Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is \F_0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in C^\l([0,T];\D((-A)^\theta) provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy \l+\theta<\frac12. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.Comment: Accepted for publication in Journal of Functional Analysi

Semilineaarisen lämpöyhtälön ratkaisun olemassaolo ja yksikäsitteisyys

This thesis is about the existence and uniqueness of a solution for the semilinear heat equation of polynomial type. The extensive study of properties for these equations started off in the 1960s, when Hiroshi Fujita published his results that the existence and uniqueness of solutions depends critically on the exponent of the nonlinear term. In this thesis we expose some of the basic methods used in the theory of linear, constant coefficient partial differential equations. These considerations lay out the groundwork for the main result of the thesis, which is the existence and uniqueness of a solution to the generalized heat equation. In Chapter 2 we expose the basics of functional analysis. We start off by defining Banach spaces and provide some examples of them. Then, we state the very useful Banach fixed point theorem, which guarantees the existence and uniqueness of a solution to certain types of integral equations. Next, we consider linear maps between normed spaces, with a focus on linear isomorphisms, which are linear maps preserving completeness. The isomorphisms prove to be very useful, when we consider weighted spaces. This is because for certain types of weights, we can identify the multiplication by weight with a linear isomorphism. In Chapter 3 we introduce the Fourier transform, which is a highly useful tool for studying linear partial differential equations. We go through its basic mapping properties, such as, interaction with derivatives and convolution. Then, we consider useful spaces in Fourier analysis. Chapter 4 is on the regular, inhomogeneous heat equation. A common method for deriving the solution to heat equation is formally applying the Fourier transform to it. This way we obtain a first order, linear ordinary differential equation, for which there is a known solution. The derived solution will serve as a motivator for how to approach the semilinear case. Also, in the end we will solve explicitly a slight generalization of the heat equation. In Chapter 5 we prove the main result of this thesis: existence and uniqueness of a generalized solution for the semilinear heat equation. The methods we use in the proof are quite elementary in the sense that we do not need heavy mathematical machinery. We reformulate the generalized semilinear heat equation using an operator and show that it satisfies the conditions of the Banach fixed point theorem in a small, closed ball of a suitable Banach space. We also include an appendix, in which we discuss differentiability properties of the generalized solution. It is possible to apply methods used in the proof of the generalized case to prove continuous differentiability. We provide some ideas on how one should approach the time differentiability of the solution by estimating the difference quotient of the integral operator

Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the study of strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution.Comment: arXiv admin note: text overlap with arXiv:1401.503

R-boundedness Approach to linear third differential equations in a UMD Space

The aim of this work is to study the existence of a periodic solutions of third order differential equations $z'''(t) = Az(t) + f(t)$ with the periodic condition $x(0) = x(2\pi), x'(0) = x'(2\pi)$ and $x''(0) = x''(2\pi)$. Our approach is based on the R-boundedness and $L^{p}$-multiplier of linear operators

Periodic solutions of integro-differential equations in Banach space having Fourier type

The aim of this work is to study the existence of a periodic solutions of integro-differential equations d dt [x(t)-- L(x t)] = A[x(t)-- L(x t)]+ G(x t)+ t --$\infty$ a(t-- s)x(s)ds+ f (t), (0 $\le$ t $\le$ 2$\pi$) with the periodic condition x(0) = x(2$\pi$), where a $\in$ L 1 (R +). Our approach is based on the M-boundedness of linear operators, Fourier type, B s p,q-multipliers and Besov spaces.Comment: arXiv admin note: text overlap with arXiv:1707.0787

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic

Periodic Motions in Banach Space and Applications to Functional-Differential Equations

In establishing the existence of periodic solutions for nonautonomous differential equations of the form x = g(x, t), where g is periodic in t of period for fixed x, it is often convenient to consider the translation operator T(x(t)) = x(t + ). If corresponding to each initial vector chosen in an appropriate region there corresponds a unique solution of our equation, then periodicity may be established by proving the existence of a fixed point under T. This same technique is also useful for more general functional equations and can be extended in a number of interesting ways. In this paper we shall consider a variable type of translation operator which is useful in investigating periodicity for autonomous differential and functional equations where the period involved is less obvious