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Martingale problem of the two-dimensional stochastic heat equation at criticality

Yu-Ting Chen

Abstract

We study the martingale formulation of the two-dimensional stochastic heat equation (SHE) at criticality. The main theorem proves an exact recursive-type equation that expresses the covariation measures of the SHE in terms of the solutions via an integro-multiplication operator. As an application, the quadratic variations of the martingale parts in the mild form are proven explicitly expressible in the solutions of the SHE and the two-dimensional two-body delta-Bose gas semigroups. The proofs are based on the standard approximations of the two-dimensional SHE at criticality, and now we analyze asymptotic expansions of the covariation measures of the approximate solutions in the limit. Also, new bounds for certain mixed moments of the fourth order of the approximate solutions are among the main tools for a priori estimates.

Martingale problem of the two-dimensional stochastic heat equation at criticality

Abstract

We study the martingale formulation of the two-dimensional stochastic heat equation (SHE) at criticality. The main theorem proves an exact recursive-type equation that expresses the covariation measures of the SHE in terms of the solutions via an integro-multiplication operator. As an application, the quadratic variations of the martingale parts in the mild form are proven explicitly expressible in the solutions of the SHE and the two-dimensional two-body delta-Bose gas semigroups. The proofs are based on the standard approximations of the two-dimensional SHE at criticality, and now we analyze asymptotic expansions of the covariation measures of the approximate solutions in the limit. Also, new bounds for certain mixed moments of the fourth order of the approximate solutions are among the main tools for a priori estimates.
Paper Structure (20 sections, 32 theorems, 405 equations, 3 figures)

This paper contains 20 sections, 32 theorems, 405 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The critical regime in two dimensions
  3. Moment duality
  4. Main results
  5. Formal expansions of the covariation measures
  6. Decomposition of the squared mild form
  7. Limiting martingale problem
  8. Skorokhod representations and weak formulations
  9. Convergence of $I^{1}_{n}(f,T)$
  10. Convergence of $I^{2}_{n}(f,T)$
  11. Convergences of $I^{3}_n(f,T)$ and $I^{4}_n(f,T)$
  12. End of the proof of Theorem \ref{['thm:main1']}
  13. Covariation measures in the mild form
  14. Mixed moments and their applications
  15. Moment bounds and $C$-tightness of $(X_{\varepsilon},\nu_{\varepsilon},\mu_{\varepsilon},\mathring{\mu}_{\varepsilon})$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume a bounded initial condition $X_0(\cdot)$. Then for every subsequential distributional limit $X_\infty$ of the approximate solutions $X_{\varepsilon}$ to SHE:vep as ${\varepsilon}\to 0$, the associated martingale measure $M_\infty$ and covariation measure $\langle M_\infty,M_\infty{\rangle} ({ where $\{P_t\}$ with kernel $P_t(x,y)$ defined by def:Pt is the semigroup of the two-dimensional st

Figures (3)

  • Figure 1: The figure illustrates the unweighted graph $\mathcal{G}_{\oslash}^{\boldsymbol \sigma}$ associated with $Q^{\lambda,\phi;\mathbf i_1,\mathbf i_2,\mathbf i_3,\mathbf i_4,\mathbf i_5;{\boldsymbol \sigma}}_{{\varepsilon};s_1,s_2,s_3,s_4,s_5,t}f(x_0)$ such that $N=4$, $x_0^\ell$ are all distinct, $\mathbf i_1=(2,1)$, $\mathbf i_2=(3,2)$, $\mathbf i_3=(2,1)$, $\mathbf i_4=(4,3)$, $\mathbf i_5=(2,1)$, and ${\boldsymbol \sigma}=(0,1,1,0,0)$. A pair of bullet points glued together represents a vertex of the form $(x^{i_\ell\prime}_\ell,x^{i_\ell}_\ell,s_\ell)$ or $(z'_\ell,z_\ell,\tau_\ell)$. Any other bullet point represents a vertex of the remaining form in \ref{['def:vertex']}. There are six subgraphs, and they are marked in six different colors (as shown in the electronic version of this paper). The same convention of marking subgraphs by different colors will apply in the other figures below.
  • Figure 2: This figure shows two graphs. The one on the left is over $[0,t'-t]$ for two particles undergoing attractive interactions. These two particles have states $\widetilde{x}_2^1,\widetilde{x}_2^2$ at time $t'-t$. The graph on the right is over $[0,t]$ for four particles with states $x^1_0,x^2_0,\widetilde{x}_2^1=x^3_0,\widetilde{x}_2^2=x^4_0$ at time $0$, where $x^2_0,\widetilde{x}_2^1,\widetilde{x}_2^2$ are distinct, and $\mathbf i_1=(2,1)$. By these states $\widetilde{x}_2^1=x^3_0$ and $\widetilde{x}_2^2=x^4_0$, the graph over $[0,t'-t]$ is glued to the graph over $[0,t]$. Note that this figure also illustrates the "extremal case" for initial states where $\widetilde{x}_0^1=\widetilde{x}_0^2$ and $x_0^1=x_0^2$, although these conditions may not be required in general.
  • Figure 4: $\mathbf i_1=(3,2)$

Theorems & Definitions (40)

  • Theorem 1.1: Main theorem
  • Theorem 1.2
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Lemma 4.5
  • Proposition 4.6
  • ...and 30 more