Uniqueness for stochastic evolution equations in Banach spaces

By Martin Ondrej'at. 2004

237 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/d00ab91c264cae0fea4add58094c01e7751f42ef

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Citations in the corpus

Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups (2008, methodology): “continuous sample paths). Hence applying [ 13, Theorem 13 ]”
Yamada-Watanabe Results for Stochastic Differential Equations with Jumps (2013, methodology): “The Yamada-Watanabe result has been generalized byOndreját [7] and Röckner et al.”
Stochastic Convolutions Driven by Martingales: Maximal Inequalities and Exponential Integrability (2007, background): “Therefore, WA has paths continuous in H−1 and it was shown in [33, 34] that such continuity is sufficient for theorems of Yamada–Watanabe type and theorems on Markov process generation to hold in H .”
Stochastic Integration in Banach Spaces - a Survey (2013, methodology): “A particularly simple proof, based on a good-λ inequality, was obtained by Ondreját [86].”
Brownian Representations of Cylindrical Local Martingales, Martingale Problem and Strong Markov Property of Weak Solutions of SPDEs in Banach Spaces (2005, methodology): “a 〈 h,G(s, e−1Π(s))· 〉 dW holds -almost surely for every t > a and every h ∈ D by the Chojnowska-Michalik theorem ([14] or [2]), so [e−1Π(t) = u(t)] = 1 for t = a and almost every t ∈ [a,∞).”
Invariant measures for stochastic nonlinear beam and wave equations (2016, background): “1) implies the condition (2) above by the Yamada– Watanabe theorem [23].”
Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains (2012, methodology): “Moreover, using the results due to Ondreját [32] we will show existence of strong solutions and uniqueness in law.”
Maximal regularity for stochastic convolutions driven by Lévy processes (2009, background): “Theorem 2.12 in [5], see also [4] and [ 26, section 5 ], where the Burkholder inequality is proved for all q ∈ (0, ∞). Condition (R2) follows directly from the definition of martingale type 2 Banach spaces.”
Stochastic nonlinear beam equations (2005, background): “... found for every deterministic initial condition u(0) = x ∈ H . (Pathwise uniqueness obviously holds for (1.3) owing to the Lipschitz continuity of F and Σ on bounded sets.) Consequently, for any Borel probability measure µ on H there exists a martingale solution (as defined in [10], Chapter 8) t”
Degenerate parabolic stochastic partial differential equations (2013, methodology): “(A detailed construction of stochastic integral for processes with values in 2-smooth Banach spaces can be found in [10] or [61].”
Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations (2008, methodology): “It should be noticed that although the equation considered in [54] is a little different from Eq.”
Stochastic NonLinear Wave Equations in Local Sobolev Spaces (2010, methodology): “If we define the processes I1(t) = v(0), I2(t) = ∫ t 0 u(s) ds, I3(t) = ∫ t 0 1Bm f (·, z(s),∇u(s)) ds, I4(t) = ∫ t 0 1Bm g(·, z(s),∇u(s)) dWs where the integral I2 converges in W 1,2 and the integrals I3, I4 converge in L 2 then P [I2(t) ∈W 2,2] = 1, P [v(t) = I1(t) +A m I2(t) + I3(t) + I4(t)] = 1,”