Mostrando recursos 1 - 20 de 429

  1. Path large deviations for interacting diffusions with local mean-field interactions in random environment

    Müller, Patrick E.
    We consider a system of $N^{d}$ spins in random environment with a random local mean-field type interaction. Each spin has a fixed spatial position on the torus $\mathbb{T} ^{d}$, an attached random environment and a spin value in $\mathbb{R} $ that evolves according to a space and environment dependent Langevin dynamic. The interaction between two spins depends on the spin values, the spatial distance and the random environment of both spins. We prove the path large deviation principle from the hydrodynamic (or local mean-field McKean-Vlasov) limit and derive different expressions of the rate function for the empirical process and for...

  2. On the chemical distance in critical percolation

    Damron, Michael; Hanson, Jack; Sosoe, Philippe
    We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the...

  3. On the chemical distance in critical percolation

    Damron, Michael; Hanson, Jack; Sosoe, Philippe
    We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the...

  4. Limiting empirical distribution of zeros and critical points of random polynomials agree in general

    Reddy, Tulasi Ram
    In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ of complex numbers whose limiting empirical measures are the same. By choosing $\xi _n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi _1)\dots (z-\xi _n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function)...

  5. Limiting empirical distribution of zeros and critical points of random polynomials agree in general

    Reddy, Tulasi Ram
    In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ of complex numbers whose limiting empirical measures are the same. By choosing $\xi _n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi _1)\dots (z-\xi _n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function)...

  6. Limiting empirical distribution of zeros and critical points of random polynomials agree in general

    Reddy, Tulasi Ram
    In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ of complex numbers whose limiting empirical measures are the same. By choosing $\xi _n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi _1)\dots (z-\xi _n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function)...

  7. On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients

    Kumar, Chaman; Sabanis, Sotirios
    Motivated by the results of [21], we propose explicit Euler-type schemes for SDEs with random coefficients driven by Lévy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Lévy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L} ^2$-convergence which is consistent with the corresponding results available...

  8. On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients

    Kumar, Chaman; Sabanis, Sotirios
    Motivated by the results of [21], we propose explicit Euler-type schemes for SDEs with random coefficients driven by Lévy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Lévy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L} ^2$-convergence which is consistent with the corresponding results available...

  9. On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients

    Kumar, Chaman; Sabanis, Sotirios
    Motivated by the results of [21], we propose explicit Euler-type schemes for SDEs with random coefficients driven by Lévy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Lévy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L} ^2$-convergence which is consistent with the corresponding results available...

  10. On uniqueness and blowup properties for a class of second order SDEs

    Gomez, Alejandro; Lee, Jong Jun; Mueller, Carl; Neuman, Eyal; Salins, Michael
    As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha <1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha $ and $(x_0,y_0)\neq (0,0)$. We also show that blowup in finite time holds if $\alpha >1$ and $(x_0,y_0)\neq (0,0)$.

  11. On uniqueness and blowup properties for a class of second order SDEs

    Gomez, Alejandro; Lee, Jong Jun; Mueller, Carl; Neuman, Eyal; Salins, Michael
    As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha <1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha $ and $(x_0,y_0)\neq (0,0)$. We also show that blowup in finite time holds if $\alpha >1$ and $(x_0,y_0)\neq (0,0)$.

  12. On uniqueness and blowup properties for a class of second order SDEs

    Gomez, Alejandro; Lee, Jong Jun; Mueller, Carl; Neuman, Eyal; Salins, Michael
    As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha <1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha $ and $(x_0,y_0)\neq (0,0)$. We also show that blowup in finite time holds if $\alpha >1$ and $(x_0,y_0)\neq (0,0)$.

  13. Harnack inequalities for SDEs driven by time-changed fractional Brownian motions

    Deng, Chang-Song; Schilling, René L.
    We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in (0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition; otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

  14. Harnack inequalities for SDEs driven by time-changed fractional Brownian motions

    Deng, Chang-Song; Schilling, René L.
    We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in (0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition; otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

  15. Harnack inequalities for SDEs driven by time-changed fractional Brownian motions

    Deng, Chang-Song; Schilling, René L.
    We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in (0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition; otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

  16. Metastability in the reversible inclusion process

    Bianchi, Alessandra; Dommers, Sander; Giardinà, Cristian
    We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_{\star }\subseteq S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_{\star }$ is irreducible, then there exists a single time-scale for the condensate...

  17. Metastability in the reversible inclusion process

    Bianchi, Alessandra; Dommers, Sander; Giardinà, Cristian
    We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_{\star }\subseteq S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_{\star }$ is irreducible, then there exists a single time-scale for the condensate...

  18. Metastability in the reversible inclusion process

    Bianchi, Alessandra; Dommers, Sander; Giardinà, Cristian
    We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_{\star }\subseteq S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_{\star }$ is irreducible, then there exists a single time-scale for the condensate...

  19. Poisson statistics for 1d Schrödinger operators with random decaying potentials

    Kotani, Shinichi; Nakano, Fumihiko
    We consider the 1d Schrödinger operators with random decaying potentials in the sub-critical case where the spectrum is pure point. We show that the point process composed of the rescaled eigenvalues in the bulk, together with those zero points of the corresponding eigenfunctions, converges to a Poisson process.

  20. Poisson statistics for 1d Schrödinger operators with random decaying potentials

    Kotani, Shinichi; Nakano, Fumihiko
    We consider the 1d Schrödinger operators with random decaying potentials in the sub-critical case where the spectrum is pure point. We show that the point process composed of the rescaled eigenvalues in the bulk, together with those zero points of the corresponding eigenfunctions, converges to a Poisson process.

Aviso de cookies: Usamos cookies propias y de terceros para mejorar nuestros servicios, para análisis estadístico y para mostrarle publicidad. Si continua navegando consideramos que acepta su uso en los términos establecidos en la Política de cookies.