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Jan van Neerven

Publications and source records attributed to Jan van Neerven.

At least 55 records · Page 3Linked to original sources
arXiv2012

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are Hölder continuous both and space.

math.PR↗
arXiv2012

Equivalence of laws and null controllability for SPDEs driven by a fractional Brownian motion

We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control problem.

math.PR↗
arXiv2012

Maximal $L^p$-regularity for stochastic evolution equations

We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12π$ on a space $L^q(\mathcal{O},μ)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space $\XAp $ we prove existence of unique strong solution with trajectories in \[L^p(0,T;\Dom(A))\cap C([0,T];\XAp),\] provided the non-linearities $F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},μ)$ and $B:[0,T]\times \Dom(A) \to \g(H,\Dom(A^{\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain $\OO\subseteq \R^d$ with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\OO))^d$ is shown.

math.PR↗
arXiv2011

A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation that ψ(x) = || x ||^p is Fréchet differentiable and its derivative satisfies the Lipschitz estimate || ψ'(x) - ψ'(y)|| \leq C(|| x || + || y ||)^{p-2} || x-y ||; the extension to 0<p<2 proceeds via Lenglart's inequality.

math.PR↗
arXiv2011

Uniformly gamma-radonifying families of operators and and the stochastic Weiss conjecture

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem $dU(t) = AU(t) dt + B dW(t); U(0) = 0$ Here, $A$ is the generator of a strongly continuous semigroup of operators on a Banach space $E$, $B$ is a bounded linear operator from a separable Hilbert space $H$ into $E$, and $W$ is an $H$-cylindrical Brownian motion. When $A$ and $B$ are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family $ \{\sqrtλ R(λ, A)B : λ> 0\} $ is uniformly gamma-radonifying. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

math.FA↗
arXiv2011

Approximating the coefficients in semilinear stochastic partial differential equations

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt + G(t,X(t))dW_H(t), X(0)=X_0, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^{tA}X_0 in the norms L^p(Ω;C^λ([0,T];E)) assuming that the approximating operators A_n are uniformly sectorial and converge to A in the strong resolvent sense, and that the approximating nonlinearities F_n and G_n are uniformly Lipschitz continuous in suitable norms and converge to F and G pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite-dimensional multiplicative noise.

math.PR↗
arXiv2011

Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators

Let (P(t)) be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the generator of a C_0-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and (W_H(t)) is an H-cylindrical Brownian motion. Assuming that (S(t)) restricts to a C_0-semigroup on H, we obtain L^p-bounds for the gradient D_H P(t). We show that if (P(t)) is analytic, then the invariance assumption is fulfilled. As an application we determine the L^p-domain of the generator of (P(t)) explicitly in the case where (S(t)) restricts to a C_0-semigroup on H which is similar to an analytic contraction semigroup. The results are applied to the 1D stochastic heat equation driven by additive space-time white noise.

math.FA↗
arXiv2010

Non-tangential maximal functions and conical square functions with respect to the Gaussian measure

We study, in $L^{1}(\R^n;γ)$ with respect to the gaussian measure, non-tangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in $L^1$-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

math.FA↗
arXiv2010

Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems

We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt + dW(t), where A is the generator of an analytic C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian motion with values in a fractional domain space E_\b associated with A. We prove that if \a,\b,\g,þ\ge 0 are such that \g + þ< 1 and max[0,(\a-\b+þ)] + \g < 1/2, then the approximate solutions U_n (where n is the number of time steps) converge to the solution U in the Holder space C^\g([0,T];E_\a), both in L^p-means and almost surely, with rate 1/n^þ.

math.NA↗
arXiv2009

Spaces of operator-valued functions measurable with respect to the strong operator topology

Let $X$ and $Y$ be Banach spaces and $(Ω,Σ,μ)$ a finite measure space. In this note we introduce the space $L^p[μ;L(X,Y)]$ consisting of all (equivalence classes of) functions $Φ:Ω\mapsto L(X,Y)$ such that $ω\mapsto Φ(ω)x$ is strongly $μ$-measurable for all $x\in X$ and $ω\mapsto Φ(ω)f(ω)$ belongs to $L^1(μ;Y)$ for all $f\in L^{p'}(μ;X)$, $1/p+1/p'=1$. We show that functions in $L^p[μ;Ł(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(μ;X)$ into $L^q(μ;Y)$, $1\le q< p<\infty$.

math.FA↗
arXiv2008

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1<p<\infty. Our main result states that the following assertions are equivalent: (1) dom((sqrt(D_V* B D_V)) = dom(D_V) and Meyer's inequalities hold for D_V* B D_V; (2) D_V D_V* B admits a bounded H-infinity calculus on the closure of the range of D_V; (3) dom(sqrt(V*BV)) = dom(V) and Meyer's inequalities hold for V*BV; (4) VV*B admits a bounded H-infinity calculus on the closure of the range of V. Moreover, if these conditions are satisfied, then dom(L) = dom(D_V^2) \cap dom(D_A). The equivalence of (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities (which correspond to the case G=H, V=I, B=I). A one-sided version of the main result, giving L^p-boundedness of the associated Riesz transforms in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C_0-contraction semigroup on a Hilbert space H and let -L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A.

math.FA↗
arXiv2008

A Clark-Ocone formula in UMD Banach spaces

Let H be a separable real Hilbert space and let F = (F_t)_{t\in [0,T]} be the augmented filtration generated by an H-cylindrical Brownian motion W_H on [0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f\in D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued processes.

math.PR↗
arXiv2007

Compactness in vector-valued Banach function spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\le p<\infty$, and extend the result to vector-valued Banach function spaces $E_X$, where $E$ is a Banach function space with order continuous norm.

math.FA↗
arXiv2007

Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\R^n;X)) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when (X=\C), our approach gives refined (p)-dependent versions of known results.

math.FA↗