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Adaptively biased sequential importance sampling for rare events in reaction networks with comparison to exact solutions from finite buffer dCME method J. Chem. Phys. 139, 025101 (2013) In this paper we present a moment closure method for stochastically modeled chemical or biochemical reaction networks. We derive a system of differential equations which describes the dynamics of means and all central moments from a chemical master equation. Truncating the system for the central moments at a certain moment term and using Taylor approximation, we obtain explicit representations of means and covariances and even higher central moments in recursive forms. This enables us to deal with the moments in successive differential equations and use conventional numerical methods for their approximations. Furthermore, we estimate the errors in the means and central moments generated by the approximation method. We also find the moments at equilibrium by solving truncated algebraic equations. We show in examples that numerical solutions based on the moment closure method are accurate and efficient by comparing the results to those of stochastic simulation algorithms.

Abstract. Let p(t, x) be the fundamental solution to the problemIf α, β ∈ (0, 1), then the kernel p(t, x) becomes the transition density of a Lévy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivativesx is a partial derivative of order n with respect to x, (−∆x) γ is a fractional Laplace operator and D σ t and I δ t are Riemann-Liouville fractional derivative and integral respectively.

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In this article we present a W n 2 -theory of stochastic parabolic partial differential systems. In particular, we focus on non-divergent type. The space domains we consider are R d , R d + and eventually general bounded C 1 -domains O. By the nature of stochastic parabolic equations we need weighted Sobolev spaces to prove the existence and the uniqueness. In our choice of spaces we allow the derivatives of the solution to blow up near the boundary and moreover the coefficients of the systems are allowed to oscillate to a great extent or blow up near the boundary.

In this paper we prove a parabolic version of the Littlewood-Paley inequality (1.3) for the operators of the type φ(−∆), where φ is a Bernstein function. As an application, we construct an Lp-theory for the stochastic integro-differential equations of the type du = (−φ(−∆)u + f ) dt + g dWt.2010 Mathematics Subject Classification. 42B25, 26D10, 60H15, 60G51, 60J35.Recently, (1.3) was proved for the fractional Laplacian ∆ α/2 , α ∈ (0, 2), in [2,8]. Also, in [17] similar result was proved for the case J = J(t, y) = m(t, y)|y| −d−α in (1.1), where α ∈ (0, 2) and m(t, y) is a bounded smooth function satisfying m(t, y) = m(t, y|y| −1 ) (i.e. homogeneous of degree zero) and m(t, y) > c > 0 on a set Γ ⊂ S d−1 of a positive Lebesgue measure. We note that even the case φ(λ) = λ α + λ β (α = β) is not covered in [17].Our motivation of studying (1.3) is that (1.3) is the key estimate for the L ptheory of the corresponding stochastic partial differential equations. For example, Krylov's result ([14, 15]) for ∆ is related to the L p -theory of the second-order stochastic partial differential equations. Below we briefly explain the reason for this. See [9,16] or Section 6 of this article for more details. Consider the stochastic integro-differential equation du = (φ(∆)u + h) dt +

Stochastic partial differential equations are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular, it can be negative and fractional. It is allowed that the coefficients of the equations blow up near the boundary.

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