the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

intermittency for stochastic partial differential equations driven by strongly inhomogeneous space time white noises abstract in this paper the main topic is to investigate the intermittent property of the one dimensional stochastic heat equation driven by an inhomogeneous brownian sheet which is a noise deduced from the study of the catalytic super brownian motion under some proper conditions on the catalytic measure of the inhomogeneous brownian sheet we show that the solution is weakly full intermittent based on the estimates of moments of the solution in particular it is proved that the second moment of the solution grows at the exponential rate the novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the lebesgue measure on r

weak amenability of fourier algebras and local synthesis of the anti diagonal abstract we show that for a connected lie group g its fourier algebra a g is weakly amenable only if g is abelian our main new idea is to show that weak amenability of a g implies that the anti diagonal g g g 1 g g is a set of local synthesis for a g g we then show that this cannot happen if g is non abelian we conclude for a locally compact group g that a g can be weakly amenable only if it contains no closed connected non abelian lie subgroups in particular for a lie group g a g is weakly amenable if and only if its connected component of the identity g e is abelian

the stochastic wave equation in two spatial dimensions keywords stochastic wave equation gaussian noise process solution reference prob article 1998 001view record in web of science record created on 2008 12 01 modified on 2017 05 12

ideals with bounded approximate identities in fourier algebras abstract we make use of the operator space structure of the fourier algebra a g of an amenable locally compact group to prove that if h is any closed subgroup of g then the ideal i h consisting of all functions in a g vanishing on h has a bounded approximate identity this result allows us to completely characterize the ideals of a g with bounded approximate identities we also show that for several classes of locally compact groups including all nilpotent groups i h has an approximate identity with norm bounded by 2 the best possible norm bound

approximation properties for group algebras and group von neumann algebras let g be a locally compact group let c g resp vn g be the c algebra resp the von neumann algebra associated with the left regular representation i of g let a g be the fourier algebra of g and let moa g be the set of completely bounded multipliers of a g with the completely bounded norm moa g is a dual space and we say that g has the approximation property ap if there is a net ua of functions in a g with compact support such that ua a 1 in the associated weak topology in particular g has the ap if g is weakly amenable a g has an approximate identity that is bounded in the completely bounded norm for a discrete group f we show that f has the ap x c f has the slice map property for subspaces of any c algebra x vn jt has the slice map property for a weakly closed subspaces of any von neumann algebra property se the semidirect product of weakly amenable groups need not be weakly amenable we show that the larger class of groups with the ap is stable with respect to semidirect products and more generally this class is stable with respect to group extensions we also obtain some results concerning crossed products for example we show that the crossed product m a g of a von neumann algebra m with property s by a group g with the ap also has property s

measurable schur multipliers and completely bounded multipliers of the fourier algebras let g be a locally compact group and a g the fourier algebra of g from 17 in 24 herz de ned the space of herz schur multipliers b2 g he claimed that this space coincided with the space he denoted v g in his article 23 but o2ered no proof of this fact the completely bounded multipliers of the fourier algebra mcba g were de ned by de canniere and haagerup in 8 in was shown by bo zejko and fendler 7 that mcba g 1 4 b2 g isometrically unfortunately this result relies on unpublished work of gilbert entitled convolution operators and banach space tensor products i ii and iii but not widely available however there is a proof of the fact that mcba g 1 4 b2 g due to jolissaint 27 which relies on the representation theorem for completely bounded maps applied to the reduced c algebra c r g in this paper we prove an analogue of the fact that mcba g 1 4 b2 g in fact our result obtains that fact in the case that g is a discrete group and improves upon it in the respect that we obtain a complete isometry where both mcba g and b2 g are given natural operator space structures moreover we show that the natural module action of mcba g on the von neumann algebra generated by the left regular representation vn g extends completely isometrically to a normal action of mcba g on all of b l g the space of bounded operators on l g in fact the image of mcba g in the completely bounded normal maps on b l g consists of l1 g bimodule maps and can be characterised as a certain intrinsic invariant part amongst those maps we feel that our approach is interesting in and of itself since it makes systematic use of modern operator space theoretic techniques pioneered by blecher e2ros haagerup paulsen ruan and smith amongst others it thus gives a starting point for obtaining representations of completely bounded multipliers of kac algebras after 29 or of multipliers of locally compact quantum groups after vaes et al see 47 to support our approach in x 3 we describe the space of measurable schur multipliers v1 x for any suitable measure space x we develop the theory of v1 x fully and systematically our main tools for doing this are the results of smith 41 and blecher and smith 5 which characterise the normal completely bounded maps on the space of all bounded operators on a hilbert space which are module maps over a von neumann algebra we make great use of the weak haagerup tensor product of two von neumann algebras which is

