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Register a new userBrownian-time Brownian motion SIEs on $RptimesRd$: Ultra Regular direct and lattice-limits solutions and fourth order SPDEs links
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Hassan Allouba
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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 Holder 1/2 regularity in d=3. In time, our solutions are locally Holder continuous with exponent $gammain(0,(4-d)/(8))$ for $1le dle3$. 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.
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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 $epsilon_1$ and $epsilon_2$ multiplied by the Laplacian and the noise, the effect of letting $epsilon_1,epsilon_2to 0$ at different speeds. More precisely, it is shown that the ratio $epsilon_2/epsilon_1^{1/4}$ determines the behavior as $epsilon_1,epsilon_2to 0$.
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.
In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.
We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of on/off moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an on/off-branching Brownian motion. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be switched off for an exponential amount of time after which they get switched on again. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed $sqrt{2}$, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off-branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by $sqrt{sqrt{5}-1}approx 1.111$ under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $Hin (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(cdot,mu) = fastmu(cdot) + g(cdot),quad f,gin B^alpha_{infty,infty}, quad alpha>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.
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