We use our earlier Brownian-time framework to formulate and establish global uniqueness and local-in-time existence of the Burgers incarnation of the Kuramoto-Sivashinsky PDE on R+×Rd, in the class of time-continuous L2p-valued solutions, p≥1, for every d<6p. We assume neither space compactness, nor spatial coordinates dependence, nor smallness of initial data. The surprising discovery of the 6p-th dimension bound, even for local solutions, is revealed by our approach and the Brownian-time kernel -- the Brownian average of an angled d-dimensional Schrödinger propagator -- at its heart. We use this kernel to give a systematic approach, for all dimensions simultaneously, including a novel formulation -- even in the well-known d=1 case -- of the KS equation. This yields the estimates leading to this article's conclusions. We achieve the stated results by fusing some of our earlier Brownian-time stochastic processes constructions and ideas -- encoded in the aforementioned kernel -- with analytic ones, including complex and harmonic analysis; by employing suitable N-ball approximations together with fixed point theory; and by an adaptation of the stochastic analytic stopping-time technique to our deterministic setting. Using a separate strategy, that is also built on our Brownian-time paradigm, we treat the global wellposedness of the multidimensional KS equation in a followup upcoming article. This work also serves as a template for another forthcoming article in which we prove similar results for the time-fractional Burgers equation in multidimensional space.
Published Work
- Allouba, Hassan
Solving the Kuramoto-Sivashinsky-Burgers equation until the 6p-th dimension: the Brownian-time paradigm.
2023 - arXiv
- Allouba, Hassan and Xiao, Yimin L-Kuramoto-Sivashinsky SPDEs vs. time-fractional SPIDEs: exact continuity and gradient moduli, 1/2-derivative criticality, and laws.
J. Differential Equations 263 (2017), no. 2, 1552–1610.
We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto- Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced—with Brownian- time-type kernel formulations—by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in d = 1, 2, 3. We show that both classes are (1/2)− Hölder continuously differentiable in space when d = 1, and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by space-time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order < 1/2 and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws gov- erning (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba’s earlier works on (1) and (2). Similar regularity results hold for Allen-Cahn nonlinear members of (1) and (2) on compacts via change of measure.
- Allouba, Hassan L-Kuramoto-Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift-Hohenberg law equivalence.
J. Differential Equations 259 (2015), no. 11, 6851–6884.
Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift-Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on $\{\Rp\times\Rd\}_{d=1}^{3}$. The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters $\vepo$ to the L-KS spatial operator and $\vept$ to the noise term, we show that the dimension-dependent critical ratio $\vept/\vepo^{d/8}$ controls the limiting behavior of the L-KS SPDE, as $\vepo,\vept\searrow0$; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence---and hence the same Hölder regularity---of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.
- Allouba, Hassan Time-fractional and memoryful ∆2k SIEs on R+ × Rd : how far can we push white noise?
Illinois J. Math. 57 (2013), no. 4, 919–963.
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs---through their fundamental solutions---on stochastic integral equations (SIEs) driven by space-time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase of the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in (Allouba 2013, Allouba 2006), where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space-time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions d=1,2,3. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in d=1,2. This gave, for the first time, an example of a space-time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space. In this paper, we introduce the 2β−1-order β-inverse-stable-Lévy-time Brownian motion (β-ISLTBM) SIEs, driven by space-time white noise. We show that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all β-ISLTBM SIEs. Furthermore, we show that increasing the order of the Laplacian β−1 beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches the temporal regularity of the Brownian sheet as β↘0.
- Allouba, Hassan Brownian-time Brownian motion SIEs on R+ × Rd: ultra regular direct and lattice- limits solutions and fourth order SPDEs links.
Discrete Contin. Dyn. Syst. (DCDS-A) 33 (2013), no. 2, 413–463.
