative function on $[0, \infty)$ with $\underset{s\rightarrow\infty}{\lim}k(s)=0$. Analogous results for the case, where $f$ is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators or the time-fractional (stochastic) $p$-Laplace equation are covered.
We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a large class of differential and pseudo-differential operators L, both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by L. We cover the cases of fractional Laplacians and Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and several types of local and nonlocal Ornstein-Uhlenbeck operators, as well as the Gross Laplacian and its negative powers, in infinite dimension.
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. The procedure is then applied to numerous examples, in particular, nonlinear PDEs that arise from control problems for infinite dimensional Ornstein-Uhlenbeck and Lévy processes.
We study the asymptotic behavior of stochastic hyperbolic parabolic equations with slow and fast time scales. Both the strong and weak convergence in the averaging principe are established, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation, which is a form of functional central limit theorem. We provide a unified proof for the above convergence by using the Poisson equation in Hilbert spaces. Moreover, sharp rates of convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable.
We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Hölder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable, and does not depend on their regularities with respect to the fast component.
By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(μ_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t μ(t)=L_{t,μ_t}^*μ_t,\ \ t\ge 0,$$ where $μ(t)$ is the image of $μ_t$ under the projection $\mathcal C\niξ\mapsto ξ(0)\in\mathbb R^d$, and $$L_{t,μ}(ξ):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,ξ,μ)\frac{\partial^2} {\partial_{ξ(0)_i} \partial_{ξ(0)_j}} +\sum_{i=1}^d b_i(t,ξ,μ)\frac{\partial}{\partial_{ξ(0)_i}},\ \ t\ge 0, ξ\in \mathcal C, μ\in \mathcal P^{\mathcal C}.$$ Under reasonable conditions on the coefficients $a_{ij}$ and $b_i$, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.
This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time $t$ and $ω$. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.
In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.
We consider a perturbation of a Hilbert space-valued Ornstein--Uhlenbeck process by a class of singular nonlinear non-autonomous maximal monotone time-dependent drifts. The only further assumption on the drift is that it is bounded on balls in the Hilbert space uniformly in time. First we introduce a new notion of generalized solutions for such equations which we call pseudo-weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation. Then we prove that their laws are absolutely continuous with respect to the law of the original Ornstein--Uhlenbeck process. In particular, we show that pseudo-weak solutions always have continuous sample paths. In addition, we obtain integrability estimates of the associated Girsanov densities. Some of our results concern non-random equations as well, while probabilistic results are new even in finite-dimensional autonomous settings.
We here consider optimal control problems governed by nonlinear stochastic equations on a Hilbert space H with nonconvex payoff, which is rewritten as a deterministic optimal control problem governed by a Kolmogorov equation in H. We prove the existence and first-order necessary condition of closed loop optimal controls for the above control problem. The strategy is based on solving a deterministic bilinear optimal control problem for the corresponding Kolmogorov equation on the space $L^2(H,ν)$, where $ν$ is the related infinitesimally invariant measure for the Kolmogorov operator.
Consider the following distribution dependent SDE: $$ {\mathrm d} X_t=σ_t(X_t,μ_{X_t}){\mathrm d} W_t+b_t(X_t,μ_{X_t}){\mathrm d} t, $$ where $μ_{X_t}$ stands for the distribution of $X_t$. In this paper for non-degenerate $σ$, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in $μ$ about $b$ and $σ$. In particular, we extend the results of Krylov-Röckner \cite{Kr-Ro} to the distribution dependent case.