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Register a new userFractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere
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This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L{e}vy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Lo`{e}ve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.
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Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-valued cylindrical Wiener process. For $alphain [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2alpha-1}:Q^{1-2alpha}(mathcal{X})subseteqmathcal{X}rightarrowmathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation begin{gather} left{begin{array}{ll} dX(t,x)=big(AX(t,x)+F(X(t,x))big)dt+ Q^{alpha}dW(t), & t>0; X(0,x)=xin mathcal{X}, end{array} right. end{gather} and in its associated transition semigroup begin{align} P(t)varphi(x):=E[varphi(X(t,x))], qquad varphiin B_b(mathcal{X}), tgeq 0, xin mathcal{X}; end{align} where $B_b(mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(mathcal{X}, u)$, where $ u$ is the unique invariant probability measure of eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincare inequalities and we study the maximal Sobolev regularity for the stationary equation [lambda u-N_2 u=f,qquad lambda>0, fin L^2(mathcal{X}, u);] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(mathcal{X}, u)$.
We study the following equation begin{equation*} frac{partial u(t,,x)}{partial t}= Delta u(t,,x)+b(u(t,,x))+sigma dot{W}(t,,x),quad t>0, end{equation*} where $sigma$ is a positive constant and $dot{W}$ is a space-time white noise. The initial condition $u(0,x)=u_0(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if begin{equation*} int_{cdot}^inftyfrac{1}{b(s)},d s<infty, end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $mathbb{R}^d$.
One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.
Recently, a number of authors have investigated the conditions under which a stochastic perturbation acting on an infinite dimensional dynamical system, e.g. a partial differential equation, makes the system ergodic and mixing. In particular, one is interested in finding minimal and physically natural conditions on the nature of the stochastic perturbation. I shall review recent results on this question; in particular, I shall discuss the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity $ u$, and grows like $ u^{-3}$ when $ u$ goes to zero. This Markov process has a unique invariant measure and is exponentially mixing in time.
We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$mathrm{d} X= u(omega,t,X), mathrm{d} t + frac12 sigma(omega,t,X)sigma(omega,t,X),mathrm{d} t + sigma(omega,t,X) , mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $sigma$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(Omega; L^infty([0,T];dot{H}^1(mathbb{R})))$ for any finite $pge 1$, $mathbb{E}left|u(t)-u(0)right|_{dot{H}^1(mathbb{R})}^2 to 0$ as $tdownarrow 0$, and $u$ satisfies the one-sided gradient bound $partial_x u(omega,t,x) le K(omega, t)$, where the process $K(omega,t )>0$ exhibits an exponential moment bound of the form $mathbb{E} expBig(pint_t^T K(s),mathrm{d} sBig) lesssim {t^{-2p}}$ for small times $t$, for some $pge1$. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter--Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation (SPDE).
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