We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, $\KBtxy$, on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE, which we recently introduced. In sharp contrast to second order heat-based SPDEs--whose real-valued mild solutions are confined to d=1--we prove the existence of solutions to the BTBM SIE in d=1,2,3 with dimension-dependent and striking Holder regularity, under both less than Lipschitz and Lipschitz conditions. In space, we show an unprecedented nearly local Lipschitz regularity for d=1,2--roughly, the SIE is spatially twice as regular as the Brownian sheet in these dimensions--and nearly local Hölder 1/2 regularity in d=3. In time, our solutions are locally Hölder continuous with exponent γ∈(0,(4−d)/(8)) for 1≤d≤3. To investigate our SIE, we (a) introduce the Brownian-time random walk and we use it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and the mild forms of many SPDEs of different orders on the lattice. Here, solutions types to our SIE are both direct and limits of their lattice version. The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that it is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $\KBtxy$ by the intimately connected kernel of our recently introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE becomes the mild form of a Kuramoto-Sivashinsky SPDE with linear PDE part. Ideas developed here are adapted in separate papers to give a new approach, via our explicit IBTBAP representation, to many KS-type SPDEs in multi spatial dimensions.
- Allouba, Hassan and Nane, Erkan Interacting time-fractional and ∆ν PDEs systems via Brownian-time and inverse-stable-Lévy-time Brownian sheets.
Stoch. Dyn. 13 (2013), no. 1, 1250012, 31 pp.(Featured Article).
Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single and the multi parameter Brownian-time PDEs. Here, we introduce a new---even in the one parameter case---proof that judiciously combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourth order system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0<βY<1. When β=1/ν, $ν\in\lbr2,3,...\rbr$, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k=1,...,ν−1. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one parameter case this condition automatically holds.
- Allouba, Hassan and Fontes, Ramiro Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock’s formulas.
Stoch. Anal. Appl. 29 (2011), no. 6, 1111–1135.
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under L2-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond Novikov's condition. This is due to the invariance under change of measure of the first author's derivative in \cite{A1}. The formulations and proofs are natural applications of the differentiation theory in \cite{A1} and standard Itô integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator F=I[K,∞)(WT), which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator.
- Allouba, Hassan From Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky- variant interacting PDEs systems.
Stoch. Anal. Appl. 29 (2011), no. 6, 933–950.
We introduce n-parameter $\Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each "time" parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of n linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) connection to fourth order linear PDEs. Our BTBS does not belong to the classical theory of random fields; and to prove our new PDEs connections, we generalize our BTP approach in \cite{Abtp1,Abtp2} and we mix it with the Brownian sheet connection to a linear PDE system, which we also give along with its corresponding nonlinear second order PDE and 2n-th order linear PDE. In addition, we introduce the n-parameter d-dimensional linear Kuramoto-Sivashinsky (KS) sheet kernel (or "transition density"); and we link it to an intimately connected system of new linear Kuramoto-Sivashinsky-variant interacting PDEs, generalizing our earlier one parameter imaginary-Brownian-time-Brownian-angle kernel and its connection to the KS PDE. The interactions here mean that our PDEs systems are to be solved for a family of functions, a feature shared with well known fluids dynamics models. The interacting PDEs connections established here open up another new fundamental front in the rapidly growing field of iterated-type processes and their connections to both new and important higher order PDEs and to some equivalent fractional Cauchy problems. We connect the BTBS fourth order interacting PDEs system given here with an interacting fractional PDE system and further study it in another article.
- Allouba, Hassan A differentiation theory for Itô’s calculus.
Stoch. Anal. Appl. 24 (2006), no. 2, 367–380.
A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.
- Allouba, Hassan A Brownian-time excursion into fourth-order PDEs, linearized Kuramoto-Sivashinsky, and BTP-SPDEs on R+ × Rd.
Stoch. Dyn. 6 (2006), no. 4, 521–534.
