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MENGÜTÜRK, Murat Cahit

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Murat Cahit

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MENGÜTÜRK
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Now showing 1 - 5 of 5
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    Stochastic sequential reduction of commutative Hamiltonians
    (AIP Publishing, 2020-10-01) Mengütürk, L. A.; Mengütürk, Murat Cahit; Business Administration; MENGÜTÜRK, Murat Cahit
    This paper introduces a class of stochastic processes constructed by conditioning cadlag processes to take a predetermined set of marginal laws at fixed points in time. We collectively refer to the elements of this class of processes as random n-bridges (RnBs), where n refers to the cardinality of the set of conditioning laws, which can be chosen arbitrarily. We prove that if the underlying cadlag process is Markov, then its RnB is Markov for n = 1 and has a particular Markov-like property for n > 1. As a canonical subclass, we construct Brownian RnBs (BRnBs) and provide their anticipative representation as well as their non-anticipative semimartingale representation. This motivates the second part of this paper, wherein we apply BRnBs to describe an energy-based state reduction in a composite quantum system involving a set of commuting observables measured successively at different times. The highly tractable so-called energy-based quantum state reduction models are significant in their role as the most important alternative to the popular CLS model. We ask the random variables composing the anticipative form of a BRnB to take the eigenvalues of a set of compatible Hamiltonians and provide a new stochastic Schrodinger evolution on a complex Hilbert space that models sequential reduction dynamics of the eigenstates through a single wave function. By doing so, we also extend the Brody-Hughston finite-time collapse model to a many-body setup with sequential measurements.
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    Captive diffusions and their applications to order-preserving dynamics
    (Royal Society Publishing, 2020-09-30) Mengütürk, L. A.; Mengütürk, Murat Cahit; Business Administration; MENGÜTÜRK, Murat Cahit
    We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of cadlag bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries-possibly in a hybrid manner over non-overlapping time periods-and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.
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    ArticlePublicationOpen Access
    On a family of coupled diffusions that can never change their initial order
    (IOP Publishing, 2022-11-18) Mengütürk, L. A.; Mengütürk, Murat Cahit; Business Administration; MENGÜTÜRK, Murat Cahit
    We introduce a real-valued family of interacting diffusions where their paths can meet but cannot cross each other in a way that would alter their initial order. Any given interacting pair is a solution to coupled stochastic differential equations with time-dependent coefficients satisfying certain regularity conditions with respect to each other. These coefficients explicitly determine whether these processes bounce away from each other or stick to one another if/when their paths collide. When all interacting diffusions in the system follow a martingale behaviour, and if all these paths ultimately come into collision, we show that the system reaches a random steady-state with zero fluctuation thereafter. We prove that in a special case when certain paths abide to a deterministic trend, the system reduces down to the topology of captive diffusions. We also show that square-root diffusions form a subclass of the proposed family of processes. Applications include order-driven interacting particle systems in physics, adhesive microbial dynamics in biology and risk-bounded quadratic optimization solutions in control theory.
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    ArticlePublicationOpen Access
    Piecewise-tunneled captive processes and corridored random particle systems
    (Springer, 2023-01) Mengütürk, L. A.; Mengütürk, Murat Cahit; Business Administration; MENGÜTÜRK, Murat Cahit
    We introduce a family of processes that generalises captive diffusions, whereby the stochastic evolution that remains within a pair of time-dependent boundaries can further be piecewise-tunneled internally. The tunneling effect on the dynamics can be random such that the process has non-zero probability to find itself within any possible tunnel at any given time. We study some properties of these processes and apply them in modelling corridored random particles that can be observed in fluid dynamics and channeled systems. We construct and simulate mean-reverting piecewise-tunneled captive models for demonstration. We also propose a doubly-stochastic system in which the tunnels themselves are generated randomly by another stochastic process that jumps at random times.
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    ArticlePublicationOpen Access
    Captive jump processes for bounded random systems with discontinuous dynamics
    (Elsevier, 2024-01) Macrina, A.; Mengütürk, L. A.; Mengütürk, Murat Cahit; Business Administration; MENGÜTÜRK, Murat Cahit
    Stochastic captive jump processes are explicitly constructed in continuous time, whose non-linear dynamics are strictly confined by bounded domains that can be time-dependent. By introducing non-anticipative path-dependency, the framework offers the possibility of generating multiple inner tunnels within a master domain, such that a captive jump process is allowed to proceed either within a single inner tunnel or jump in between tunnels without ever penetrating the outermost shell. If a captive jump process is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Under certain conditions, it can be shown that captive jump processes are invariant under monotonic transformations, enabling one to construct and study systems of increasing complexity using simpler building blocks. Amongst many applications, captive jump processes may be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, quantum tunnelling where stochastic wave-functions can “penetrate” inner boundaries (i.e., walls) of potential energy, non-linear dynamical systems involving multiple attractors, and sticky concentration behaviour of pathogens in epidemics. We provide concrete, worked-out examples, and numerical simulations for the dynamics of captive jump processes within different geometries as demonstrations.