Pith Number

pith:MBHPHYDZ

pith:2026:MBHPHYDZ4OPMFD3CEBM74SGQS5

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Limit theorems for random walks with spatio-temporal drift

Ngo P.N. Ngoc, Tuan-Minh Nguyen

Random walks with position-and-time dependent polynomial drift show three distinct asymptotic regimes.

arxiv:2605.17725 v1 · 2026-05-18 · math.PR · cond-mat.stat-mech · math-ph · math.MP

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Claims

C1strongest claim

In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere.

C2weakest assumption

The increments have moments of order p for some p>2 (stated in the abstract as the standing assumption under which all asymptotic results are derived).

C3one line summary

The authors derive asymptotic behaviors for discrete-time random walks in R^d with polynomial spatio-temporal drifts, finding phase transitions to Gaussian limits in the linear case and three regimes (Gaussian, non-Gaussian SDE stationary, and hypersphere localization) in the nonlinear case under p>

References

37 extracted · 37 resolved · 0 Pith anchors

[1] David Aldous,Stopping times and tightness, Ann. Probability6(1978), no. 2, 335–340. MR 474446 1978

[2] 1709, Springer, Berlin, 1999, pp 1999

[3] Bernard Bercu,A martingale approach for the elephant random walk, J. Phys. A51(2018), no. 1, 015201, 16. MR 3741953 2018

[4] Bernard Bercu and Lucile Laulin,On the multi-dimensional elephant random walk, J. Stat. Phys.175(2019), no. 6, 1146–1163. MR 3962977 2019

[5] Nils Berglund,Long-time dynamics of stochastic differential equations, arXiv preprint arXiv:2106.12998 (2021) 2021

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:54.900382Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

604ef3e079e39ec28f622059fe48d09753d7367eceac6263c5db6e877594bd46

Aliases

arxiv: 2605.17725 · arxiv_version: 2605.17725v1 · doi: 10.48550/arxiv.2605.17725 · pith_short_12: MBHPHYDZ4OPM · pith_short_16: MBHPHYDZ4OPMFD3C · pith_short_8: MBHPHYDZ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MBHPHYDZ4OPMFD3CEBM74SGQS5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 604ef3e079e39ec28f622059fe48d09753d7367eceac6263c5db6e877594bd46

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "18b6d75677179004f532e9463bc629ea22805c83e6c9e7d74a13cefe4af858f8",
    "cross_cats_sorted": [
      "cond-mat.stat-mech",
      "math-ph",
      "math.MP"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-18T01:06:11Z",
    "title_canon_sha256": "31c8f0e459daba8024ce233dab12d917da305e32c49f3054a71b045d42df2bdc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17725",
    "kind": "arxiv",
    "version": 1
  }
}