In recent articles we have introduced the class of Brownian-time processes (BTPs) and the Linearized Kuramoto–Sivashinsky process (LKSP). Probabilistically, BTPs represent a unifying class for some different exciting processes like the iterated Brownian motion (IBM) of Burdzy (a process with fourth-order properties) and the Brownian–snake of Le Gall (a second-order process); they also include many additional new and quite interesting processes. The LKSP is closely connected to the Kuramoto–Sivashinsky PDEs, one of the most celebrated PDEs in modern applied mathematics. We start by surveying the fourth-order PDE connections to BTPs and the LKSP that we uncovered in two recent articles. In the second part of this paper we introduce BTP-SPDEs, these are SPDEs in which the PDE part is that solved by running a BTP. We consider a BTP-SPDE driven by an additive spacetime white noise on the time-space set ℝ+ × ℝd; and we prove the existence of a unique real-valued, Lp(Ω,ℙ) for all p ≥ 1, BTP solution to such BTP-SPDEs for 1 ≤ d ≤ 3. This contrasts sharply with the standard theory of reaction-diffusion type SPDEs driven by spacetime white noise, in which real-valued solutions are confined to one spatial dimension. Like the PDEs case, BTP-SPDEs also provide a valuable insight into other fourth-order SPDEs of applied science. We carry out such a program in forthcoming articles.
- Allouba, Hassan and Langa, José A. Semimartingale attractors for Allen-Cahn SPDEs driven by space- time white noise. I. Existence and finite dimensional asymptotic behavior.
Stoch. Dyn. 4 (2004), no. 2, 223–244.
We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa \cite{AL0}. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determining modes, both in the forward and the pullback sense.
- Allouba, Hassan SDDEs limits solutions to sublinear reaction-diffusion SPDEs.
Electron. J. Differen- tial Equations 2003, No. 111, 21 pp.
We start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations) limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs approach to SPDEs from earlier work, is entirely based on the approximating SDDEs. It is applicable to, and gives a multiscale view of, a variety of SPDEs. We extend this approach in related work to other heat-based SPDEs (Burgers, Allen-Cahn, and others) and to the difficult case of SPDEs with multi-dimensional spacial variable. We focus here on one-spacial-dimensional reaction-diffusion SPDEs; and we prove the existence of a SDDEs limit solution to these equations under less-than-Lipschitz conditions on the drift and the diffusion coefficients, thus extending our earlier SDDEs work to the nonzero drift case. The regularity of this solution is obtained as a by-product of the existence estimates. The uniqueness in law of our SPDEs follows, for a large class of such drifts/diffusions, as a simple extension of our recent Allen-Cahn uniqueness result. We also examine briefly, through order parameters ε1 and ε2 multiplied by the Laplacian and the noise, the effect of letting ε1,ε2→0 at different speeds. More precisely, it is shown that the ratio ε2/ε1/41 determines the behavior as ε1,ε2→0.
- Allouba, Hassan and Langa, José A. Semimartingale attractors for generalized Allen-Cahn SPDEs driven by space-time white noise.
C. R. Math. Acad. Sci. Paris 337 (2003), no. 3, 201–206.
We introduce the notion of a semimartingale attractor associated with space–time white noise driven generalized Allen–Cahn SPDEs. We treat the driving noise in the martingale measure setting, and we give an existence result for this type of random attractors for these generalized Allen–Cahn SPDEs in our setting. This Note focuses on semimartingale functional attractors, but our noise setting also leads naturally to a related type of random attractors that we call semimartingale measure attractors and which we detail in an upcoming article. Detailed proofs and extensions of our result, as well as other properties of semimartingale attractors, for different types of SPDEs are also furnished in the follow-up article.
- Allouba, Hassan A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process.
C. R. Math. Acad. Sci. Paris 336 (2003), no. 4, 309–314.
We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto-Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order PDEs, we give an explicit solution to a linearized Kuramoto-Sivashinsky PDE in d-dimensional space: $\scriptstyle{\eight{u_t=-\frac18Δ^2u-\frac12Δu-\frac12u}}$. The solution is given in terms of a functional of our LKSP.
- Allouba, Hassan and Goodman, Victor Market price of risk and random field driven models of term structure: a space-time change of measure look.
Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, LA, 2001), 37–44, Contemp. Math., 317, Amer. Math. Soc., Providence, RI, 2003.