rank one subspaces of bimodules over maximal abelian selfadjoint algebras spaces of operators that are left and right modules over maximal abelian selfadjoint algebras masa bimodules for short are natural generalizations of algebras with commutative subspace lattices this paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules it is shown that every finite rank operator of a norm closed masa bimodule m is in the trace norm closure of the rank one subspace of m an important consequence is that the rank one subspace of a strongly reflexive masa bimodule that is one which is the reflexive hull of its rank one operators is dense in the module in the weak operator topology however in contrast to the situation for algebras it is shown that such density need not hold in the ultraweak topology a new method of representing masa bimodules is introduced this uses a novel concept of an topology with the appropriate notion of support a correspondence is established between reflexive masa bimodules and their supports it is shown that if a c2 closed masa bimodule contains a trace class operator then it must contain rank one operators indeed every such operator is in the c2 norm closure of the rank one subspace of the module consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive however the trace norm closure of the rank one subspace need not contain all trace class operators of the module also it is shown that there exists a csl algebra which contains no trace class operators yet contains an operator belonging to cpfor allp 1 from this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in cpfor 1 p as an application it is shown that there exists a commutative subspace lattice l such that l is non synthetic but every weakly closed algebra which contains a masa and has invariant lattice l coincides with alg l

geometric characterization of intermittency in the parabolic anderson model we consider the parabolic anderson problem tu u x u on r z d with localized initial condition u 0 x 0 x and random i i d potential under the assumption that the distribution of 0 has a double exponential or slightly heavier tail we prove the following geometric characterisation of intermittency with probability one as t 1 the overwhelming contribution to the total mass p x u t x comes from a slowly increasing number of islands which are located far from each other these islands are local regions of those high exceedances of the eld in a box of side length 2t log 2 t for which the local principal dirichlet eigenvalue of the random operator is close to the top of the spectrum in the box we also prove that the shape of in these regions is non random and that u t is close to the corresponding positive eigenfunction this is the geometric picture suggested by localization theory for the anderson hamiltonian

operator synthesis i synthetic sets bilattices and tensor algebras abstract the interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by arveson ann of math 2 100 1974 433 is extended to include other topics as harmonic analysis for varopoulos algebras and approximation by projection valued measures we propose a coordinate approach which nevertheless does not use the technique of pseudo integral operators as well as a coordinate free one which allows to extend to non separable spaces some important results and constructions of arveson we solve some problems posed in arveson 1974

a local time correspondence for stochastic partial differential equations it is frequently the case that a white noise driven parabolic and or hyperbolic stochastic partial differential equation spde can have random field solutions only in spatial dimension one here we show that in many cases where the spatial operator is the l generator of a levy process x a linear spde has a random field solution if and only if the symmetrization of x possesses local times this result gives a probabilistic reason for the lack of existence of random field solutions in dimensions strictly bigger than one in addition we prove that the solution to the spde is holder continuous in its spatial variable if and only if the said local time is holder continuous in its spatial variable we also produce examples where the random field solution exists but is almost surely unbounded in every open subset of space time our results are based on first establishing a quasi isometry between the linear l space of the weak solutions of a family of linear spdes on one hand and the dirichlet space generated by the symmetrization of x on the other hand we study mainly linear equations in order to present the local time correspondence at a modest technical level however some of our work has consequences for nonlinear spdes as well we demonstrate this assertion by studying a family of parabolic spdes that have additive nonlinearities for those equations we prove that if the linearized problem has a random field solution then so does the nonlinear spde moreover the solution to the linearized equation is holder continuous if and only if the solution to the nonlinear equation is and the solutions are bounded and unbounded together as well finally we prove that in the cases that the solutions are unbounded they almost surely blow up at exactly the same points date october 7 2007 2000 mathematics subject classification primary 60h15 60j55 secondary 35r60 35d05

parabolic problems for the anderson model the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

