Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach

By Sandra Cerrai. 2001

384 citations recorded by the source corpus.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/0765c07a305cf46659a71b27ed3902c80e48a081

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Citations in the corpus

Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, background): “[3, 5, 15, 26]) since the essential supremum norm with respect to the time variable is taken in the expectation.”
A Khasminskii type averaging principle for stochastic reaction–diffusion equations (2008, methodology): “…proved that an averaging principle is satisfied by using a completely different approach based on Kolmogorov equations and martingale solutions of stochastic equations, which is more in the spirit of the general method introduced by Papanicolaou, Strook and Varadhan in their paper [24] of 1977.”
Strong and weak orders in averaging for SPDEs (2012, methodology): “When working with SDEs, this strategy can be entirely followed; nevertheless in the case of SPDEs, the Kolmogorov equations involve the unbounded operators A and B, and quantities like Tr(D(2)yyu), and this leads to technical problems—see [3] or [5].”
Flow of diffeomorphisms for SDEs with unbounded H (2009, methodology): “Probabilistic formulae for the spatial derivatives of Markov semigroups have been much studied for different classes of degenerate and non-degenerate diffusion processes even with jumps (see [3], [21], [6], [5], [4], [12] [26], [27], [34] and the references therein).”
Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces (2004, background): “4) are obtained, smoothing properties on the semigroup (Pt ) associated to the operator L are also required and, consequently, nondegeneracy assumptions on G are needed; see, for example, [4] and [14].”
Estimates of the derivatives for parabolic operators with unbounded coefficients (2005, background): “The problem of estimating the derivatives of T (t)f has already been studied in literature by both analytic ([2, 4, 6, 12]) and probabilistic methods ([5, 19]).”
Well Posedness and Asymptotic Behavior for Stochastic Reaction-Diffusion Equations with Multiplicative Poisson Noise (2009, background): “[9, 11, 12] and references therein), a corresponding body of results for equations driven by jump noise seems to be missing.”
Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations (2010, methodology): “Then, again by [3], μ is equivalent to each Pt(x, ·) with x ∈ D(A), t > 0 and consequently (i) holds.”
Gradient estimates in parabolic problems with unbounded coefficients (2004, methodology): “In the case Ω = RN the previous estimates were proved in [15] with the same method and in [5] with probabilistic methods.”
Measuring Sample Quality with Diffusions (2016, background): “The work of Cerrai [9] also suggests that the Lipschitz condition on our drift and diffusion coefficients can be relaxed.”