In this article we prove a weak invariance principle for a strictly stationary φ-mixing sequence ... more In this article we prove a weak invariance principle for a strictly stationary φ-mixing sequence {X j } j≥1 , whose truncated variance function L(x) := EX 2 1 1 {|X1|≤x} is slowly varying at ∞ and mixing coefficients satisfy the logarithmic growth condition: n≥1 φ 1/2 (2 n ) < ∞. This will be done under the condition that lim n Var( n j=1X j )/[ n j=1 Var(X j )] = β 2 exists in (0, ∞), whereX j = X j I {|Xj |≤ηj } and η 2 n ∼ nL(η n ).
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d−2, which coincides with the condition obtained in , when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
In this paper we generalize Yu's strong invariance principle for associated sequences to the mult... more In this paper we generalize Yu's strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n → ∞. The main tools will be the Berkes-Morrow multi-parameter blocking technique, the Csörgő-Révész quantile transform method and the Bulinski rate of convergence in the CLT for associated random fields.
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d−2, which coincides with the condition obtained in , when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong invariance principle f... more In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n\to \infty. The main tools that we use are the following: the Berkes and Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15-37] multi-parameter blocking technique, the Csorgo and Revesz [Z. Wahrsch. Verw. Gebiete 31 (1975) 255-260] quantile transform method and the Bulinski [Theory Probab. Appl. 40 (1995) 136-144] rate of convergence in the CLT.
We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which i... more We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index H > 1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in [10], when the noise is white in time. Under this condition, we show that the solution is L 2 (Ω)-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.
We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\d... more We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\dot{W}$ in ℝ+×ℝd , whose solution is interpreted in the mild sense. The noise $\dot{W}$ is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.
We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R d ,... more We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R d , whose solution is interpreted in the mild sense. The noiseẆ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f ). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the sufficient condition for the existence of the solution is d ≤ 2 + α (if H > 1/2), respectively d < 2 + α (if H = 1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the k-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of k independent d-dimensional Brownian motions.
; t ∈ [0, T ], x ∈ R d } be the process solution of the stochastic heat equation ut = ∆u +Ḟ , u(0... more ; t ∈ [0, T ], x ∈ R d } be the process solution of the stochastic heat equation ut = ∆u +Ḟ , u(0, ·) = 0 driven by a Gaussian noisė F , which is white in time and has spatial covariance induced by the kernel f . In this paper we prove that the process u is locally germ Markov, if f is the Bessel kernel of order α = 2k, k ∈ N + , or f is the Riesz kernel of order α = 4k, k ∈ N + .
We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which i... more We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index H > 1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in [10], when the noise is white in time. Under this condition, we show that the solution is L 2 (Ω)-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.
In this article we prove a weak invariance principle for a strictly stationary φ-mixing sequence ... more In this article we prove a weak invariance principle for a strictly stationary φ-mixing sequence {X j } j≥1 , whose truncated variance function L(x) := EX 2 1 1 {|X1|≤x} is slowly varying at ∞ and mixing coefficients satisfy the logarithmic growth condition: n≥1 φ 1/2 (2 n ) < ∞. This will be done under the condition that lim n Var( n j=1X j )/[ n j=1 Var(X j )] = β 2 exists in (0, ∞), whereX j = X j I {|Xj |≤ηj } and η 2 n ∼ nL(η n ).
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d−2, which coincides with the condition obtained in , when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
In this paper we generalize Yu's strong invariance principle for associated sequences to the mult... more In this paper we generalize Yu's strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n → ∞. The main tools will be the Berkes-Morrow multi-parameter blocking technique, the Csörgő-Révész quantile transform method and the Bulinski rate of convergence in the CLT for associated random fields.
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
We consider the stochastic wave equation with multiplicative noise, which is fractional in time w... more We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d−2, which coincides with the condition obtained in , when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.
In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong invariance principle f... more In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n\to \infty. The main tools that we use are the following: the Berkes and Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15-37] multi-parameter blocking technique, the Csorgo and Revesz [Z. Wahrsch. Verw. Gebiete 31 (1975) 255-260] quantile transform method and the Bulinski [Theory Probab. Appl. 40 (1995) 136-144] rate of convergence in the CLT.
We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which i... more We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index H > 1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in [10], when the noise is white in time. Under this condition, we show that the solution is L 2 (Ω)-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.
We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\d... more We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\dot{W}$ in ℝ+×ℝd , whose solution is interpreted in the mild sense. The noise $\dot{W}$ is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.
We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R d ,... more We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R d , whose solution is interpreted in the mild sense. The noiseẆ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f ). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the sufficient condition for the existence of the solution is d ≤ 2 + α (if H > 1/2), respectively d < 2 + α (if H = 1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the k-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of k independent d-dimensional Brownian motions.
; t ∈ [0, T ], x ∈ R d } be the process solution of the stochastic heat equation ut = ∆u +Ḟ , u(0... more ; t ∈ [0, T ], x ∈ R d } be the process solution of the stochastic heat equation ut = ∆u +Ḟ , u(0, ·) = 0 driven by a Gaussian noisė F , which is white in time and has spatial covariance induced by the kernel f . In this paper we prove that the process u is locally germ Markov, if f is the Bessel kernel of order α = 2k, k ∈ N + , or f is the Riesz kernel of order α = 4k, k ∈ N + .
We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which i... more We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index H > 1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in [10], when the noise is white in time. Under this condition, we show that the solution is L 2 (Ω)-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.
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