on the existence and position of the farthest peaks of a family of stochastic heat and wave equations we study the stochastic heat equation partial t u mathcal l u sigma u dot w in 1 1 dimensions where dot w is space time white noise r r is lipschitz continuous and mathcal l is the generator of a symmetric levy process that has finite exponential moments and u0 has exponential decay at we prove that under natural conditions on i the th absolute moment of the solution to our stochastic heat equation grows exponentially with time and ii the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting see however gartner et al in probab theory relat fields 111 17 55 1998 gartner et al in ann probab 35 439 499 2007 for the analysis of the location of the peaks in a different model finally we show that these results extend to the stochastic wave equation driven by laplacian

conditional expectations on cv_ p g applications keywords locally compact groups amenable groups fourier algebra figa talamanca algebra convolution operators pseudomeasures reference cahru article 2007 001doi 10 1016 j jfa 2007 03 003view record in web of science record created on 2008 12 10 modified on 2016 08 08

completely bounded module maps and the haagerup tensor product abstract we obtain a representation theorem for module maps defined on the algebra k h of compact operators this is used to show that the haagerup tensor product of two operator algebras enjoys strong versions of tomita s commutant theorem and the slice map property we also give a general theorem concerning the automatic complete boundedness of module maps

moment bounds for a class of fractional stochastic heat equations we consider fractional stochastic heat equations of the form frac partial u t x partial t delta alpha 2 u t x lambda sigma u t x dot f t x here dot f denotes the noise term under suitable assumptions we show that the second moment of the solution grows exponentially with time in particular this answers an open problem in cite cokh along the way we prove a number of other interesting properties which extend and complement results in cite foonjose cite khoshnevisan 2013aa and cite khoshnevisan 2013ab

on the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups we extend the results by froelich and spronk and turowska on the connection between operator synthesis and spectral synthesis for a g to second countable locally compact groups g this gives us another proof that one point subset of g is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis furthermore we show that non triangular sets are strong operator ditkin sets and we establish a connection between operator ditkin sets and ditkin sets these results are applied to prove that any closed subgroup of g is a local ditkin set

compact operators invariant subspaces and spectral synthesis let l be a subspace lattice which contains a sequence pn of commuting projections such that for any subsequence pni vpni i and pni 0 then alg l k h 0 suppose g is a compact abelian group and e g is a closed set there is a family le of commutative subspace lattices for which alg le k h 0 precisely when e is a set of multiplicity in the sense of harmonic analysis by showing that the graph of is a set of uniqueness in 2 2 we obtain a thin set proof that alg 2 m12 contains no nonzero compact operators for an important class of sets k g gamin k amax k iff k is a set of spectral synthesis in the group g x g it follows that if e1and e2 are sets of spectral synthesis in g then alg pe1 bo alg pe2 alg pe1 pe2iff e1 e2 is a set of spectral synthesis in g x g and if semigroups 1 and 2 are sets of spectral synthesis in g then alg 1 bo alg 2 alg 1 2iff 1 2 is a set of spectral synthesis in g g the operator algebra alg p1 p2 is synthetic iff for all n alg p1 pn is synthetic this implies that the operator algebras alg 2 mp are synthetic

ideals of a g and bimodules over maximal abelian selfadjoint algebras this paper is concerned with weak closed masa bimodules generated by a g invariant subspaces of vn g an annihilator formula is established which is used to characterise the weak closed subspaces of b l2 g which are invariant under both schur multipliers and a canonical action of m g on b l2 g via completely bounded maps we study the special cases of extremal ideals with a given null set and for a large class of groups we establish a link between relative spectral synthesis and relative operator synthesis

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

nonlinear stochastic wave and heat equations we study nonlinear wave and heat equations on d driven by a spatially homogeneous wiener process for the wave equation we consider the cases of d 1 2 3 the heat equation is considered on an arbitrary d space we give necessary and sufficient conditions for the existence of a function valued solution in terms of the covariance kernel of the noise

reduced spectral synthesis and compact operator synthesis we introduce and study the notion of reduced spectral synthesis which unifies the concepts of spectral synthesis and uniqueness in locally compact groups we exhibit a number of examples and prove that every non discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis we introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory studied previously we show that a closed subset e of a second countable locally compact group g satisfies reduced local spectral synthesis if and only if the subset e s t ts 1 is an element of e of g x g satisfies compact operator synthesis we apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on schatten p classes c 2020 elsevier inc all rights reserved

