A Stochastic Approach to Evolution Equations on Nilpotent Quantum Groups
نویسندگان
چکیده
Evolution equations like the heat or diffusion equation or the Schrödinger equation can be associated with stochastic processes. We introduce a notion of nilpotency for quantum groups / braided groups that extends that for (simply-connected) Lie groups, and study the relations between evolution equations and (quantum) stochastic processes on these structures. We show on some examples how polynomial solutions can be calculated in terms of Appell systems. We use a generalisation of Weyl correspondence and Wigner distributions to give an interpretation of the processes on the classical (‘unquantized’) group or space.
منابع مشابه
Continuous dependence on coefficients for stochastic evolution equations with multiplicative Levy Noise and monotone nonlinearity
Semilinear stochastic evolution equations with multiplicative L'evy noise are considered. The drift term is assumed to be monotone nonlinear and with linear growth. Unlike other similar works, we do not impose coercivity conditions on coefficients. We establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. As corollaries of ...
متن کاملStochastic evolution equations with multiplicative Poisson noise and monotone nonlinearity
Semilinear stochastic evolution equations with multiplicative Poisson noise and monotone nonlinear drift in Hilbert spaces are considered. The coefficients are assumed to have linear growth. We do not impose coercivity conditions on coefficients. A novel method of proof for establishing existence and uniqueness of the mild solution is proposed. Examples on stochastic partial differentia...
متن کاملOn time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays
In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory
متن کاملQuantum Trajectories in Random Environment: the Statistical Model for a Heat Bath
In this article, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the ...
متن کاملStochastic differential inclusions of semimonotone type in Hilbert spaces
In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...
متن کامل