Semilinear Stochastic Partial Differential Equations Research Articles

On generators of transition semigroups associated to semilinear stochastic partial differential equations

Let X be a real separable Hilbert space. Let Q be a linear, bounded, positive and compact operator on X and let A:Dom(A)⊆X→X be a linear, self-adjoint operator generating a strongly continuous semigroup on X. Let F:X→X be a (smooth enough) function and let {W(t)}t≥0 be a X-valued cylindrical Wiener process. For any α≥0, we are interested in the mild solution X(t,x) of the semilinear stochastic partial differential equation{dX(t,x)=(AX(t,x)+F(X(t,x)))dt+QαdW(t),t>0;X(0,x)=x∈X, and in its associated transition semigroup(0.1)P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t≥0,x∈X; where Bb(X) denotes the space of the real-valued, bounded and Borel measurable functions on X. In this paper we study the behavior of the semigroup P(t) in the space L2(X,ν), where ν is the unique invariant probability measure of (0.1), when F is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincaré inequalities and we study the maximal Sobolev regularity for the stationary equationλu−N2u=f,λ>0,f∈L2(X,ν); where N2 is the infinitesimal generator of P(t) in L2(X,ν).

Strong solutions of semilinear SPDEs with unbounded diffusion

We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions. In the case of “critical unboundedness”, we show how to use the stochastic compactness method to obtain a martingale solution.

Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients

We provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations in the framework of the semigroup approach with locally monotone coefficients. An important component of the proof is an application of the dilation theorem of Nagy, which allows us to reduce the problem to infinite dimensional stochastic differential equations on a larger Hilbert space. Properties of the solutions like the Markov property are discussed as well.

On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

AbstractWe consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces

Let \(\mathcal {X}\) be a separable Hilbert space with norm \(\left \|{\cdot }\right \|\) and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on \(\mathcal {X}\), let \(F:\mathcal {X}\rightarrow \mathcal {X}\) be a (smooth enough) function and let W(t) be a \(\mathcal {X}\)-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator \(A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}\). We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation $$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$ and in its associated transition semigroup $$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$ where \(B_{b}(\mathcal {X})\) is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of \(\mathcal {X}\). More precisely there exists K := K(F, T) > 0 such that for every \(\varphi \in B_{b}(\mathcal {X})\), \(x\in \mathcal {X}\), t ∈ (0, T] and \(h\in Q^{\alpha }(\mathcal {X})\) it holds $$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$

Diffusivity Estimation for Activator\u2013Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton

A theory for diffusivity estimation for spatially extended activator–inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction–diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in Pasemann and Stannat (Electron J Stat 14(1):547–579, 2020), to the problem of joint estimation of diffusivity and parametrized reaction terms. Our theoretical findings are applied to the estimation of effective diffusivity of signaling components contributing to intracellular dynamics of the actin cytoskeleton in the model organism Dictyostelium discoideum.

Semilinear stochastic partial differential equations: Central limit theorem and moderate deviations

In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to stochastic partial differential equations of various types such as the stochastic Burgers equation and the reaction‐diffusion equations perturbed by space‐time white noise. The Garsia lemma is crucial to our results.

Peng's Maximum Principle for Stochastic Partial Differential Equations

We extend Peng's maximum principle for semilinear stochastic partial differential equations (SPDEs) in one space-dimension with non-convex control domains and control-dependent diffusion coefficients to the case of general cost functionals with Nemytskii-type coefficients. Our analysis is based on a new approach to the characterization of the second order adjoint state as the solution of a function-valued backward SPDE.

Strong Convergence of a Fully Discrete Finite Element Method for a Class of Semilinear Stochastic Partial Differential Equations with Multiplicative Noise

Strong Convergence of a Fully Discrete Finite Element Method for a Class of Semilinear Stochastic Partial Differential Equations with Multiplicative Noise

Freidlin--Wentzell Type Large Deviation Principle for Multiscale Locally Monotone SPDEs

This work is concerned with a Freidlin--Wentzell type large deviation principle for a family of multiscale quasilinear and semilinear stochastic partial differential equations. Employing the weak c...

On the equivalence of pathwise mild and weak solutions for quasilinear SPDEs

The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter H>frac {1}{2} and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order mathcal {O}(h^{2}+varDelta t) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

Central Limit Theorem and Moderate Deviations for a Class of Semilinear Stochastic Partial Differential Equations

In this paper we prove a central limit theorem and a moderate deviation principle for a class of semilinear stochastic partial differential equations, which contain the stochastic Burgers’ equation and the stochastic reaction-diffusion equation. The weak convergence method plays an important role.

Deterministic control of stochastic reaction-diffusion equations

<p style='text-indent:20px;'>We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.</p>

Backward Doubly SDEs with weak Monotonicity and General Growth Generators

We deal with backward doubly stochastic differential equations (BDSDEs) with a weak monotonicity and general growth generators and a square integrable terminal datum. We show the existence and uniqueness of solutions. As application, we establish the existenceand uniqueness of Sobolev solutions to some semilinear stochastic partial differential equations (SPDEs) with a general growth and a weak monotonicity generators. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.

A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equations

We discrete the ergodic semilinear stochastic partial differential equations in space dimension d≤3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is 1−ϵ in space and 12−ϵ in time for the space-time white noise case and 2−ϵ in space and 1−ϵ in time for the trace class noise case in space dimension d=1, with arbitrarily small ϵ>0. Numerical results are finally reported to confirm these theoretical findings.

Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space

In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: $$ du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum _{k = 1}^m g^k (u(t, x)) dw^k (t). $$ We prove the convergence of a Wong–Zakai type approximation scheme of the above equation in the space $$ C^{\theta } ([0, T], H^{\gamma }_p ({\mathbb {R}}^d)) $$ in probability, for some $$ \theta \in (0,1/2), \gamma \in (1, 2)$$ , and $$p > 2$$ . We also prove a Stroock–Varadhan’s type support theorem. To prove the results we combine V. Mackevicius’s ideas from his papers on Wong–Zakai theorem and the support theorem for diffusion processes with N. V. Krylov’s $$L_p$$ -theory of SPDEs.

Stochastic analysis with modelled distributions

Using a Besov topology on spaces of modelled distributions in the framework of Hairer’s regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type 2. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness results for mild solutions of semilinear stochastic partial differential equations in these spaces of modelled distributions and for distribution-valued SDEs. Furthermore, we provide a Fubini type theorem allowing to interchange the order of stochastic integration and reconstruction.

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.

Small time asymptotics for SPDEs with locally monotone coefficients

<p style='text-indent:20px;'>This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.