intermittency for the wave and heat equations with fractional noise in time in this article we consider the stochastic wave and heat equations driven by a gaussian noise which is spatially homogeneous and behaves in time like a fractional brownian motion with hurst index h 1 2 the solutions of these equations are interpreted in the skorohod sense using malliavin calculus techniques we obtain an upper bound for the moments of order p 2 of the solution in the case of the wave equation we derive a feynman kac type formula for the second moment of the solution based on the points of a planar poisson process this is an extension of the formula given by dalang mueller and tribe trans amer math soc 360 2008 4681 4703 in the case h 1 2 and allows us to obtain a lower bound for the second moment of the solution these results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time

ranges of bimodule projections and reflexivity abstract we develop a general framework for reflexivity in dual banach spaces motivated by the question of when the weak closed linear span of two reflexive masa bimodules is automatically reflexive we establish an affirmative answer to this question in a number of cases by examining two new classes of masa bimodules defined in terms of ranges of masa bimodule projections we give a number of corollaries of our results concerning operator and spectral synthesis and show that the classes of masa bimodules we study are operator synthetic if and only if they are strong operator ditkin

on the stochastic burgers equation in the real line this is the publisher s version also available electronically from http projecteuclid org euclid aop 1022677386 abstract

random nonlinear wave equations propagation of singularities on etudie les proprietes de lissite des solutions d equations d ondes unidimensionnelles avec forcing aleatoire non lineaire on definit les singularites comme anomalies dans le modulo local de continuite des solutions

effective equations for two phase flow with trapping on the micro scale in this paper we consider water drive for recovering oil from a strongly heterogeneous porous column the two phase model uses corey relative permeabilities and brooks corey capillary pressure the heterogeneities are perpendicular to the flow and have a periodic structure this results in one dimensional flow and a space periodic absolute permeability reflecting alternating coarse and fine layers assuming many or thin layers we use homogenization techniques to derive the effective transport equations the form of these equations depends critically on the capillary number the analysis is confirmed by numerical experiments

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

operator synthesis and tensor products we show that kraus property s is preserved under taking weak closed sums with masa bimodules of finite width and establish an intersection formula for weak closed spans of tensor products one of whose terms is a masabimodule of finite width we initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form where is a set of finite width while is operator synthetic is under a necessary restriction on the sets again operator synthetic we show that property s is preserved under spatial morita subordinance

sets of multiplicity and closable multipliers on group algebras we undertake a detailed study of the sets of multiplicity in a second countable locally compact group g and their operator versions we establish a symbolic calculus for normal completely bounded maps from the space b l2 g of bounded linear operators on l2 g into the von neumann algebra vn g of g and use it to show that a closed subset e g is a set of multiplicity if and only if the set e s t g g ts 1 e is a set of operator multiplicity analogous results are established for m1 sets and m0 sets we show that the property of being a set of multiplicity is preserved under various operations including taking direct products and establish an inverse image theorem for such sets we characterise the sets of finite width that are also sets of operator multiplicity and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set we show that if g satisfies a mild approximation condition pointwise multiplication by a given measurable function g c defines a closable multiplier on the reduced c algebra view the mathml source of g if and only if schur multiplication by the function n g g c given by n s t ts 1 is a closable operator when viewed as a densely defined linear map on the space of compact operators on l2 g similar results are obtained for multipliers on vn g

on the chaotic character of the stochastic heat equation ii consider the stochastic heat equation partial t u varkappa 2 delta u sigma u dot f where the solution u u t x is indexed by t x in 0 infty times bf r d and dot f is a centered gaussian noise that is white in time and has spatially correlated coordinates we analyze the large x fixed t behavior of the solution u in different regimes thereby study the effect of noise on the solution in various cases among other things we show that if the spatial correlation function f of the noise is of riesz type that is f x propto x alpha then the fluctuation exponents of the solution are psi for the spatial variable and 2 psi 1 for the time variable where psi 2 4 alpha moreover these exponent relations hold as long as alpha in 0 d wedge 2 that is precisely when dalang s theory dalang electron j probab 4 paper no 6 29 1999 implies the existence of a solution to our stochastic pde these findings bolster earlier physical predictions kardar et al phys rev lett 58 20 889 892 1985 kardar and zhang phys rev lett 58 20 2087 2090 1987