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  • 1.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Quasi-invariance under flows generated by non-linear PDEs2023In: Analysis and Applications, ISSN 0219-5305, E-ISSN 1793-6861 Article in journal (Refereed)
    Abstract [en]

    The paper is concerned with the change of probability measures μμ along non-random probability measure-valued trajectories νtνt, t∈[−1,1]t∈[−1,1]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map ν≡ν0↦νtν≡ν0↦νt does not exit the state space, for t∈[−1,0]t∈[−1,0] or for t∈[0,1]t∈[0,1], the Radon–Nikodym derivative dμ∘νt/dμdμ∘νt/dμ is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula.

  • 2.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Knudsen type group for time in R and related Boltzmann type equations2021In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 25, no 3, article id 2150072Article in journal (Refereed)
    Abstract [en]

    We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time t R. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time t ≥ 0 or for time t [τ0,∞) for some τ0 < 0 which is independent of the initial value at time 0. Depending on the collision kernel, τ0 can be arbitrarily small.

  • 3.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Boundedness of the Stationary Solution to the Boltzmann Equation with Spatial Smearing, Diffusive Boundary Conditions, and Lions’ Collision Kernel2018In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 50, no 6, p. 5761-5782Article in journal (Refereed)
    Abstract [en]

    We investigate the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions’ collision kernel. Both the physical as well as the velocity space, are assumed to be bounded. Existence and uniqueness of a stationary solution, which is a probability density, has been demonstrated in [S. Caprino, M. Pulvirenti, and W. Wagner, SIAM J. Math. Anal., 29 (1998), pp. 913–934] under a certain smallness assumption on the collision term. We prove that whenever there is a stationary solution then it is a.e. positively bounded from below and above.

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  • 4.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus2017In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 249, no 1185, p. 1-135Article in journal (Refereed)
    Abstract [en]

    The text is concerned with a class of two-sided stochastic processes of the form . Here is a two-sided Brownian motion with random initial data at time zero and is a function of . Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when is a jump process. Absolute continuity of under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, , and on with we verifya.e. where the product is taken over all coordinates. Here is the divergence of with respect to the initial position. Crucial for this is the temporal homogeneity of in the sense that , , where is the trajectory taking the constant value .By means of such a density, partial integration relative to a generator type operator of the process is established. Relative compactness of sequences of such processes is established.

  • 5.
    Karlsson, John
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula2016In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, no 17-18, p. 2192-2222Article in journal (Refereed)
    Abstract [en]

    The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron- Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

  • 6.
    Karlsson, John
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    A class of infinite dimensional stochastic processes with unbounded diffusion2015In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, no 3, p. 424-457Article in journal (Refereed)
    Abstract [en]

    The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron-Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

  • 7.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Mosco Type Convergence of Bilinear Forms and Weak Convergence of n-Particle Systems2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 2, p. 241-267Article in journal (Refereed)
    Abstract [en]

    It is well known that Mosco (type) convergence is a tool in order to verify weak convergence of finite dimensional distributions of sequences of stochastic processes. In the present paper we are concerned with the concept of Mosco type convergence for non-symmetric stochastic processes and, in particular, n-particle systems in order to establish relative compactness.

  • 8.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    ABSOLUTE CONTINUITY UNDER TIME SHIFT FOR ORNSTEIN-UHLENBECK TYPE PROCESSES WITH DELAY OR ANTICIPATION2014In: Communications on Stochastic Analysis, ISSN 0973-9599, Vol. 8, no 4, p. 439-448Article in journal (Refereed)
    Abstract [en]

    The paper is concerned with one-dimensional two-sided Ornstein-Uhlenbeck type processes with delay or anticipation. We prove existence and uniqueness requiring almost sure boundedness on the left half-axis in case of delay and almost sure boundedness on the right half-axis in case of anticipation. For those stochastic processes (X, Pμ) we calculate the Radon-Nikodym density under time shift of trajectories, Pμ(dX·−t)/Pμ(dX), t 2 R.

  • 9.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Weak Convergence of n-Particle Systems Using Bilinear Forms2013In: Milan Journal of Mathematics, ISSN 1424-9286, E-ISSN 1424-9294, Vol. 81, no 1, p. 37-77Article in journal (Refereed)
    Abstract [en]

    The paper is concerned with the weak convergence of n-particle processes to deterministic stationary paths as n -andgt; infinity. A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the n-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary n-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion. less thanbrgreater than less thanbrgreater thanThe present paper is in close relation to the paper [9]. Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system.

  • 10.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    A Stationary Fleming-Viot type Brownian particle system2009In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 263, no 3, p. 541-581Article in journal (Refereed)
    Abstract [en]

    We consider a system {X(1),...,X(N)} of N particles in a bounded d-dimensional domain D. During periods in which none of the particles X(1),...,X(N) hit the boundary. partial derivative D, the system behaves like N independent d-dimensional Brownian motions. When one of the particles hits the boundary partial derivative D, then it instantaneously jumps to the site of one of the remaining N - 1 particles with probability (N - 1)(-1). For the system {X(1),..., X(N)}, the existence of an invariant measure w has been demonstrated in Burdzy et al. [Comm Math Phys 214(3): 679-703, 2000]. We provide a structural formula for this invariant measure w in terms of the invariant measure m of the Markov chain xi which returns the sites the process X := (X(1),...,X(N)) jumps to after hitting the boundary partial derivative D(N). In addition, we characterize the asymptotic behavior of the invariant measure m of xi when N -> infinity. Using the methods of the paper, we provide a rigorous proof of the fact that the stationary empirical measure processes 1/N Sigma(N)(i=1) (delta)X(i) converge weakly as N -> infinity to a deterministic constant motion. This motion is concentrated on the probability measure whose density with respect to the Lebesgue measure is the first eigenfunction of the Dirichlet Laplacian on D. This result can be regarded as a complement to a previous one in Grigorescu and Kang [Stoch Process Appl 110(1): 111 - 143, 2004].

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