No-arbitrage models of term structure have the feature that the return on zero-coupon bonds is the sum of the short rate and the product of volatility and market price of risk. Well known models restrict the behavior of the market price of risk so that it is not dependent on the type of asset being modeled. We show that the models recently proposed by Goldstein and Santa-Clara and Sornette, among others, allow the market price of risk to depend on characteristics of each asset, and we quantify this dependence. A key tool in our analysis is a very general space-time change of measure theorem, proved by the first author in earlier work, and covers continuous orthogonal local martingale measures including space-time white noise.
- Allouba, Hassan Brownian-time processes: the PDE connection. II. And the corresponding Feynman-Kac formula.
Trans. Amer. Math. Soc. 354 (2002), no. 11, 4627–4637.
We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes, and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.
- Allouba, Hassan and Zheng, Weian Brownian-time processes: the PDE connection and the half-derivative generator.
Ann. Probab. 29 (2001), no. 4, 1780–1795.
We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic PDEs. We then study their exit problem as they exit nice domains in $\Rd$, and connect it to elliptic PDEs. We show that these processes have the peculiar property that they solve fourth order parabolic PDEs, but their exit distribution - at least in the standard Brownian-time process case - solves the usual second order Dirichlet problem. We recover fourth order PDEs in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.
- Allouba, Hassan SPDEs law equivalence and the compact support property: applications to the Allen-Cahn SPDE.
C. R. Acad. Sci., 331 (2000), no. 3, 245–250.
Using our uniqueness in law transfer result for SPDEs, described in a recent note, we prove the equivalence of laws of SPDEs differing by a drift, under vastly applicable conditions. This gives us the equivalence in the compact support property among a large class of SPDEs. As an important application, we prove the equivalence in law of the Allen-Cahn and the associated heat SPDEs; and we give a criterion for the compact support property to hold for the Allen-Cahn SPDE with diffusion function a(t,x,u)=Cuγ, with C≠0 and 1/2≤γ<1.
- Allouba, Hassan Uniqueness in law for the Allen-Cahn SPDE via change of measure.
C. R. Acad. Sci., 330 (2000), no. 5, 371–376.
We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure L2 condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function a(t,x,u)=Cuγ, 1/2≤γ≤1 and C≠0. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.
- Allouba, Hassan Different types of SPDEs in the eyes of Girsanov’s theorem.
Stochastic Anal. Appl., 16 (1998), no. 5, 787–810.
We prove Girsanov's theorem for continuous orthogonal martingale measures. We then define space-time SDEs, and use Girsanov's theorem to establish a one-to-one correspondence between solutions of two space-time SDEs differing only by a drift coefficient. For such stochastic equations, we give necessary conditions under which the laws of their solutions are absolutely continuous with respect to each other. Using Girsanov's theorem again, we prove additional existence and uniqueness results for space-time SDEs. The same one-to-one correspondence and absolute continuity theorems are also proved for the stochastic heat and wave equations.
- Allouba, Hassan A non-nonstandard proof of Reimers’ existence result for heat SPDEs.
J. Appl. Math. Stochastic Anal., 11 (1998), no. 1, 29–41.
In 1989, Reimers gave a nonstandard proof of the existence of a solution to heat SPDEs, driven by space-time white noise, when the diffusion coefficient is continuous and satisfies a linear growth condition. Using the martingale problem approach, we give a non-nonstandard proof of this fact, and with the aid of Girsanov's theorem for continuous orthogonal martingale measures (proved in a separate paper by the author), the result is extended to the case of a measurable drift.
- Allouba, Hassan; Durrett, Rick; Hawkes, John; and Perkins, Edwin Super-tree random measures.
J. Theoret. Probab., 10 (1997), no. 3, 773–794.
We use supercritical branching processes with random walk steps of geometrically decreasing size to construct random measures. Special cases of our construction give close relatives of the super-(spherically symmetric stable) processes. However, other cases can produce measures with very smooth densities in any dimension.